A Maple session that accompanies the paper ON THE STATIONARY DISTRIBUTION OF REFLECTED BROWNIAN MOTION IN A WEDGE: DIFFERENTIAL PROPERTIESby Bousquet-M\303\251lou, Elvey Price, Franceschi, Hardouin, and Raschel restart: libname:="/home/bousquet/Calculs/Gfun/lib", libname

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with(gfun): gfun[version]

JCIkJVEhIiM=

Some expressions derived from Section 2.1We denote b=beta, t=theta and so on. s12expr:=-cos(b)*sqrt(s11*s22); mu1expr:=-sin(b-t)*sqrt(s11*Delta/det); mu2expr:=-sin(t)*sqrt(s22*Delta/det)

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r12expr:=r22*sqrt(s11/s22)*sin(b-d)/sin(d); r21expr:=r11*sqrt(s22/s11)*sin(b-e)/sin(e);

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detexpr:=sin(b)^2*s11*s22

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Masses phi10expr:=simplify(subs(mu1=mu1expr, mu2=mu2expr, r12=r12expr,r21=r21expr,det=detexpr, (mu1*r22-mu2*r12)/(r12*r21-r11*r22))) assuming (s11>0,s22>0,sin(b)>0)

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Simplification (sin(t)*sin(b-d)-sin(d)*sin(b-t))/(sin(b-d)*sin(b-e)-sin(e)*sin(d))

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A bit of trigonometry... (simplify(sin(t)*sin(b-d)-sin(d)*sin(b-t) - sin(b)*sin(t-d)))

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simplify(sin(b-d)*sin(b-e)-sin(e)*sin(d)-sin(b)*sin(b-d-e))

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phi10expr_bis:=normal(phi10expr/ ((sin(t)*sin(b-d)-sin(d)*sin(b-t))/(sin(b-d)*sin(b-e)-sin(e)*sin(d))) *sin(t-d)/sin(b-d-e))

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simplify(phi10expr-phi10expr_bis)

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phi10expr:=phi10expr_bis: phi20exr:=subs({s22=s11,e=d,d=e, t=b-t}, phi10expr)

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We introduce normal variables denoted u and v here (x and y, with an unusual font, in the paper) xn:=x*det/sqrt(s22*Delta); yn:=y*det/sqrt(s11*Delta);

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xform:=solve(xn-u,x); yform:=solve(yn-v,y)

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The kernel gxy:=1/2*(s11*x^2+s22*y^2+2*s12*x*y)+mu1*x+mu2*y

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simplify(subs(mu1=mu1expr, mu2=mu2expr, s12=s12expr,r12=r12expr,r21=r21expr,x=xform,y=yform,det=detexpr, gxy)) assuming(s11>0, s22>0, Delta>0,sin(b)>0)

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gxynorm:=factor(%*(detexpr/det))

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The polynomials gamma1 and gamma2 g1xy:=r11*x+r21*y

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simplify(subs(mu1=mu1expr, mu2=mu2expr, s12=s12expr,r12=r12expr,r21=r21expr,x=xform,y=yform,det=detexpr, g1xy)) assuming(s11>0, s22>0, Delta>0,sin(b)>0)

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g1xynorm:=factor(simplify(%*(detexpr/det)) assuming(s11>0, s22>0, Delta>0,sin(b)>0))

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g2xynorm:=subs({u=v,v=u,e=d,d=e,r11=r22,s11=s22},g1xynorm)

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g1xynorm*phi10expr

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phi10g1expr:=simplify(g1xynorm*phi10expr) assuming(det>0,sin(b)>0,s11>0,s22>0)

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phi20g2expr:=subs({u=v,v=u,d=e,e=d,t=b-t}, phi10g1expr)

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up:=(1+cos(t)); vp:=(1+cos(b-t));

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Roots of the kernel (Section 3.2) factor([solve(gxynorm,v)])

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simplify(%)

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Parametrization of the curve gamma=0, in normal form us:=1/2*(s+exp(I*t))*(1+exp(-I*t)/s); vs:=1/2*(s+exp(I*t))*(exp(-I*b)+exp(I*(b-t))/s);

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We begin with gamma1 op(4,numer(g1xynorm))

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subs(u=us,v=vs,%)

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s1a:=op(2,[solve(%,s)])

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s1b:=factor(convert(s1a,exp))

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s1:=-exp(I*(2*b-2*e-t))

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simplify(s1b-s1)

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Now gamma 2 and s2 op(4,numer(g2xynorm))

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subs(u=us,v=vs,%)

LCYqKiMiIiIiIiNGJS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCZJImJHRixGJUkiZEdGLCEiIkYlLCZJInNHRixGJS1JJGV4cEdGKTYjKiZeI0YlRiVJInRHRixGJUYlRiUsJkYlRiUqJi1GNTYjLCQqJiwkRjhGJUYlRjlGJUYxRiVGM0YxRiVGJUYlKipGJEYlLUYoNiNGMEYlRjJGJSwmLUY1NiMsJComRkBGJUYvRiVGMUYlKiYtRjU2IyomRjhGJSwmRi9GJUY5RjFGJUYlRjNGMUYlRiVGJQ==

s2a:=op(2,[solve(%,s)])

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRzMmFHRigsJComLCYqJi1JJHNpbkdGJTYjLCZJImJHRigiIiJJImRHRighIiJGNy1JJGV4cEdGJTYjLCQqJiwkXiNGN0Y3RjdJInRHRihGN0Y5RjdGNyomLUYzNiNGOEY3LUY7NiMqJkZARjcsJkY2RjdGQUY5RjdGN0Y3RjcsJiomLUY7NiMsJComRj9GN0Y2RjdGOUY3RkNGN0Y3RjJGN0Y5Rjk3I0Yu

s2b:=factor(convert(s2a,exp))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRzMmJHRigsJComLCYtSSRleHBHRiU2IywkKiYsJF4jIiIiRjhGOCwoSSJ0R0YoRjhJImJHRihGOEkiZEdGKCEiIkY4Rj1GOC1GMjYjKiZGN0Y4LChGO0Y4RjxGOEY6Rj1GOEY9RjgsJi1GMjYjLCQqJkY2RjgsJkY7RjhGPEY4RjhGPUY4LUYyNiMqJkY3RjgsJkY7RjhGPEY9RjhGPUY9Rj03I0Yu

s2:=-exp(I*(2*d-t))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNzMkdGKCwkLUkkZXhwR0YlNiMqJl4jIiIiRjQsJiomIiIjRjRJImRHRihGNEY0SSJ0R0YoISIiRjRGOjcjRi4=

simplify(s2b-s2)

IiIh

Lemma 3.6 cos(t)-1+sin(b-e)/sin(e)*(cos(t-b)-cos(b))

LCgtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kidEdGKCIiIkYrISIiKigtSSRzaW5HRiU2IywmSSJiR0YoRitJImVHRihGLEYrLUYvNiNGM0YsLCYtRiQ2IywmRjJGK0YqRixGKy1GJDYjRjJGLEYrRis=

simplify(%-2*sin(b)/sin(e)*sin(t/2)*sin((2*b-2*e-t)/2))

IiIh

T:=proc(a,x) local n: sum(a/(a+n)*binomial(a+n,2*n)*2^n*(x-1)^n,n=0..infinity): end: T(a,x);series(T(a,x),x=1,3)

LUkkc3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqLEkiYUdGJyIiIiwmRipGK0kibkc2Iy9JJ19sb2NhbEdGJUkyX24xNDAxNTMxODczMjIzNjhHRidGKyEiIi1JKWJpbm9taWFsR0YkNiRGLCwkKiYiIiNGK0YtRitGK0YrKUY4Ri1GKyksJkkieEdGJ0YrRitGMkYtRisvRi07IiIhSSlpbmZpbml0eUdGJQ==

KyssJkkieEc2IiIiIkYmISIiRiYiIiEsJCoqIiIjRiZJImFHRiVGJiwmRiZGJkYsRiZGJy1JKWJpbm9taWFsRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiRGLUYrRiZGJkYmLCQqKiIiJUYmRixGJiwmRixGJkYrRiZGJy1GLzYkRjdGNkYmRiZGKy1JIk9HRjA2I0YmIiIk

Another expression expr:=hypergeom([-a,a],[1/2],(1-x)/2);series (%,x=1,3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVleHByR0YoLUkqaHlwZXJnZW9tR0YnNiU3JEkiYUdGKCwkRjIhIiI3IyMiIiIiIiMsJiomRjZGN0kieEdGKEY3RjRGNkY3NyNGLg==

KyssJkkieEc2IiIiIkYmISIiRiYiIiEqJClJImFHRiUiIiNGJkYmLCQqKiNGJiIiJ0YmRipGJiwmRiZGJkYrRiZGJiwmRitGJ0YmRiZGJkYnRiwtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJiIiJA==

map(simplify,series(T(a,x)-expr,x=1,10),GAMMA);

KyUsJkkieEc2IiIiIkYmISIiLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRiYiIzU=

non linear DE ed1:=(1-x^2)/a^2*diff(Ta(x),x)^2+Ta(x)^2-1

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlZDFHRigsKCooLCYqJClJInhHRigiIiMiIiIhIiJGNUY1RjUpSSJhR0YoRjRGNiktSSVkaWZmR0YmNiQtSSNUYUdGKDYjRjNGM0Y0RjVGNSokKUY9RjRGNUY1RjVGNjcjRi4=

simplify(series(eval(subs(Ta(x)=T(a,x),ed1)),x=1,10),GAMMA)

KyUsJkkieEc2IiIiIkYmISIiLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRiYiIzU=

Linear DE edT:=(1-x^2)*diff(Ta(x),x$2)-x*diff(Ta(x),x)+a^2*Ta(x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlZFRHRigsKComLCYqJClJInhHRigiIiMiIiIhIiJGNUY1RjUtSSVkaWZmR0YmNiQtSSNUYUdGKDYjRjMtSSIkR0YmNiRGM0Y0RjVGNSomRjNGNS1GODYkRjpGM0Y1RjYqJilJImFHRihGNEY1RjpGNUY1NyMsKComRjBGNS1GODYkRkFGM0Y1RjVGQEY2RkNGNQ==

simplify(series(subs(Ta(x)=T(a,x),edT),x=1,10),GAMMA);

KyUsJkkieEc2IiIiIkYmISIiLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRiYiIzU=

factor(eval(subs(Ta(x)=cos(a*arccos(x)),edT)));

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This coincides with Ta(x) simplify(series(T(a,x)-cos(a*arccos(x)),x=1,6),GAMMA) assuming x>1

KyUsJkkieEc2IiIiIkYmISIiLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRiYiIic=

Another solution to the differential equation factor(eval(subs(Ta(x)=sin(a*arccos(x)),edT)));

IiIh

The algebraic expression Taalt:=1/2*((x+sqrt(x^2-1))^a+(x-sqrt(x^2-1))^a)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZUYWFsdEdGKCwmKiYjIiIiIiIjRjEpLCZJInhHRihGMSokKSwmKiQpRjVGMkYxRjFGMSEiIkYwRjFGMUkiYUdGKEYxRjEqJkYwRjEpLCZGNUYxRjZGO0Y8RjFGMTcjRi4=

simplify(series(Taalt-T(a,x),x=1,7),GAMMA)

KyUsJkkieEc2IiIiIkYmISIiLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRiYiIig=

factor(series(cos(a*arccos(x)),x=1,5));

Ky8sJkkieEc2IiIiIkYmISIiRiYiIiEqJClJImFHRiUiIiNGJkYmLCQqKiNGJiIiJ0YmRipGJiwmRitGJkYmRidGJiwmRitGJkYmRiZGJkYmRiwsJCouI0YmIiMhKkYmRipGJkYxRiYsJkYrRiZGLEYnRiYsJkYrRiZGLEYmRiZGMkYmRiYiIiQsJCoyI0YmIiU/REYmRipGJkYxRiZGN0YmLCZGK0YmRjlGJ0YmLCZGK0YmRjlGJkYmRjhGJkYyRiZGJiIiJS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YmIiIm

We now want to express sqrt(1+Ta) and sqrt(1-Ta) simplify(series(sqrt(1+T(a,x))/T(a/2,x),x=1,7),GAMMA)

KycsJkkieEc2IiIiIkYmISIiKiQpIiIjI0YmRipGJiIiIS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YmIiIo

factor(series(1/a/sqrt(2)/sqrt(epsilon)*sin(a*arccos(1-epsilon)),epsilon,5));

Ky9JKGVwc2lsb25HNiIiIiIiIiEsJCooI0YlIiM3RiUsJiomIiIjRiVJImFHRiRGJUYlRiUhIiJGJSwmRixGJUYlRiVGJUYvRiUsJCosI0YlIiQhW0YlLCZGLEYlIiIkRiVGJUYwRiVGK0YlLCZGLEYlRjZGL0YlRiVGLSwkKjAjRiUiJj8uJUYlLCZGLEYlIiImRiVGJUY1RiVGMEYlRitGJUY3RiUsJkYsRiVGPUYvRiVGL0Y2LCQqNCNGJSIoITMxZUYlLCZGLEYlIiIoRiVGJUY8RiVGNUYlRjBGJUYrRiVGN0YlRj5GJSwmRixGJUZERi9GJUYlIiIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiVGPQ==

map(factor,series(hypergeom([-a+1/2,a+1/2],[3/2],epsilon/2),epsilon=0,5));

Ky9JKGVwc2lsb25HNiIiIiIiIiEsJCooI0YlIiM3RiUsJiomIiIjRiVJImFHRiRGJUYlRiUhIiJGJSwmRixGJUYlRiVGJUYvRiUsJCosI0YlIiQhW0YlLCZGLEYlIiIkRiVGJUYwRiVGK0YlLCZGLEYlRjZGL0YlRiVGLSwkKjAjRiUiJj8uJUYlLCZGLEYlIiImRiVGJUY1RiVGMEYlRitGJUY3RiUsJkYsRiVGPUYvRiVGL0Y2LCQqNCNGJSIoITMxZUYlLCZGLEYlIiIoRiVGJUY8RiVGNUYlRjBGJUYrRiVGN0YlRj5GJSwmRixGJUZERi9GJUYlIiIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiVGPQ==

simplify(simplify(series(1/a*sqrt((1-T(a,x))/(1-x))-hypergeom([(1-a)/2,(1+a)/2],[3/2], (1-x)/2),x=1,8),GAMMA),symbolic);

KyUsJkkieEc2IiIiIkYmISIiLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRiYiIik=

simplify(convert(simplify(series(1/a*sqrt((1-T(a,x))/(1-x))- hypergeom([(1-a)/2,(1+a)/2],[3/2],(1-x)/2),x=1,5)),GAMMA)) assuming a>0;

KyUsJkkieEc2IiIiIkYmISIiLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRiYiIiY=

The basic conditions that hold throughout the paper (divided by pi) conds:={0<t,0<d,0<e,0<b,b<1, d<1,e<1, t<b,d-1<b-e, b-e<t, t<d};

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZjb25kc0dGKDwtMiIiIUkiYkdGKDJGMEkiZEdGKDJGMEkiZUdGKDJGMEkidEdGKDJGMSIiIjJGM0Y5MkY1RjkyRjdGMTJGN0YzMiwmRjFGOUY1ISIiRjcyLCZGM0Y5RjlGQEY/NyNGLg==

We want alpha=-1, r=2, k=0 conds1:=conds union {d+e-1+b=0}

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdjb25kczFHRig8Li8sKkkiZEdGKCIiIkkiZUdGKEYyRjIhIiJJImJHRihGMiIiITJGNkY1MkY2RjEyRjZGMzJGNkkidEdGKDJGNUYyMkYxRjIyRjNGMjJGO0Y1MkY7RjEyLCZGNUYyRjNGNEY7MiwmRjFGMkYyRjRGQjcjRi4=

and also a double root conds2:=conds1 union {2*e+t-1}

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdjb25kczJHRig8LywoKiYiIiMiIiJJImVHRihGMkYySSJ0R0YoRjJGMiEiIi8sKkkiZEdGKEYyRjNGMkYyRjVJImJHRihGMiIiITJGOkY5MkY6RjgyRjpGMzJGOkY0MkY5RjIyRjhGMjJGM0YyMkY0RjkyRjRGODIsJkY5RjJGM0Y1RjQyLCZGOEYyRjJGNUZFNyNGLg==

no solution solve(conds2);

PCgvSSJkRzYiLChJImVHRiUhIiIiIiJGKUkiYkdGJUYoL0kidEdGJSwmKiYiIiNGKUYnRilGKEYpRikyI0YpIiIkRicyRipGJzJGJyNGKUYvMkYtRio=

solve(conds2 union{e=3/8})

PCcvSSJkRzYiLCYjIiImIiIpIiIiSSJiR0YlISIiL0kiZUdGJSMiIiRGKS9JInRHRiUjRioiIiUyRjNGKzJGK0Yv

sol:=solve(conds2 union{e=3/8, b=5/16})

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRzb2xHRig8Ji9JImJHRigjIiImIiM7L0kiZEdGKEYxL0kiZUdGKCMiIiQiIikvSSJ0R0YoIyIiIiIiJTcjRi4=

eval(subs(sol,-exp(I*(2*b-2*e-t)*pi)))

LCQtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKiYsJComIyIiJCIiKSIiIl4jRjFGMUYxRjFJI3BpR0YoRjEhIiJGNA==

simplify(int(exp(y*z)*z*exp(-a*z),z=0..infinity) assuming y<a)

KiQpLCZJImFHNiIiIiJJInlHRiYhIiIiIiNGKQ==

Taalt

LCYqJiMiIiIiIiNGJSksJkkieEc2IkYlKiQpLCYqJClGKUYmRiVGJUYlISIiRiRGJUYlSSJhR0YqRiVGJSomRiRGJSksJkYpRiVGK0YwRjFGJUYl

beta=Pi/2, Pi/3, etc. (ie n=1): not interesting (forces alpha to be an integer) Explore beta=2Pi/3 n:=10: ser:=series(subs(a=3/2,x=1+y,Taalt),y,n+1)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRzZXJHRigrO0kieUdGKCIiIiIiISMiIioiIiVGMCMiIzoiI0siIiMjISIoIiRHIiIiJCMiI0YiJVs/RjQjISNMIiUjPikiIiYjIiMiKiImT2InIiInIyEkTiIiJ1dARSIiKCMiJSRvIiIoMycpUSkiIikjISU8RiIpS1diTEYzIyIlNCEqIipjYVZvIyIjNS1JIk9HRiU2I0YwIiM2NyNGLg==

with(gfun):seriestoalgeq(ser,TT(y))

NyQsLComIiIlIiIiKUkieUc2IiIiJEYmRiYqJiIjN0YmKUYoIiIjRiZGJiomIiIqRiZGKEYmRiZGLkYmKiZGLkYmKS1JI1RUR0YpNiNGKEYuRiYhIiJJJG9nZkdGKQ==

algT32:=collect(subs(TT(y)=TT(x),y=x-1,op(1,%)),TT,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdUMzJHRigsJiomIiIjIiIiKS1JI1RUR0YoNiNJInhHRihGMEYxISIiKiYsJkYxRjFGNkYxRjEpLCZGMUY3KiZGMEYxRjZGMUYxRjBGMUYxNyNGLg==

[solve(algT32,TT(x))];subs(x=y+1,%)

NyQsJCooIyIiIiIiI0YmKSwmKiZGJ0YmSSJ4RzYiRiZGJkYnRiZGJUYmLCZGJiEiIkYqRiZGJkYmLCRGJEYu

NyQsJCooIyIiIiIiI0YmKSwmIiIlRiYqJkYnRiZJInlHNiJGJkYmRiVGJiwmRiZGJkYrRiZGJkYmLCRGJCEiIg==

T32sol:=(1/2)*sqrt(2*x+2)*(-1+2*x)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdUMzJzb2xHRigsJCooIyIiIiIiI0YxKSwmKiZGMkYxSSJ4R0YoRjFGMUYyRjFGMEYxLCZGMSEiIkY1RjFGMUYxNyNGLg==

factor(simplify(factor(subs(x=2*u^2-1,T32sol)),symbolic))

KiZJInVHNiIiIiIsJiomIiIlRiUpRiMiIiNGJUYlIiIkISIiRiU=

factor(%+1)

KiYsJkkidUc2IiIiIkYmRiZGJiksJkYmISIiKiYiIiNGJkYkRiZGJkYrRiY=

solve(T32sol+1,x)

IyEiIiIiIw==

factor(subs(TT(x)=-1,algT32))

KiYsJkkieEc2IiIiIkYmISIiRiYpLCZGJkYmKiYiIiNGJkYkRiZGJkYrRiY=

factor(expand(rationalize(subs(z=sqrt((1+x)/2),(1+2*z)/(1-2*z)/(1+z)))))

LCQqKCwoKigiIiMiIiIpLCYqJkYmRidJInhHNiJGJ0YnRiZGJyNGJ0YmRidGK0YnRicqJEYoRichIiJGJkYvRicsJkYqRidGJ0YnRi8sJkYrRidGJ0YvRi9GLw==

with(algcurves): genus(subs(TT(x)=TT,algT32),x,TT)

IiIh

factor(parametrization(subs(TT(x)=TT,algT32),x,TT,S))

NyQsJCooIyIiIiIiI0YmLCgqJClJIlNHNiJGJ0YmRiYqJiIiJUYmRitGJiEiIkYnRiZGJiksJkYrRiZGJkYvRidGL0YvLCQqKkYlRiZGK0YmLCgqJkYnRiZGKkYmRiYqJiIiJ0YmRitGJkYvIiIkRiZGJilGMUY4Ri9GJg==

factor(T32sol-subs(x=x0,T32sol))

LCoqJiksJiomIiIjIiIiSSJ4RzYiRihGKEYnRigjRihGJ0YoRilGKEYoKiZGK0YoRiRGKCEiIiomKSwmKiZGJ0YoSSN4MEdGKkYoRihGJ0YoRitGKEYyRihGLSomRitGKEYvRihGKA==

n:=20: ser:=series(subs(a=5/2,x=1+y,Taalt),y,n+1)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRzZXJHRigrT0kieUdGKCIiIiIiISMiI0QiIiVGMCMiJHYiIiNLIiIjIyIkMCIiJEciIiIkIyEkbCIiJVs/RjQjIiRWIiIlIz4pIiImIyEkRCQiJk9iJyIiJyMiJEQlIidXQEUiIigjISVYWyIoMycpUSkiIikjIiU6dCIpS1diTCIiKiMhJkJJIyIqY2FWbyMiIzUjIiZ2dCQiK0M9dXQ1IiM2IyEnRCdbIyIsJT1wKXpyIiIjNyMiJzo6VSIsT25aPihvIiM4IyEoJik9WCIiLSkpUSJldlwmIiM5IyIoKkdMRCIuX2JEQiEqPiMiIzojISlEVF1yIjBHYE4pW1AyOSIjOyMiKnYzTEYiIjA3OFVgKlxIYyIjPCMhKnYleXJYIjEnXHF0aSpmLlgiIz0jIip2SWNFKSIyJSk+WzQmKVI5IT0iIz4jISs6ZiRbLCciM1c8cl5oUEkjKUciIz8tSSJPR0YlNiNGMCIjQDcjRi4=

seriestoalgeq(ser,TT(y))

NyQsMComIiM7IiIiKUkieUc2IiIiJkYmISIiKiYiIyEpRiYpRigiIiVGJkYrKiYiJFMiRiYpRigiIiRGJkYrKiYiJCsiRiYpRigiIiNGJkYrKiYiI0RGJkYoRiZGK0Y3RisqJkY3RiYpLUkjVFRHRik2I0YoRjdGJkYmSSRvZ2ZHRik=

algT25:=collect(subs(TT(y)=TT(x),y=x-1,op(1,%)),TT,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdUMjVHRigsJiomIiIjIiIiKS1JI1RUR0YoNiNJInhHRihGMEYxRjEqJiwmRjZGMUYxRjFGMSksKComIiIlRjEpRjZGMEYxRjEqJkYwRjFGNkYxISIiRjFGP0YwRjFGPzcjRi4=

solve(%,TT(x))

NiQsJCooIyIiIiIiI0YmKSwmKiZGJ0YmSSJ4RzYiRiZGJkYnRiZGJUYmLCgqJiIiJUYmKUYrRidGJkYmRiohIiJGJkYxRiZGJiwkRiRGMQ==

n:=30: ser:=series(subs(a=5/4,x=1+y,Taalt),y,n+1) : seriestoalgeq(ser,TT(y))

NyQsMComIiM7IiIiKUkieUc2IiIiJkYmISIiKiYiIyEpRiYpRigiIiVGJkYrKiYiJFMiRiYpRigiIiRGJkYrKiYiJCsiRiYpRigiIiNGJkYrKiYiI0RGJkYoRiZGKyomIiIpRiYpLUkjVFRHRik2I0YoRjdGJkYrKiZGO0YmKUY9Ri9GJkYmSSRvZ2ZHRik=

A differential equation for w(y) can be obtained from the DE for Ta eqwT:=w(y)-Ta(-(2*y-yp-ym)/(yp-ym));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcXdUR0YoLCYtSSJ3R0YoNiNJInlHRigiIiItSSNUYUdGKDYjLCQqJiwoKiYiIiNGM0YyRjNGM0kjeXBHRighIiJJI3ltR0YoRj1GMywmRjxGM0Y+Rj1GPUY9Rj03I0Yu

DTa:=solve(diff(eqwT,y),(D(Ta))(-(2*y-yp-ym)/(yp-ym)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSREVGFHRigsJCooIyIiIiIiI0YxLUklZGlmZkdGJjYkLUkid0dGKDYjSSJ5R0YoRjlGMSwmSSN5cEdGKCEiIkkjeW1HRihGMUYxRjE3I0Yu

D2Ta:=solve(diff(eqwT,y$2),(D(D(Ta)))(-(2*y-yp-ym)/(yp-ym)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVEMlRhR0YoLCQqKCMiIiIiIiVGMS1JJWRpZmZHRiY2JC1JIndHRig2I0kieUdGKC1JIiRHRiY2JEY5IiIjRjEpLCZJI3lwR0YoISIiSSN5bUdGKEYxRj1GMUYxNyMsJCooRjBGMS1GNDYkLUY0NiRGNkY5RjlGMUY+RjFGMQ==

Here is the DE satisfied by w(y) edw:=subs(diff(Ta(x), x, x)=D2Ta,diff(Ta(x), x)=DTa,Ta(x)=w(y),x=-(2*y-yp-ym)/(yp-ym),edT);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlZHdHRigsKCoqIyIiIiIiJUYxLCYqJiksKComIiIjRjFJInlHRihGMUYxSSN5cEdGKCEiIkkjeW1HRihGO0Y4RjEpLCZGOkYxRjxGO0Y4RjtGO0YxRjFGMS1JJWRpZmZHRiY2JC1JIndHRig2I0Y5LUkiJEdGJjYkRjlGOEYxKSwmRjpGO0Y8RjFGOEYxRjEqLCNGMUY4RjFGNkYxRj5GOy1GQDYkRkJGOUYxRklGMUYxKiYpSSJhR0YoRjhGMUZCRjFGMTcjLCgqKkYwRjFGM0YxLUZANiRGTEY5RjFGSEYxRjFGSkYxRk5GMQ==

We now move to the final example in Section 6.3 The basic conditions that hold throughout the paper (divided by pi) conds:={0<t,0<d,0<e,0<b,b<1, d<1,e<1, t<b,d-1<b-e, b-e<t, t<d};

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZjb25kc0dGKDwtMiIiIUkiYkdGKDJGMEkiZEdGKDJGMEkiZUdGKDJGMEkidEdGKDJGMSIiIjJGM0Y5MkY1RjkyRjdGMTJGN0YzMiwmRjFGOUY1ISIiRjcyLCZGM0Y5RjlGQEY/NyNGLg==

The additional condition in this example cond:={d+e+b=2};

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVjb25kR0YoPCMvLChJImRHRigiIiJJImVHRihGMkkiYkdGKEYyIiIjNyNGLg==

Some (sub)examples map(evalb,subs(b=2/3,d=5/6,e=1/2,t=1/4,conds union cond));

PCNJJXRydWVHJSpwcm90ZWN0ZWRH

subs(b=2/3,d=5/6,e=1/2,t=1/4,2-2*e-t);

IyIiJCIiJQ==

map(evalb,subs(b=2/3,d=4/9,e=8/9,t=1/4,conds union cond));

PCNJJXRydWVHJSpwcm90ZWN0ZWRH

subs(b=2/3,d=4/9,e=8/9,t=1/4,2-2*e-t);

IyEiIiIjTw==

map(evalb,subs(b=2/3,d=7/12,e=3/4,t=1/2,conds union cond));

PCNJJXRydWVHJSpwcm90ZWN0ZWRH

subs(b=2/3,d=7/12,e=3/4,t=1/2,2-2*e-t);

IiIh

Expression of the Laplace transform phi_1 phiexpr:=kappa*(w(y)-w0)/(y-b)/(y-c);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShwaGlleHByR0YoKipJJmthcHBhR0YoIiIiLCYtSSJ3R0YoNiNJInlHRihGMEkjdzBHRighIiJGMCwmRjVGMEkiYkdGKEY3RjcsJkY1RjBJImNHRihGN0Y3NyNGLg==

The DE satisfied by phi_1 (denoted here by phi) edphi:=collect(eval(subs(isolate(phiexpr-phi(y),w(y)),kappa*edw*2)),[diff(phi(y), y, y),diff(phi(y), y),phi(y),y],factor);

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

nops(numer(factor(edphi)));

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We use the package GFUN to determine the corresponding recurrence relation with(gfun);gfun[version]

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

JCIkPyQhIiM=

here, co(n) is the coefficient of x^n, that is, M_n/n! recco:=collect(op(1,diffeqtorec(edphi,phi(y),co(n))),[co,n],factor);

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

All coefficients are quadratic in n for i from 0 to 4 do degree(coeff(recco,co(n+i)),n) od;

IiIj

IiIj

IiIj

IiIj

IiIj

how many initial conditions? four nops(diffeqtorec(edphi,phi(y),co(n)))

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Initial conditions init:=[seq(op(i,diffeqtorec(edphi,phi(y),co(n))),i=2..5)]: init[1]; init[2];

Ly1JI2NvRzYiNiMiIiEmSSNfQ0dGJUYm

Ly1JI2NvRzYiNiMiIiImSSNfQ0dGJUYm

init[3]; init[4];

Ly1JI2NvRzYiNiMiIiMsJCouIyIiIiIiJUYrLDgqLEYnRispSSJhR0YlRidGK0kiYkdGJUYrSSJjR0YlRismSSNfQ0dGJTYjIiIhRitGKyoqRidGK0YvRitJJmthcHBhR0YlRitJI3cwR0YlRitGKyoqRjFGK0YyRitJI3ltR0YlRismRjQ2I0YrRitGKyoqRjFGK0YyRitJI3lwR0YlRitGPEYrRisqLEYsRitGMUYrRjtGK0Y/RitGPEYrRisqLEYsRitGMkYrRjtGK0Y/RitGPEYrRisqKEYxRitGO0YrRjNGKyEiIiooRjFGK0Y/RitGM0YrRkMqKEYyRitGO0YrRjNGK0ZDKihGMkYrRj9GK0YzRitGQyoqRixGK0Y7RitGP0YrRjNGK0ZDRitGP0ZDRjtGQ0YxRkNGMkZDRis=

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

In fact, co(3) is given by the recurrence relation, with co(n)=0 if n<0 subs(co(-1)=0,factor(subs(n=-1,op(init),recco)))

IiIh

and for co(2) this is not the case, but the difference is small collect(subs(co(-1)=0,co(-2)=0,subs(n=-2,op(init),recco)),[_C[0],_C[1]])

LCQqKiIiIyIiIkkma2FwcGFHNiJGJSlJImFHRidGJEYlSSN3MEdGJ0YlISIi

Here we introduce the sum and difference of y^+ and y^- rec:=collect(subs(yp=(s+d)/2, ym=(s-d)/2,recco*2),[co],factor);

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

Order 4, degree 2. Let us look at the degree of the coefficients when we write co(n)=M_n/n! recM:=map(factor,collect(subs([seq(co(n+i)=M(n+i)/(n+i)!,i=0..4)], rec*(n+2)!), M,simplify));

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

Let us check that this does not factor nops(factor(recM))

IiNf

seq(degree(coeff(recM,M(n+i)),n),i=0..4);

NiciIiUiIiQiIiMiIiIiIiE=

op(2,coeff(recM,M(n+1))); collect(%,n,factor)

LDgqKCIiIyIiIilJImFHNiJGJEYlSSJiR0YoRiVGJSooRiRGJUYmRiVJImNHRihGJUYlKihGJEYlRilGJSlJIm5HRihGJEYlISIiKihGJEYlRitGJUYtRiVGLyooRiRGJUYtRiVJInNHRihGJUYvKigiIilGJUYpRiVGLkYlRi8qKEY0RiVGK0YlRi5GJUYvKigiIzZGJUYuRiVGMkYlRi8qJkY0RiVGKUYlRi8qJkY0RiVGK0YlRi8qJiIjOkYlRjJGJUYv

LDAqJiwoKiYiIiMiIiJJImJHNiJGJyEiIiomRiZGJ0kiY0dGKUYnRioqJkYmRidJInNHRilGJ0YqRicpSSJuR0YpRiZGJ0YnKiYsKComIiIpRidGKEYnRioqJkY0RidGLEYnRioqJiIjNkYnRi5GJ0YqRidGMEYnRicqKEYmRicpSSJhR0YpRiZGJ0YoRidGJyooRiZGJ0Y5RidGLEYnRidGM0YqRjVGKiomIiM6RidGLkYnRio=

The simplest case is when the poles b and c are the same, and w0=-1 edphisimple:=subs(c=b,w0=-1,edphi);

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

recsimpleco:=subs(c=b,rec);

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

nops(factor(%));

IiNS

recsimpleM:=subs(c=b,recM);

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

We now introduce the modified moments M tilda, and the cosines c1 and c2, as denoted in the paper rectMsimple:=map(factor,collect(subs (s=d*c2,b=(c2-c1)*d/2,seq(M(n+i)=2^(n+i)/d^(n+i)*tM(n+i), i=0..4),recsimpleM/2^n*d^n/4),tM,simplify));

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

As expected, d has disappeared indets(%)

PCtJImFHNiJJI2MxR0YkSSNjMkdGJEkibkdGJC1JI3RNR0YkNiNGJy1GKTYjLCZGJyIiIkYuRi4tRik2IywmRidGLiIiI0YuLUYpNiMsJkYnRi4iIiRGLi1GKTYjLCZGJ0YuIiIlRi4=

Since the above recurrence is also valid for n=-1 (so that is computes tM(3)), the following one is valid for n equal to 1 at least rectMshifted:=collect(subs(n=n-2,-rectMsimple),tM,factor);

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

The version in the paper rectMpap:=-(1-c2^2)*(c1-c2)^2*tM(n+2)+(c2-c1)*(2*(c1*c2-2*c2^2+1)*n+c1*c2-5*c2^2+4)*tM(n+1) +((c1^2-6*c2*c1+6*c2^2-1)*n^2-3*(2*c1*c2-3*c2^2+1)*n-(c1-c2)^2*a^2-2*c1*c2+4*c2^2-2)*tM(n) -n*(2*(2*c2-c1)*n^2+3*c2*n+2*(c1-c2)*a^2+c2)*tM(n-1)+n*(n-1)*(n^2-a^2)*tM(n-2)

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

collect(rectMshifted-rectMpap,tM,factor)

IiIh

What should we add for this to hold at n=0 as well? subs(n=0,rectMpap)

LCgqKCwmKiQpSSNjMkc2IiIiIyIiIiEiIkYqRipGKiksJkkjYzFHRihGKkYnRitGKUYqLUkjdE1HRig2I0YpRipGKyooLCZGJ0YqRi5GK0YqLCgqJkYuRipGJ0YqRioqJiIiJkYqRiZGKkYrIiIlRipGKi1GMDYjRipGKkYqKiYsKiomRixGKilJImFHRihGKUYqRisqKEYpRipGJ0YqRi5GKkYrKiZGOEYqRiZGKkYqRilGK0YqLUYwNiMiIiFGKkYq

collect(subs(c=b,yp=(s+d)/2,ym=(s-d)/2, s=d*c2,b=(c2-c1)*d/2,w0=-1,init[3]),[_C[0],_C[1]],factor)

Ly1JI2NvRzYiNiMiIiMsKCowRiciIiIpSSJkR0YlRichIiIsLiomKUkiYUdGJUYnRiopSSNjMUdGJUYnRipGKioqRidGKkYwRipGM0YqSSNjMkdGJUYqRi0qJkYwRiopRjVGJ0YqRioqKEYnRipGM0YqRjVGKkYqKiYiIiVGKkY3RipGLUYnRipGKiwmRjVGKkYqRipGLSwmRjVGKkYqRi1GLSksJkYzRipGNUYtRidGLSZJI19DR0YlNiMiIiFGKkYqKi4sKComRjNGKkY1RipGKiomIiImRipGN0YqRi1GOkYqRipGLEYtRj5GLUY8Ri1GO0YtJkZANiNGKkYqRioqMCIiKUYqSSZrYXBwYUdGJUYqRjBGKilGLEY6Ri1GO0YtRjxGLUY9Ri1GLQ==

collect(subs(n=0,tM(0)=_C[0], tM(1)=_C[1]/2*d, tM(2)=d^2/2*co(2),%, rectMpap),[_C[0],_C[1]],factor)

LCQqKiIiJSIiIkkma2FwcGFHNiJGJSlJImFHRiciIiNGJSlJImRHRidGKiEiIkYt

The first moments phiexprz:=kp*(Ta(c2-z)+1)/(c2-c1-z)^2;;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlwaGlleHByekdGKCooSSNrcEdGKCIiIiwmLUkjVGFHRig2IywmSSNjMkdGKEYwSSJ6R0YoISIiRjBGMEYwRjApLChGNkYwSSNjMUdGKEY4RjdGOCIiI0Y4NyNGLg==

kpexpr:=factor(solve(subs(z=0,phiexprz-phi0),kp));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdrcGV4cHJHRigqKEklcGhpMEdGKCIiIiksJkkjYzFHRihGMEkjYzJHRighIiIiIiNGMCwmLUkjVGFHRig2I0Y0RjBGMEYwRjU3I0Yu

phiexprz:=subs(kp=%,phiexprz);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlwaGlleHByekdGKCosSSVwaGkwR0YoIiIiKSwmSSNjMUdGKEYwSSNjMkdGKCEiIiIiI0YwLCYtSSNUYUdGKDYjRjRGMEYwRjBGNSwmLUY5NiMsJkY0RjBJInpHRihGNUYwRjBGMEYwKSwoRjRGMEYzRjVGP0Y1RjZGNTcjRi4=

factor(series(phiexprz,z,2));

KylJInpHNiJJJXBoaTBHRiQiIiEsJCoqRiUiIiIsKiomLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kjVGFHRiQ2I0kjYzJHRiRGKUkjYzFHRiRGKUYpKiZGLEYpRjVGKSEiIiomIiIjRiktRjNGNEYpRilGOkYpRiksJkY2RilGNUY4RjgsJkY7RilGKUYpRjhGOEYpLUkiT0dGLzYjRilGOg==

tM1:=collect(coeff(%,z,1),D(Ta)(c2),factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR0TTFHRigsJiooIiIjIiIiSSVwaGkwR0YoRjEsJkkjYzFHRihGMUkjYzJHRighIiJGNkY2KihGMkYxLCYtSSNUYUdGKDYjRjVGMUYxRjFGNi0tSSJER0YlNiNGOkY7RjFGNjcjRi4=

Let us express the above derivative diff(cos(a*arccos(x)),x);

KihJImFHNiIiIiIpLCYqJClJInhHRiQiIiNGJSEiIkYlRiUjRiVGK0YsLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMqJkYjRiUtSSdhcmNjb3NHRjA2I0YqRiVGJQ==

plot(subs(a=0.8,cos(a*arccos(x))),x=-1..1);

6*-%'CURVESG6$7dw7$$!#5!""$!1t%\P%*p,4)!#;7$$!1)[(\**y%[*)*!#;$!1')=!\z0ZN(!#;7$$!1xa4>cN.)*!#;$!1/VsX04eq!#;7$$!1BX!4)HY+(*!#;$!17[SN7P)y'!#;7$$!1)e<NI))of*!#;$!1.6\3W"Gb'!#;7$$!1iC\)*e!Q\*!#;$!1j<*=yC5M'!#;7$$!0%yc8`B)R*!#:$!1^p'omG!fh!#;7$$!19Fa3vF*H*!#;$!12<-Keo")f!#;7$$!1D^-0_$p>*!#;$!1C:\)oZx!e!#;7$$!1lGd96#\4*!#;$!1y:#\44Ak&!#;7$$!1'3<M3()**)*)!#;$!1pdkE'e(ya!#;7$$!1"He;8hv*))!#;$!1ASo!)4qR`!#;7$$!1D]+,9^$z)!#;$!1WE_)QMz=&!#;7$$!1/3;KW.*o)!#;$!18%)R?m-S]!#;7$$!1Ryc8:N)e)!#;$!2W.(*Hsb7!\!#<7$$!1[%*)yx@p\)!#;$!1mS/C[6yZ!#;7$$!1sW*)yH?)Q)!#;$!2$pxk.(4\j%!#<7$$!1(Qxa*Q5'H)!#;$!2<t%)\6%4;X!#<7$$!0<Mowq*)=)!#:$!1cPnJf^!Q%!#;7$$!1v[(\zRT4)!#;$!2XfnwY$oiU!#<7$$!1CZ%*)=&4!*z!#;$!1n<zULdNT!#;7$$!0*zf>,-"*y!#:$!1"=$[r]\;S!#;7$$!1f>Ryck(y(!#;$!1E%4;0YT*Q!#;7$$!1$yc8n:Fp(!#;$!29&)\h=(Q$y$!#<7$$!1a3<M3K!f(!#;$!2d+nRXCbm$!#<7$$!1)RzePhR[(!#;$!1up&o8cZa$!#;7$$!1pQxa^P"R(!#;$!2.17_?B4W$!#<7$$!1)[(\*\z8H(!#;$!1eBO8;1IL!#;7$$!1w],.U2)=(!#;$!2NA=">Y&o@$!#<7$$!1:JiC,,(3(!#;$!2<:"*G=Qt5$!#<7$$!1/3;KaA*)p!#;$!2NnGy!*zC+$!#<7$$!1v],.Al!)o!#;$!2Ps#zkLE()G!#<7$$!0>QwK%4$y'!#:$!2OE0RAsZy#!#<7$$!13=Os/$*ym!#;$!2'*QI%*ytjn#!#<7$$!1V%)oPDa%e'!#;$!2E2fFeI!zD!#<7$$!1\)pRf^8['!#;$!0#G+!zFNZ#!#:7$$!1X*)ydpD%Q'!#;$!1">yB\*4vB!#;7$$!0;Kk3lFG'!#:$!2_wCs<^IF#!#<7$$!1yd:J#QN='!#;$!1$e%4hk2u@!#;7$$!1Rxa4plzg!#;$!2;2PYbq72#!#<7$$!1c7D]ogzf!#;$!1pr;+@,t>!#;7$$!1^.297Hxe!#;$!2aQ,b$zEt=!#<7$$!1v],.G#ex&!#;$!28&yl)>i]x"!#<7$$!1$yc8(Qe#o&!#;$!2KHW8*fU&o"!#<7$$!1T$oO`@dd&!#;$!2;;t)\9Q$e"!#<7$$!1BY#\eU,[&!#;$!2iQz$4]r#\"!#<7$$!1hAX!\N#y`!#;$!1$\?=H`mR"!#;7$$!0&**)zR(p!G&!#:$!1CA)ocz_I"!#;7$$!1NqS")QRs^!#;$!2^.)=LMX/7!#<7$$!0&)pR*4hy]!#:$!2al=*[um<6!#<7$$!2vW*)ydm@(\!#<$!2X')HSGM(>5!#<7$$!117C[S:v[!#;$!1Zg1`q"*4$*!#<7$$!2NnMpeP!pZ!#<$!1(RWz<aYM)!#<7$$!1zc8FqZxY!#;$!1j>;HJC;v!#<7$$!1Hf=P-?tX!#;$!1z8*)RPmxl!#<7$$!2/:Ig+uCZ%!#<$!1.I0;R&en&!#<7$$!28U%)od8=P%!#<$!1"pkL&QBzZ!#<7$$!2C['HfM_rU!#<$!11eHi2P!*Q!#<7$$!1KjE`g<vT!#;$!1tTDn0aSI!#<7$$!1%*)yd&>-rS!#;$!2Y&>\u%\i7#!#=7$$!2%=Pu[bvrR!#<$!24eW[G:!f7!#=7$$!21C['HBGnQ!#<$!1T4X%3Mc]$!#=7$$!2NrU&3*3Fx$!#<$"1c\^`38"o%!#=7$$!1Z$pQd(=oO!#;$"2G(Gz+m"*o8!#=7$$!2$=Pu[V2oN!#<$"2)RelX4*yA#!#=7$$!005?![?oM!#:$"2u?wdV:63$!#=7$$!1MoOt1*QO$!#;$"1*f3wc%\oR!#<7$$!2'=Qw_N"yE$!#<$"1p=J#yDCy%!#<7$$!2&zf>RuTpJ!#<$"2Ej%4>cr7c!#=7$$!1a4>QkxgI!#;$"1a$>9aqc_'!#<7$$!1kFb54TiH!#;$"1&>c*f:#*[t!#<7$$!29MoOt8='G!#<$"1rkib&>w=)!#<7$$!1**)zf**3&fF!#;$"1En4O&ps.*!#<7$$!1`3<M5ZlE!#;$"14vv@yR:)*!#<7$$!28Ig?hH_c#!#<$"2"y%yN?*=k5!#<7$$!2V&3<M7vlC!#<$"1*))\Wa7f9"!#;7$$!2fGd9HS$fB!#<$"2y/jk&)4IB"!#<7$$!2d5@UCPaE#!#<$"2;8`?o)f48!#<7$$!2v_06i#4d@!#<$"1"*H7-nl(R"!#;7$$!2Yze<&*>$f?!#<$"2&)ep:iUoZ"!#<7$$!22:Ig+zD'>!#<$"2Cu:_XO\b"!#<7$$!1x_069be=!#;$"1=A]UEjQ;!#;7$$!1v],.I5a<!#;$"0DT()y!QA<!#:7$$!2wU&3</2f;!#<$"2\y'=ZRL)z"!#<7$$!2L!3;K+:e:!#<$"2ENf,ET(y=!#<7$$!2'*******fh'f9!#<$"2odIi9lp&>!#<7$$!2E^-05**GN"!#<$"1wbPs4\T?!#;7$$!1#Qw_&)>0E"!#;$"2O:"e9lS9@!#<7$$!2>MoO$*=W:"!#<$"1'=&Q[<!z>#!#;7$$!2hBZ%*[yY0"!#<$"1mAQ`3:wA!#;7$$!0Hd9H9!f&*!#;$"1`!R)*y<MN#!#;7$$!1Nxa4zXq&)!#<$"2Un()f3C0V#!#<7$$!1rV([(H>iv!#<$"2())3P)>U*3D!#<7$$!0)e<N!\M\'!#;$"1mp'fM>=f#!#;7$$!1Uu[(\:N^&!#<$"2/ZS)z:fnE!#<7$$!1l;Lm_>YX!#<$"2B)30&)p=UF!#<7$$!2%*['Hfe97N!#=$"1UBxH+r@G!#;7$$!2X=Pu[Z.[#!#=$"1_/Tk%Q3!H!#;7$$!2jQze<ZUb"!#=$"/c1&=v;(H!#97$$!1tJkGd9sX!#=$"10OcwAObI!#;7$$"1R#f=P9tg%!#=$"1mfOGN?DJ!#;7$$"2C/5?SsV`"!#=$"2XJ*[h2o1K!#<7$$"1JtY$pe!fD!#<$"2vF'4aYB%G$!#<7$$"1Lu[(\")RZ$!#<$"1Dj,0:J`L!#;7$$".(Rzy!H]%!#9$"2&)*=7"423V$!#<7$$"1GjE`YlQb!#<$"1(4BsS<'3N!#;7$$"1%e<Nqy%pl!#<$"1^qV!Qiee$!#;7$$"13Qw_X=Dv!#<$"2.x/mP1tl$!#<7$$"1"4:IgiZ^)!#<$"2/=.WL56t$!#<7$$"1b4>Qc=Q&*!#<$"1<'**)flD2Q!#;7$$"2fNrUlKe0"!#<$"0K*>@o(H)Q!#:7$$"2[8Faom2;"!#<$"1)*\77vngR!#;7$$"2#Hf=PE>`7!#<$"145yd7'*GS!#;7$$"2a>RyOUsN"!#<$"12"eB$4m0T!#;7$$"1=MoO$><Y"!#;$"10?&*Q[\#=%!#;7$$"2GQw_D-Cc"!#<$"1kf^L$pjD%!#;7$$"2Fxa4*>$Ql"!#<$"1J***eD7LK%!#;7$$"2$[(\**y]Dw"!#<$"1E'f'*yRFS%!#;7$$"2P$oOt)\Y&=!#<$"1e5@b!z)pW!#;7$$"2805?+$yh>!#<$"2DT:/=6ya%!#<7$$"2zMpQ(Rhc?!#<$"1YIJGjk;Y!#;7$$"1(\**)z&e1;#!#;$"2l$3%yO6?p%!#<7$$"2<BY#\OtfA!#<$"1'*=%*QZijZ!#;7$$"2<E_/43JO#!#<$"1QOVW&*=Q[!#;7$$"2$Qu[(4Q!eC!#<$"2ED]JKCl!\!#<7$$"1nLnMHVgD!#;$"2v-a+3&3!)\!#<7$$"1D['HR#zmE!#;$"1P$\(*pLj0&!#;7$$"2ONqShy$fF!#<$"1cxJ!ywD7&!#;7$$"1LnMpUPfG!#;$"1c/.U])R>&!#;7$$"1X"HeczE'H!#;$"0/tA"*4wE&!#:7$$"2j5@UkVP1$!#<$"1iq95Y\R`!#;7$$"1<MoO$G:;$!#;$"1#Q5lm8*3a!#;7$$"2m9Hec,,F$!#<$"1e/Lw*Re[&!#;7$$"0.17WfwO$!#:$"1>[;Mp#[b&!#;7$$"2DT#['HB=Z$!#<$"1GS>DfMGc!#;7$$"1yd:J7@mN!#;$"0#e&*[?%[p&!#:7$$"0Pu[<-%pO!#:$"1&z2#3uSnd!#;7$$"2xFb5"o\mP!#<$"1[R?Y1cNe!#;7$$"017Co))z'Q!#:$"1Ji$Q/sm!f!#;7$$"1U%)oPb@nR!#;$"13_y:32wf!#;7$$"2O['Hfo4rS!#<$"10*\D9$f[g!#;7$$"1lHf=p9rT!#;$"1&))*[*f9$=h!#;7$$"1oQxaDYtU!#;$"1Q5"o3)[*='!#;7$$"1Z"He'4$\P%!#;$"1&46>%o%*fi!#;7$$"1Nu[(*)p"oW!#;$"0^RS2#eCj!#:7$$"2w(e<NA.vX!#<$"0rIu6M&)R'!#:7$$"2'*f>R=61n%!#<$"1<!**HkjXY'!#;7$$"2&f>Ry#=Dx%!#<$"1p'RpOZ[`'!#;7$$"1sU&3Pc+([!#;$"0E(>)G0?g'!#:7$$"1&=Pu))f$y\!#;$"0yq_tZkn'!#:7$$"1rV([xU@2&!#;$"1e#4*o<!3u'!#;7$$"1tZ&4>(ey^!#;$"1"yJ%HUs8o!#;7$$"1:Ig?(*fv_!#;$"1y^(G/v+)o!#;7$$"1Y&4>=;<Q&!#;$"1)4u%yK`_p!#;7$$"1W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A case that is only DA (not in the paper) eqwp:=wp(y)-(w(y)-w(0))/y/D(w)(0); wwp:=solve(%,w(y));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcXdwR0YoLCYtSSN3cEdGKDYjSSJ5R0YoIiIiKigsJi1JIndHRihGMUYzLUY3NiMiIiEhIiJGM0YyRjstLUkiREdGJTYjRjdGOUY7Rjs3I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR3d3BHRigsJiooLUkjd3BHRig2I0kieUdGKCIiIkYzRjQtLUkiREdGJTYjSSJ3R0YoNiMiIiFGNEY0LUY5RjpGNDcjRi4=

edwp:=eval(subs(isolate(eqwp,w(y)),edw));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlZHdwR0YoLCgqKiMiIiIiIiVGMSwmKiYpLCgqJiIiI0YxSSJ5R0YoRjFGMUkjeXBHRighIiJJI3ltR0YoRjtGOEYxKSwmRjpGMUY8RjtGOEY7RjtGMUYxRjEsJiooLUklZGlmZkdGJjYkLUkjd3BHRig2I0Y5LUkiJEdGJjYkRjlGOEYxRjlGMS0tSSJER0YlNiNJIndHRig2IyIiIUYxRjEqKEY4RjEtRkI2JEZERjlGMUZKRjFGMUYxKSwmRjpGO0Y8RjFGOEYxRjEqLCNGMUY4RjFGNkYxRj5GOywmKihGUkYxRjlGMUZKRjFGMSomRkRGMUZKRjFGMUYxRlVGMUYxKiYpSSJhR0YoRjhGMSwmKihGREYxRjlGMUZKRjFGMS1GTkZPRjFGMUYxNyMsKCoqRjBGMUYzRjEsJiooLUZCNiRGUkY5RjFGOUYxRkpGMUYxRlFGMUYxRlRGMUYxRlZGMUZlbkYx

edphi1:=eval(subs(wp(y)=-1/phi(y),edwp));

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

edphi2:=collect(edphi1*phi(y)^3/D(w)(0),[diff(phi(y), y, y),diff(phi(y), y),phi(y)],factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdlZHBoaTJHRigsLCosLCZJInlHRigiIiJJI3lwR0YoISIiRjIsJkYxRjJJI3ltR0YoRjRGMkYxRjItSSRwaGlHRig2I0YxRjItSSVkaWZmR0YmNiRGNy1JIiRHRiY2JEYxIiIjRjJGNCosRkBGMkYwRjJGNUYyRjFGMiktRjs2JEY3RjFGQEYyRjIqKCwqKiYiIiRGMilGMUZARjJGNCooIyIiJkZARjJGMUYyRjZGMkYyKihGS0YyRjFGMkYzRjJGMiooRkBGMkYzRjJGNkYyRjRGMkY3RjJGQ0YyRjIqKilJImFHRihGQEYyLUkid0dGKDYjIiIhRjItLUkiREdGJTYjRlNGVEY0KUY3RkhGMkYyKiYsKkYxRjIqJiNGMkZARjJGM0YyRjQqJkZobkYyRjZGMkY0KiZGUEYyRjFGMkY0RjIpRjdGQEYyRjI3IywsKixGMEYyRjVGMkYxRjJGN0YyLUY7NiRGQ0YxRjJGNEZBRjJGRUYyRk9GMkZlbkYy

map(factor,subs(yp=(s+d)/2, ym=(s-d)/2,%));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsLCouIyIiIiIiJUYvLCgqJiIiI0YvSSJ5R0YoRi8hIiJJInNHRihGL0kiZEdGKEYvRi8sKEYyRi9GNkY1RjdGL0YvRjRGLy1JJHBoaUdGKDYjRjRGLy1JJWRpZmZHRiY2JEY5LUkiJEdGJjYkRjRGM0YvRi8qLCNGL0YzRi9GMUYvRjhGL0Y0Ri8pLUY9NiRGOUY0RjNGL0Y1KipGQ0YvLCoqJClGN0YzRi9GLyokKUY2RjNGL0Y1KigiIiZGL0Y2Ri9GNEYvRi8qJiIiJ0YvKUY0RjNGL0Y1Ri9GOUYvRkVGL0YvKiopSSJhR0YoRjNGLy1JIndHRig2IyIiIUYvLS1JIkRHRiU2I0ZWRldGNSlGOSIiJEYvRi8qKEZDRi8sKCooRjNGL0ZTRi9GNEYvRi9GNkYvRjJGNUYvKUY5RjNGL0Y1NyMsLCouRi5GL0YxRi9GOEYvRjRGL0Y5Ri8tRj02JEZFRjRGL0YvRkJGNUZHRi9GUkYvRmluRjU=

indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KUkiYUdGKEkiZEdGKEkic0dGKEkieUdGKC1JJWRpZmZHRiY2JC1JJHBoaUdGKDYjRjBGMC1GMjYkRjQtSSIkR0YmNiRGMCIiI0Y0NyM8KUYtRi5GL0YwLUYyNiRGMUYwRjFGNA==

We start by the Laplace transform of p_0 int(sqrt(r)*exp(-a*r),r=0..infinity) assuming a>0

LCQqKCMiIiIiIiNGJSlJI1BpRyUqcHJvdGVjdGVkR0YkRiUpSSJhRzYiIyIiJEYmISIiRiU=

Ts:=tan(s)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNUc0dGKC1JJHRhbkdGJTYjSSJzR0YoNyNGLg==

simplify(1/sqrt(1+Ts^2)) assuming (s>-Pi/2, s<Pi/2)

LUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJInNHRic=

simplify(Ts/sqrt(1+Ts^2)) assuming (s>-Pi/2, s<Pi/2)

LUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJInNHRic=

diff(Ts,s)/(1+Ts^2)

IiIi

factor(int(1/(2*sinus-(1-t^2)*A-2*t*B)^(3/2),t))

LCQqKCwmKiZJIkFHNiIiIiJJInRHRidGKEYoSSJCR0YnISIiRigpLCoqJkYmRigpRikiIiNGKEYoKihGMEYoRipGKEYpRihGK0YmRisqJkYwRihJJnNpbnVzR0YnRihGKCNGKEYwRissKCokKUYmRjBGKEYoKihGMEYoRiZGKEYzRihGKyokKUYqRjBGKEYoRitGKw==

We want to translate alpha1=0=alpha2 in terms of the original quadrant model tant:=sin(b)/(mu1/mu2*sqrt(s22/s11)+cos(b))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSV0YW50R0YoKiYtSSRzaW5HRiU2I0kiYkdGKCIiIiwmKihJJG11MUdGKEYzSSRtdTJHRighIiIpKiZJJHMyMkdGKEYzSSRzMTFHRihGOCNGMyIiI0YzRjMtSSRjb3NHRiVGMUYzRjg3I0Yu

tand:=sin(b)/(r12/r22*sqrt(s22/s11)+cos(b))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSV0YW5kR0YnKiYtSSRzaW5HRiU2I0kiYkdGKCIiIiwmKihJJHIxMkdGKEYzSSRyMjJHRighIiIpKiZJJHMyMkdGKEYzSSRzMTFHRihGOCNGMyIiI0YzRjMtSSRjb3NHRiVGMUYzRjg3I0Yu

tantbis:=expand(tan(2*d-Pi))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSh0YW50YmlzR0YoLCQqKCIiIyIiIi1JJHRhbkdGJTYjSSJkR0YoRjEsJkYxRjEqJClGMkYwRjEhIiJGOUYxNyNGLg==

factor(simplify(op(3,numer(factor(subs(tan(d)=tand,tantbis-tant)))),trig))

LCoqLCIiIyIiIkkkczIyRzYiRiVJJG11MUdGJ0YlSSRyMTJHRidGJUkkcjIyR0YnRiVGJSooRiZGJUkkbXUyR0YnRiUpRilGJEYlISIiKi5GJEYlKSomRiZGJUkkczExR0YnRi4jRiVGJEYlLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YnNiNJImJHRidGJUYoRiUpRipGJEYlRjJGJUYlKihGLEYlRjtGJUYyRiVGJQ==

cosb:=-s12/sqrt(s11*s22)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVjb3NiR0YoLCQqJkkkczEyR0YoIiIiKSomSSRzMTFHRihGMUkkczIyR0YoRjEjRjEiIiMhIiJGODcjRi4=

numer(factor(subs(cos(b)=cosb,%%)))

LCoqLiIiIyIiIikqJkkkczIyRzYiRiVJJHMxMUdGKSEiIiNGJUYkRiVJJHMxMkdGKUYlSSRtdTFHRilGJSlJJHIyMkdGKUYkRiVGKkYlRisqLkYkRiVGKEYlRi5GJUkkcjEyR0YpRiVGMEYlKSomRipGJUYoRiVGLEYlRiUqKkYoRiVJJG11MkdGKUYlKUYyRiRGJUYzRiVGKyoqRjZGJUYvRiVGKkYlRjNGJUYl

simplify(%) assuming s11>0 assuming s22>0

KigpSSRzMTFHNiIjIiIiIiIjRicpSSRzMjJHRiVGJkYnLCoqLEYoRidJJG11MUdGJUYnSSRyMTJHRiVGJ0kkcjIyR0YlRidGKkYnRicqKkYoRidGLUYnKUYvRihGJ0kkczEyR0YlRichIiIqKEkkbXUyR0YlRicpRi5GKEYnRipGJ0YzKihGNUYnRjFGJ0YkRidGJ0Yn

rel1:=collect(op(3,%),[mu1,mu2],factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVyZWwxR0YoLCYqKiIiIyIiIkkkcjIyR0YoRjEsJiomSSRyMTJHRihGMUkkczIyR0YoRjFGMSomRjJGMUkkczEyR0YoRjEhIiJGMUkkbXUxR0YoRjFGMSomLCYqJilGNUYwRjFGNkYxRjkqJilGMkYwRjFJJHMxMUdGKEYxRjFGMUkkbXUyR0YoRjFGMTcjRi4=

tane:=sin(b)/(r21/r11*sqrt(s11/s22)+cos(b))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSV0YW5lR0YoKiYtSSRzaW5HRiU2I0kiYkdGKCIiIiwmKihJJHIyMUdGKEYzSSRyMTFHRighIiIpKiZJJHMxMUdGKEYzSSRzMjJHRihGOCNGMyIiI0YzRjMtSSRjb3NHRiVGMUYzRjg3I0Yu

tanbmt:=sin(b)/(mu2/mu1*sqrt(s11/s22)+cos(b))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSd0YW5ibXRHRigqJi1JJHNpbkdGJTYjSSJiR0YoIiIiLCYqKEkkbXUyR0YoRjNJJG11MUdGKCEiIikqJkkkczExR0YoRjNJJHMyMkdGKEY4I0YzIiIjRjNGMy1JJGNvc0dGJUYxRjNGODcjRi4=

tanbmtbis:=expand(tan(2*e-Pi))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSp0YW5ibXRiaXNHRigsJCooIiIjIiIiLUkkdGFuR0YlNiNJImVHRihGMSwmRjFGMSokKUYyRjBGMSEiIkY5RjE3I0Yu

factor(simplify(op(2,numer(factor(subs(tan(e)=tane,tanbmtbis-tanbmt)))),trig))

LCoqKEkkczExRzYiIiIiSSRtdTFHRiVGJilJJHIyMUdGJSIiI0YmISIiKixGKkYmRiRGJkkkbXUyR0YlRiZJJHIxMUdGJUYmRilGJkYmKi5GKkYmKSomRiRGJkkkczIyR0YlRisjRiZGKkYmLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNJImJHRiVGJkYtRiYpRi5GKkYmRjJGJkYmKihGJ0YmRjtGJkYyRiZGJg==

numer(factor(subs(cos(b)=cosb,%)))

LCoqLiIiIyIiIikqJkkkczExRzYiRiVJJHMyMkdGKSEiIiNGJUYkRiVJJHMxMkdGKUYlSSRtdTJHRilGJSlJJHIxMUdGKUYkRiVGKkYlRisqKkkkbXUxR0YpRiVGL0YlRipGJSkqJkYoRiVGKkYlRixGJUYlKipGKEYlRjJGJSlJJHIyMUdGKUYkRiVGM0YlRisqLkYkRiVGKEYlRi5GJUYwRiVGN0YlRjNGJUYl

simplify(%) assuming s11>0 assuming s22>0

KigsKiooSSRtdTFHNiIiIiIpSSRyMTFHRiYiIiNGJ0kkczIyR0YmRidGJyooRiVGJylJJHIyMUdGJkYqRidJJHMxMUdGJkYnISIiKipGKkYnSSRtdTJHRiZGJ0YoRidJJHMxMkdGJkYnRjAqLEYqRidGMkYnRilGJ0YuRidGL0YnRidGJylGKyNGJ0YqRicpRi9GNkYn

rel2:=collect(op(1,%),[mu1,mu2],factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVyZWwyR0YoLCYqJiwmKiYpSSRyMTFHRigiIiMiIiJJJHMyMkdGKEY1RjUqJilJJHIyMUdGKEY0RjVJJHMxMUdGKEY1ISIiRjVJJG11MUdGKEY1RjUqKkY0RjVGM0Y1LCYqJkYzRjVJJHMxMkdGKEY1RjUqJkY5RjVGOkY1RjtGNUkkbXUyR0YoRjVGOzcjRi4=

rel1;

LCYqKiIiIyIiIkkkcjIyRzYiRiUsJiomSSRyMTJHRidGJUkkczIyR0YnRiVGJSomRiZGJUkkczEyR0YnRiUhIiJGJUkkbXUxR0YnRiVGJSomLCYqJilGKkYkRiVGK0YlRi4qJilGJkYkRiVJJHMxMUdGJ0YlRiVGJUkkbXUyR0YnRiVGJQ==

solve({rel1,rel2},[mu1,mu2])

NyM3JC9JJG11MUc2IiIiIS9JJG11Mkc2IiIiIQ==

factor([solve(rel2,r21)])

NyQqKiwmKiZJJHMxMUc2IiIiIkkkbXUyR0YnRihGKCokKSomRiZGKCwoKiYpSSRtdTFHRiciIiNGKEkkczIyR0YnRihGKCoqRjFGKEYwRihGKUYoSSRzMTJHRidGKCEiIiomKUYpRjFGKEYmRihGKEYoI0YoRjFGKEYoRihGMEY1RiZGNUkkcjExR0YnRigsJCoqLCZGJUY1RipGKEYoRjBGNUYmRjVGOUYoRjU=

factor([solve(rel1,r12)]);

NyQqKiwmKiZJJG11MUc2IiIiIkkkczIyR0YnRihGKCokKSomRilGKCwoKiYpRiYiIiNGKEYpRihGKCoqRjBGKEYmRihJJG11MkdGJ0YoSSRzMTJHRidGKCEiIiomKUYyRjBGKEkkczExR0YnRihGKEYoI0YoRjBGKEYoRihGMkY0RilGNEkkcjIyR0YnRigsJCoqLCZGJUY0RipGKEYoRjJGNEYpRjRGOUYoRjQ=

rel2bis:=-r21/r11+mu2/mu1+1/mu1*sqrt(Delta/s11); rel1bis:=-r12/r22+mu1/mu2+1/mu2*sqrt(Delta/s22);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShyZWwyYmlzR0YoLCgqJkkkcjIxR0YoIiIiSSRyMTFHRighIiJGMyomSSRtdTJHRihGMUkkbXUxR0YoRjNGMSomRjZGMykqJkkmRGVsdGFHRihGMUkkczExR0YoRjMjRjEiIiNGMUYxNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShyZWwxYmlzR0YoLCgqJkkkcjEyR0YoIiIiSSRyMjJHRighIiJGMyomSSRtdTFHRihGMUkkbXUyR0YoRjNGMSomRjZGMykqJkkmRGVsdGFHRihGMUkkczIyR0YoRjMjRjEiIiNGMUYxNyNGLg==

latex(rel1bis)

-\134frac{\134mathit{r12}}{\134mathit{r22}}+\134frac{\134mu1}{\134mu2}+\134frac{\134sqrt{\134frac{\134Delta}{\134mathit{s22}}}}{\134mu2}

factor(subs(isolate(rel2bis,r21),rel2)); factor(subs(isolate(rel1bis,r12),rel1))

LCQqKClJJHIxMUc2IiIiIyIiIkkkbXUxR0YmISIiLCoqJilGKUYnRihJJHMyMkdGJkYoRioqKkYnRihGKUYoSSRtdTJHRiZGKEkkczEyR0YmRihGKComKUYwRidGKEkkczExR0YmRihGKkkmRGVsdGFHRiZGKEYoRio=

LCQqKClJJHIyMkc2IiIiIyIiIkkkbXUyR0YmISIiLCoqJilJJG11MUdGJkYnRihJJHMyMkdGJkYoRioqKkYnRihGLkYoRilGKEkkczEyR0YmRihGKComKUYpRidGKEkkczExR0YmRihGKkkmRGVsdGFHRiZGKEYoRio=

Relation entre sigma et r relrs:=op(3,numer(factor(subs(isolate(rel2,mu2),rel1))))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZyZWxyc0dGKCwyKigpSSRyMTFHRigiIiMiIiIpSSRyMTJHRihGMkYzKUkkczIyR0YoRjJGM0YzKi4iIiVGM0YwRjNGNUYzSSRyMjJHRihGM0kkczEyR0YoRjNGN0YzISIiKipGMEYzKUY6RjJGM0kkczExR0YoRjNGN0YzRjwqKkY5RjNGMEYzRj5GMylGO0YyRjNGMyowRjlGM0YxRjNGNUYzSSRyMjFHRihGM0Y6RjNGP0YzRjdGM0YzKi5GOUYzRjFGM0ZDRjNGPkYzRj9GM0Y7RjNGPCoqRjRGMylGQ0YyRjNGP0YzRjdGM0Y8KihGRkYzRj5GMylGP0YyRjNGMzcjRi4=

collect(relrs,[s11,s22,s12],distributed,factor)

LC4qKClJJHIyMUc2IiIiIyIiIilJJHIyMkdGJkYnRigpSSRzMTFHRiZGJ0YoRigqLiIiJUYoSSRyMTFHRiZGKEYlRihGKUYoRixGKEkkczEyR0YmRighIiIqKCwoKiYpRi9GJ0YoRilGKEYxKixGLkYoRi9GKEkkcjEyR0YmRihGJUYoRipGKEYoKiYpRjdGJ0YoRiRGKEYxRihGLEYoSSRzMjJHRiZGKEYoKipGLkYoRjVGKEYpRigpRjBGJ0YoRigqLkYuRihGNUYoRjdGKEYqRihGMEYoRjpGKEYxKihGNUYoRjlGKClGOkYnRihGKA==

factor(expand(sin(b)*sin(d)+sin(t)*sin(b-d)-sin(d)*sin(b-t)))

LCQqJi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJiR0YpIiIiLCgqJi1GJTYjSSJkR0YpRiwtSSRjb3NHRiY2I0kidEdGKUYsRiwqJi1GJUY0RiwtRjNGMEYsISIiRi9GOUYsRjk=

simplify(subs(t=2*d-Pi,%))

IiIh

gxynorm; gxyspe:=eval(subs(b=t+2*e-Pi,t=2*d-Pi,%))

LCQqLCMiIiIiIiNGJSwsKipGJkYlLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJImJHRi5GJUkidUdGLkYlSSJ2R0YuRiVGJSoqRiZGJS1JJHNpbkdGKzYjLCZGMEYlSSJ0R0YuISIiRiUtRjVGL0YlRjFGJUYlKipGJkYlRjpGJS1GNTYjRjhGJUYyRiVGJSokKUYxRiZGJUY5KiQpRjJGJkYlRjlGJUkmRGVsdGFHRi5GJSlGOkYmRjlJJGRldEdGLkY5Rjk=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdneHlzcGVHRigsJCosIyIiIiIiI0YxLCwqKkYyRjEtSSRjb3NHRiU2IywmKiZGMkYxSSJkR0YoRjFGMSomRjJGMUkiZUdGKEYxRjFGMUkidUdGKEYxSSJ2R0YoRjFGMSoqRjJGMS1JJHNpbkdGJTYjLCRGO0YxRjEtRkFGN0YxRj1GMSEiIioqRjJGMUZERjEtRkE2IywkRjlGMUYxRj5GMUZFKiQpRj1GMkYxRkUqJClGPkYyRjFGRUYxSSZEZWx0YUdGKEYxKUZERjJGRUkkZGV0R0YoRkVGRTcjRi4=

uvpspe:=eval(subs(b=t+2*e-Pi,t=2*d-Pi,[up,vp]))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSd1dnBzcGVHRig3JCwmIiIiRjAtSSRjb3NHRiU2IywkKiYiIiNGMEkiZEdGKEYwRjAhIiIsJkYwRjAtRjI2IywkKiZGNkYwSSJlR0YoRjBGMEY4NyNGLg==

uvpspe[1]

LCYiIiJGIy1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiI0YjSSJkR0YpRiNGIyEiIg==

We introduce t1=tan(d) and t2=tan(e) t1e:=tan(d);t2e:=tan(e);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR0MWVHRigtSSR0YW5HRiU2I0kiZEdGKDcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR0MmVHRigtSSR0YW5HRiU2I0kiZUdGKDcjRi4=

simplify(cos(2*d)-(1-t1e^2)/(1+t1e^2)); simplify(sin(2*d)-2*t1e/(1+t1e^2))

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Factorization of the kernel gxyfact:=factor(subs(cos(dt2)=(1-t1^2)/(1+t1^2),sin(dt2)=2*t1/(1+t1^2), cos(et2)=(1-t2^2)/(1+t2^2),sin(et2)=2*t2/(1+t2^2), expand(subs(u=uvpspe[1]*(1-r1^2), v=uvpspe[2]*(1-r2^2),d=dt2/2,e=et2/2, gxyspe))))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShneHlmYWN0R0YoLCQqMCMiIiIiIiNGMUkmRGVsdGFHRihGMSw0KigpSSNyMUdGKEYyRjEpSSN0MUdGKEYyRjEpSSN0MkdGKEYyRjFGMSosRjJGMUY3RjFJI3IyR0YoRjFGOEYxRjpGMSEiIiooKUY9RjJGMUY4RjFGOkYxRjEqJkY2RjFGOEYxRjEqLEYyRjFGN0YxRj1GMUY5RjFGO0YxRjEqJkZARjFGOkYxRjEqJEY4RjFGPiooRjJGMUY5RjFGO0YxRj4qJEY6RjFGPkYxLDRGNUYxRjxGMUY/RjFGQUYxRkJGPkZDRjFGREY+RkVGPkZGRj5GMSksJkY7RjFGOUYxRjJGPiksJiomRjlGMUY7RjFGMUYxRj5GMkY+SSRkZXRHRihGPkYxNyNGLg==

Now the numerator num:=eval(subs(b=t+2*e-Pi,t=2*d-Pi,-(phi10g1expr/r2+phi20g2expr/r1)))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRudW1HRigsJiowSSZEZWx0YUdGKCIiIiwmKiYtSSRzaW5HRiU2IywmKiYiIiNGMUkiZEdGKEYxRjFJImVHRihGMUYxSSJ2R0YoRjFGMSomLUY1NiNGO0YxSSJ1R0YoRjFGMUYxSSRkZXRHRighIiItRjU2IywmRjhGMSomRjlGMUY7RjFGMUZCLUY1NiNGOkYxLUY1NiMsJkY6RjFGO0YxRkJJI3IyR0YoRkJGMSowRjBGMSwmKiYtRjU2IywmRjpGMUZGRjFGMUZARjFGMSomRkdGMUY8RjFGMUYxRkFGQkZDRkJGPkYxRklGQkkjcjFHRihGQkYxNyNGLg==

num1:=subs(u=uvpspe[1]*(1-r1^2), v=uvpspe[2]*(1-r2^2),num)

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

num1:=factor(num1)

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

The numerator and denominator of num1 have expressions in terms of cos(2d), sin(2d), cos(2e), sin(2e) once multiplied by sin(d) sin(e) factor(subs(cos(dt2)=(1-t1^2)/(1+t1^2),sin(dt2)=2*t1/(1+t1^2), cos(et2)=(1-t2^2)/(1+t2^2),sin(et2)=2*t2/(1+t2^2), expand(subs(d=dt2/2,e=et2/2,combine(sin(d)*sin(e)*denom(num1))))))

LCQqNiIiIyIiIkkkZGV0RzYiRiVJI3IyR0YnRiVJI3IxR0YnRiVJI3QxR0YnRiVJI3QyR0YnRiUsJiomRipGJUYrRiVGJUYlISIiRiUpLCZGK0YlRipGJUYkRiUpLCYqJClGKkYkRiVGJUYlRiVGJEYuKSwmKiQpRitGJEYlRiVGJUYlRiRGLkYu

factor(subs(cos(dt2)=(1-t1^2)/(1+t1^2),sin(dt2)=2*t1/(1+t1^2), cos(et2)=(1-t2^2)/(1+t2^2),sin(et2)=2*t2/(1+t2^2), expand(subs(d=dt2/2,e=et2/2,combine(sin(d)*sin(e)*numer(num1))))))

LCQqMiIiIyIiIkkmRGVsdGFHNiJGJSlJI3QxR0YnRiRGJSlJI3QyR0YnRiRGJSwmSSNyMUdGJ0YlSSNyMkdGJ0YlRiUsNCooKUYtRiRGJUYoRiVGKkYlRiUqLEYkRiVGLUYlRi5GJUYoRiVGKkYlISIiKigpRi5GJEYlRihGJUYqRiVGJSomRjFGJUYoRiVGJSosRiRGJUYtRiVGLkYlRilGJUYrRiVGJSomRjVGJUYqRiVGJSokRihGJUYzKihGJEYlRilGJUYrRiVGMyokRipGJUYzRiUpLCZGOUYlRiVGJUYkRjMpLCZGO0YlRiVGJUYkRjNGMw==

num2:=factor(%/%%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVudW0yR0YoKjZJJkRlbHRhR0YoIiIiSSN0MUdGKEYwSSN0MkdGKEYwLCZJI3IxR0YoRjBJI3IyR0YoRjBGMCw0KigpRjQiIiNGMClGMUY5RjApRjJGOUYwRjAqLEY5RjBGNEYwRjVGMEY6RjBGO0YwISIiKigpRjVGOUYwRjpGMEY7RjBGMComRjhGMEY6RjBGMCosRjlGMEY0RjBGNUYwRjFGMEYyRjBGMComRj9GMEY7RjBGMCokRjpGMEY9KihGOUYwRjFGMEYyRjBGPSokRjtGMEY9RjBJJGRldEdGKEY9RjVGPUY0Rj0sJiomRjFGMEYyRjBGMEYwRj1GPSksJkYyRjBGMUYwRjlGPTcjRi4=

phixyspe:=factor(num2/gxyfact)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlwaGl4eXNwZUdGKCwkKjIiIiMiIiJJI3QxR0YoRjFJI3QyR0YoRjEsJkkjcjFHRihGMUkjcjJHRihGMUYxRjYhIiJGNUY3LCYqJkYyRjFGM0YxRjFGMUY3RjEsNCooKUY1RjBGMSlGMkYwRjEpRjNGMEYxRjEqLEYwRjFGNUYxRjZGMUY9RjFGPkYxRjEqKClGNkYwRjFGPUYxRj5GMUYxKiZGPEYxRj1GMUYxKixGMEYxRjVGMUY2RjFGMkYxRjNGMUY3KiZGQUYxRj5GMUYxKiRGPUYxRjcqKEYwRjFGMkYxRjNGMUY3KiRGPkYxRjdGN0YxNyNGLg==

We can now restore expressions in terms of delta and epsilon pol:=op(3,denom(phixyspe))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRwb2xHRigsNCooKUkjcjFHRigiIiMiIiIpSSN0MUdGKEYyRjMpSSN0MkdGKEYyRjNGMyosRjJGM0YxRjNJI3IyR0YoRjNGNEYzRjZGM0YzKigpRjlGMkYzRjRGM0Y2RjNGMyomRjBGM0Y0RjNGMyosRjJGM0YxRjNGOUYzRjVGM0Y3RjMhIiIqJkY7RjNGNkYzRjMqJEY0RjNGPiooRjJGM0Y1RjNGN0YzRj4qJEY2RjNGPjcjRi4=

collect(pol/(1+t1^2)/(1+t2^2),[r1,r2],distributed,factor)

LCoqKClJI3QxRzYiIiIjIiIiLCYqJEYkRihGKEYoRighIiIpSSNyMUdGJkYnRihGKCoyRidGKEYlRihJI3QyR0YmRigsJiomRiVGKEYvRihGKEYoRitGKEYpRissJiokKUYvRidGKEYoRihGKEYrRi1GKEkjcjJHRiZGKEYoKihGNEYoRjJGKylGNUYnRihGKCooKSwmRi9GKEYlRihGJ0YoRilGK0YyRitGKw==

subs(t1=t1e,t2=t2e,%)

LCoqKCktSSR0YW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kiZEdGKiIiIyIiIiwmKiRGJEYuRi5GLkYuISIiKUkjcjFHRipGLUYuRi4qMkYtRi5GJUYuLUYmNiNJImVHRipGLiwmKiZGJUYuRjVGLkYuRi5GMUYuRi9GMSwmKiQpRjVGLUYuRi5GLkYuRjFGM0YuSSNyMkdGKkYuRi4qKEY8Ri5GOkYxKUY9Ri1GLkYuKigpLCZGNUYuRiVGLkYtRi5GL0YxRjpGMUYx

map(simplify,%)

LCoqJilJI3IxRzYiIiIjIiIiKS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjSSJkR0YmRidGKEYoKi5GJ0YoRiVGKEkjcjJHRiZGKEYqRigtRis2I0kiZUdGJkYoLCYqJi1JJGNvc0dGLEYvRigtRjlGNEYoRigqJkYqRihGM0YoISIiRihGPComKUYyRidGKClGM0YnRihGKCokKSwmKiZGKkYoRjpGKEYoKiZGM0YoRjhGKEYoRidGKEY8

simplify(r1^2*sin(d)^2+r2^2*sin(e)^2-2*r2*r1*sin(d)*sin(e)*cos(d+e)-sin(d+e)^2-%)

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subs(t1=t1e,t2=t2e,numer(phixyspe)/(1+t1^2)/(1+t2^2))

LCQqMCIiIyIiIi1JJHRhbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJkR0YrRiUtRic2I0kiZUdGK0YlLCZJI3IxR0YrRiVJI3IyR0YrRiVGJSwmKiZGJkYlRi5GJUYlRiUhIiJGJSwmKiQpRiZGJEYlRiVGJUYlRjYsJiokKUYuRiRGJUYlRiVGJUY2RiU=

simplify(%)

LCQqLCIiIyIiIiwmSSNyMUc2IkYlSSNyMkdGKEYlRiUtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRig2I0kiZEdGKEYlLUYrNiNJImVHRihGJSwmKiYtSSRjb3NHRixGL0YlLUY3RjJGJUYlKiZGKkYlRjFGJSEiIkYlRjo=

phixyspetrig:=-(2*(r1+r2))*sin(d)*sin(e)*cos(d+e)/r1/r2/(r1^2*sin(d)^2+r2^2*sin(e)^2-2*r2*r1*sin(d)*sin(e)*cos(d+e)-sin(d+e)^2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SS1waGl4eXNwZXRyaWdHRigsJCoyIiIjIiIiLCZJI3IxR0YoRjFJI3IyR0YoRjFGMS1JJHNpbkdGJTYjSSJkR0YoRjEtRjY2I0kiZUdGKEYxLUkkY29zR0YlNiMsJkY4RjFGO0YxRjFGMyEiIkY0RkAsKiomKUYzRjBGMSlGNUYwRjFGMSomKUY0RjBGMSlGOUYwRjFGMSouRjBGMUY0RjFGM0YxRjVGMUY5RjFGPEYxRkAqJCktRjZGPkYwRjFGQEZARkA3I0Yu

Let us check factor(simplify(expand(subs(t1=t1e,t2=t2e,phixyspe-phixyspetrig))))

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The value when x=y=0 simplify(expand(subs(r1=1,r2=1,phixyspetrig)))

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The denominator den:=-op(3,denom(phixyspetrig))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRkZW5HRigsKiomKUkjcjFHRigiIiMiIiIpLUkkc2luR0YlNiNJImRHRihGMkYzRjMqJilJI3IyR0YoRjJGMyktRjY2I0kiZUdGKEYyRjNGMyouRjJGM0Y7RjNGMUYzRjVGM0Y9RjMtSSRjb3NHRiU2IywmRjhGM0Y/RjNGMyEiIiokKS1GNkZDRjJGM0ZFNyNGLg==

Parametrization of the denominator subs(r1=1/2/sin(d)*(v+1/v),den)

LCoqJiMiIiIiIiVGJSksJkkidkc2IkYlKiRGKSEiIkYlIiIjRiVGJSomKUkjcjJHRipGLUYlKS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKjYjSSJlR0YqRi1GJUYlKipGMEYlRihGJUYyRiUtSSRjb3NHRjQ2IywmSSJkR0YqRiVGOEYlRiVGLCokKS1GM0Y8Ri1GJUYs

factor(discrim(%,r2))

KigpLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJImVHRikiIiMiIiIpSSJ2R0YpRiwhIiIsMComKS1JJGNvc0dGJjYjLCZJImRHRilGLUYrRi1GLEYtKUYvIiIlRi1GLSooRixGLUYzRi1GLkYtRi0qKEY6Ri0pLUYlRjZGLEYtRi5GLUYtKiRGOUYtRjAqJEYzRi1GLSomRixGLUYuRi1GMEYtRjBGLQ==

simplify(%,trig)

LCQqLClJInZHNiIiIiMhIiIpLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNJImVHRiZGJyIiIiktRis2IywmSSJkR0YmRjFGMEYxRidGMSksJkYlRjFGMUYoRidGMSksJkYlRjFGMUYxRidGMUYo

r2sol:=simplify(factor([solve(subs(r1=1/2/sin(d)*(v+1/v),den),r2)]),symbolic)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZyMnNvbEdGKDckLCQqKiMiIiIiIiNGMiwmKiYsJiomXiNGMkYyKUkidkdGKEYzRjJGMkY4ISIiRjItSSRzaW5HRiU2IywmSSJkR0YoRjJJImVHRihGMkYyRjIqJiwmKiRGOUYyRjJGMkYyRjItSSRjb3NHRiVGPkYyRjJGMkY6RjstRj02I0ZBRjtGMiwkKipGMUYyLCYqJiwmKiYsJEY4RjJGMkY5RjJGO0Y4RjJGMkY8RjJGMkZCRjJGMkY6RjtGR0Y7RjI3I0Yu

collect(op(1,%),v)

LCYqKiMiIiIiIiNGJSwmKiZeI0YlRiUtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywmSSJkR0YvRiVJImVHRi9GJUYlRiUtSSRjb3NHRixGMEYlRiUtRis2I0YzISIiSSJ2R0YvRiVGJSoqRiRGJSwmKiYsJEYpRiVGJUYqRiVGOEY0RiVGJUY2RjhGOUY4RiU=

Let us check convert(subs(r1=1/2/sin(d)*(v+1/v),r2=1/2/sin(e)*(v*exp(-I*(d+e))+exp(I*(d+e))/v), den),exp)

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normal(subs(v=1/v,r2sol[1])-r2sol[2])

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The ellipse den(r1,r2)=0 with(plots): uvpspe

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simplify(factor(subs(r1=1,r2=-1,den)))

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La courbe den(r1,r2)=0, et la m\303\252me courbe sur les variables normales u et v dexpl:=3*Pi/4: eexpl:=11*Pi/20: evalf(3*Pi/2-dexpl-eexpl); r10:=-sin(dexpl+eexpl)/sin(dexpl): r20:=-sin(dexpl+eexpl)/sin(eexpl): P1:=implicitplot(subs(d=dexpl, e=eexpl, den),r1=-0..r10+1/2, r2=-0..r20+1/2,thickness=3): P2:=implicitplot(subs(d=dexpl, e=eexpl, den),r1=-3..3, r2=-3..3,linestyle=dot): P3:=implicitplot(r2,r1=-sin(dexpl+eexpl)/sin(dexpl)..3,r2=-3..3,thickness=3): P4:=implicitplot(r1,r1=-3..3,r2=-sin(dexpl+eexpl)/sin(eexpl)..3,thickness=3): display(P1,P2,P3,P4,labels=[r[1],r[2]],view=[-r10-1/2..r10+1/2,-r20-1/2..r20+1/2],scaling=constrained);

JCIrM2A9JEcnISM1

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umin:=-1: uplus:=subs(d=dexpl, e=eexpl,uvpspe[1]): evalf(%); vplus:=subs(d=dexpl, e=eexpl,uvpspe[2]):evalf(%); Q0:=implicitplot(subs( r1=sqrt(1-u/uvpspe[1]), r2=sqrt(1-v/uvpspe[2]),d=dexpl, e=eexpl,den),u=-5..uplus,v=-5..uplus): Q1:=implicitplot(subs( r1=sqrt(1-u/uvpspe[1]), r2=sqrt(1-v/uvpspe[2]),d=dexpl, e=eexpl,den), u=uplus*(1-r10^2)..uplus+1/10, v=-1/10+vplus*(1-r20^2)..vplus,numpoints=100000): Q2:=implicitplot([v-vplus], u=umin..uplus*(1-r10^2), v=umin..vplus+1/10): Q3:=implicitplot([u-uplus], u=umin..uplus+1/10, v=umin..1/11+vplus*(1-r20^2)): display(Q1,Q2,Q3,labelfont=[Helvetica, roman,14],labels=[x,y],view=[umin..uplus+1/10,umin..vplus+1/10],thickness=3,scaling=constrained);

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dexpl:=3*Pi/4: eexpl:7*Pi/6: evalf(3*Pi/2-dexpl-eexpl); r10:=-sin(dexpl+eexpl)/sin(dexpl): r20:=-sin(dexpl+eexpl)/sin(eexpl): P1:=implicitplot(subs(d=dexpl, e=eexpl, den),r1=-0..r10+1/2, r2=-0..r20+1/2,thickness=3): P2:=implicitplot(subs(d=dexpl, e=eexpl, den),r1=-3..3, r2=-3..3,linestyle=dot): P3:=implicitplot(r2,r1=-sin(dexpl+eexpl)/sin(dexpl)..3,r2=-3..3,thickness=3): P4:=implicitplot(r1,r1=-3..3,r2=-sin(dexpl+eexpl)/sin(eexpl)..3,thickness=3): display(P1,P2,P3,P4,labels=[r[1],r[2]],view=[-r10-1/2..r10+1/2,-r20-1/2..r20+1/2],scaling=constrained);

JCIrM2A9JEcnISM1

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umin:=-1: uplus:=subs(d=dexpl, e=eexpl,uvpspe[1]): evalf(%); vplus:=subs(d=dexpl, e=eexpl,uvpspe[2]):evalf(%); Q0:=implicitplot(subs( r1=sqrt(1-u/uvpspe[1]), r2=sqrt(1-v/uvpspe[2]),d=dexpl, e=eexpl,den),u=-5..uplus,v=-5..uplus): Q1:=implicitplot(subs( r1=sqrt(1-u/uvpspe[1]), r2=sqrt(1-v/uvpspe[2]),d=dexpl, e=eexpl,den), u=uplus*(1-r10^2)..uplus+1/10, v=-1/10+vplus*(1-r20^2)..vplus,numpoints=100000): Q2:=implicitplot([v-vplus], u=umin..uplus*(1-r10^2), v=umin..vplus+1/10): Q3:=implicitplot([u-uplus], u=umin..uplus+1/10, v=umin..vplus*(1-r20^2)): display(Q1,Q2,Q3,labelfont=[Helvetica, roman,14],labels=[x,y],view=[umin..uplus+1/10,umin..vplus+1/10],thickness=3,scaling=constrained);

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An inequality that is needed in Lemma C.1 plot3d(Re(sqrt(1+I*u1)*sqrt(1+I*u2)),u1=-3..3,u2=-3..3)

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IiIh

plot3d(cos((t1+t2)/2)/sqrt(cos(t1))/sqrt(cos(t2)),t1=-Pi/2+1/10..Pi/2-1/10,t2=-Pi/2+1/10..Pi/2-1/10,view=0..3)

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factorization of den (with the notation of the paper, z1=sqrt(v1), z2=sqrt(v2)) denf:=simplify(eval(subs(r1=1/2/sin(d)*(exp(I*Pi/4)/z1+ z1*exp(-I*Pi/4)),r2=1/2/sin(e)*(exp(I*Pi/4)/z2+ z2*exp(-I*Pi/4)), d=-e+Pi+b/2,den)))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVkZW5mR0YoLCQqLCMiIiIiIiVGMSlJI3oxR0YoIiIjISIiKUkjejJHRihGNUY2LCgqKkY1RjEtSSRjb3NHRiU2IywkKiYjRjFGNUYxSSJiR0YoRjFGMUYxRjRGMUY4RjFGMSokRjNGMUYxKiRGN0YxRjFGMSwoKigsJF4jRjFGMUYxRjNGMUY3RjFGNkY6RjFGR0YxRjFGMTcjRi4=

simplify(factor([solve(denf,z1)]),symbolic)

NyYqJiwmKiYsJF4jIiIiRihGKC1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiNGKCIiI0YoSSJiR0YuRihGKEYoISIiLUkkc2luR0YrRi9GKEYoSSN6MkdGLkY1KiYsJkYlRjVGNkY1RihGOEY1KiYsJkYpRjUqJkYnRihGNkYoRihGKEY4RigsJComLCZGKUYoRj1GKEYoRjhGKEY1

Let us check the formula in the paper denp:=(z1+z2*exp(I*b/2))*(z1+z2*exp(-I*b/2))*(z1*z2+I*exp(I*b/2))*(z1*z2+I*exp(-I*b/2))/4/I/z1^2/z2^2

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVkZW5wR0YoLCQqMCwkKiYjIiIiIiIlRjNeI0YzRjNGM0YzLCZJI3oxR0YoRjMqJkkjejJHRihGMy1JJGV4cEdGJTYjKiYqJiNGMyIiI0YzRjVGM0YzSSJiR0YoRjNGM0YzRjMsJkY3RjMqJkY5RjMtRjs2IywkKiYsJEY+RjNGM0ZBRjMhIiJGM0YzRjMsJiomRjdGM0Y5RjNGMyomRjVGM0Y6RjNGM0YzLCZGS0YzKiZGNUYzRkRGM0YzRjMpRjdGQEZJKUY5RkBGSUZJNyNGLg==

simplify(convert(denf-denp,exp))

IiIh

The final integral factor(int(1/sqrt(v)*exp(-(v*zb+z/v)/2),v=0..infinity))

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factor(int(1/sqrt(v)/v*exp(-(v*zb+z/v)/2),v=0..infinity))

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Checking the integration constant (numerically), in the case studied by Harrison. We take mu_1=-1. p0:=exp(-x-sqrt(x^2+y^2))*cos(arctan(y/x)/2)/(x^2+y^2)^(1/4)

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simplify(subs(x=rho*cos(t),y=rho*sin(t),p0) ) assuming (rho>0,0<t, t<Pi/2)

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simplify(int(%*rho,rho=0..infinity)) assuming (0<t, t<Pi/2)

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int(%,t=0..Pi/2); evalf(%)

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evalf(int(int(p0,x=0..infinity),y=0..infinity))

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Checking the constant occurring in the beta-density q_0 q0:=kp*cos((t-a)/2)/sqrt(rho)*exp(-2*rho*sqrt(Delta/det)*cos((t-a)/2)^2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMEdGKCoqSSNrcEdGKCIiIi1JJGNvc0dGJTYjLCYqJiNGMCIiI0YwSSJhR0YoRjBGMComRjZGMEkidEdGKEYwISIiRjApSSRyaG9HRihGNkY7LUkkZXhwR0YlNiMsJCoqRjdGMEY9RjApKiZJJkRlbHRhR0YoRjBJJGRldEdGKEY7RjZGMClGMUY3RjBGO0YwNyNGLg==

I1:=simplify(int(q0*rho,rho=0..infinity)) assuming (t>0,a>0,a<Pi,Delta>0,det>0,a-t<Pi,t<Pi)

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int(%,a=0..beta) assuming (t>0,t<Pi,Delta>0,det>0,beta<Pi,beta>0)

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expand(tan(2*x))

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Testing special values of delta and epsilon modul:=sqrt(z1^2+z2^2+2*z1*z2*cos(b));

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cosinus:=sqrt((z1*cos(t)+z2*cos(b-t)+modul)/2/modul)

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p0tilde:=kappa*cosinus/sqrt(modul)*exp(-2*modul*cosinus^2)

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dexpl; eexpl

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Changement de variable modul_s:=simplify(expand(subs(z1=u-v*cos(b)/sin(b), z2=v/sin(b), modul)))

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cosinus_s:=simplify(expand(subs(z1=u-v*cos(b)/sin(b), z2=v/sin(b), cosinus)))

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p0tilde_uv:=kappa*cosinus_s/sqrt(modul_s)*exp(-2*modul_s*cosinus_s^2)

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p0tilde_r:=simplify(subs(u=rho*cos(2*a-t), v=rho*sin(2*a-t),p0tilde_uv)) assuming rho>0; combine(%,trig);

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simplify(subs(cos(2*a-2*t)=2*cos(a-t)^2-1,%)) assuming cos(a-t)>0

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int(%*rho,rho=0..infinity) assuming cos(a-t)>0

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int(%,a=-t/2..t/2)

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factor(simplify(%))

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int(cos(b)*exp(-2*rho*cos(b)^2)*sqrt(rho),rho=0..infinity) assuming cos(b)>0

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int(1/cos(b)^2,b=Pi/2-e..d-Pi/2) assuming(e>Pi/2, e<Pi, d>Pi/2, d<Pi)

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int(erf(sqrt(b*z))*exp(z*(y-a)),z=0..infinity)

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int(erf(sqrt(b*z))/z*exp(z*(y-a)),z=0..infinity)

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The real part of x(lambda(u+iv)), up to a multiplicative positive real Rxtilde:=proc(u,v,theta) cos(theta)+cos(u)*cosh(v): end: The imaginary part of x(lambda(u+iv)), up to a multiplicative positive real (not needed, in fact) Ixtilde:=proc(u,v) -sin(u)*sinh(v): end: The real part of y(lambda(u+iv)), up to a multiplicative positive real Rytilde:=proc(u,v,beta,theta) cos(theta-beta)+cos(u-beta)*cosh(v): end: The imaginary part of y(lambda(u+iv)), up to a multiplicative positive real (not needed, in fact) Iytilde:=proc(u,v,beta) -sin(u-beta)*sinh(v): end: with(plots): Give values to all four angles. This is an example where theta <pi/2 and beta-theta <pi/2 beta:=Pi/3: epsilon:=Pi/2: delta:=Pi/5: theta:=Pi/7: evalf(theta-Pi/2); evalf(beta-theta-Pi/2);

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Are our conditions fulfilled? test1:= evalb(evalf(theta-beta)<0) and (evalf(theta)>0) and (evalf(beta-Pi)<0);

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test2:= evalb(evalf(beta-epsilon-delta+Pi)>0) and evalb(evalf(theta-beta+epsilon)>0) and evalb(evalf(delta-theta)>0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZ0ZXN0MkdGKEkldHJ1ZUdGJjcjRi4=

vmax:=3: Vertical lines at zero, 2pi, etc (not really needed either) Pxline:=implicitplot([u=0,u=Pi,u=2*Pi],u=0..2*Pi,v=-vmax..vmax,color=green): Pyline:=implicitplot([u=beta,u=beta+Pi],u=0..2*Pi,v=-vmax..vmax,color=green,linestyle=dash): The curve Ix, with its asymptotes (first the asymptotes) Ax:=implicitplot([u=Pi/2,u=3*Pi/2],u=0..2*Pi,v=-vmax..vmax,view=[0..2*Pi,-vmax..vmax],color=red,linestyle=dash);

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

Ix:=implicitplot([Rxtilde(u,v,theta)=0],u=0..2*Pi,v=-vmax..vmax,view=[0..2*Pi,-vmax..vmax],color=red);

NiotJSdDVVJWRVNHNiU3Z243JCQiMjgnPWd1KFw0bSIhIzskISNJISIiNyQkIjJwRD5BOT1hbCIhIzskITInZWM/Iil5ITMhSCEjOzckJCIyT0kiUkFEOyZvIiEjOyQhJHcjISIjNyQkIjFqKkcmKjMzNW8iISM6JCExKFEhKTNbV19vIyEjOjckJCIyL3RqXTJoZHIiISM7JCEkXyMhIiM3JCQiMk8mMzklKmVMODwhIzskITBUTi5FOmhaIyEjOTckJCIyMzlCZC8qR2E8ISM7JCEkRyMhIiM3JCQiMmBtPW4pSCxhPCEjOyQhMngpZl1BIWZcRiMhIzs3JCQiKycpPUhmPCEiKiQhMTxhZjJ3UVxBISM6NyQkIjJ4RSM9QHo5IXoiISM7JCExQFg8VmBZcD8hIzo3JCQiMmA4OGY4X1khPSEjOyQhJC8jISIjNyQkIjJGOk0hUT9GTD0hIzskITJwRi05JWVrcT0hIzs3JCQiMm4heSQqbygzdSc9ISM7JCEjPSEiIjckJCIyVmRzI0dRIXApPSEjOyQhMkhuKydHLyc9byIhIzs3JCQiMmJURCo9WTBXPiEjOyQhJGMiISIjNyQkIjI5bFcmKlw8RyY+ISM7JCEydkl1I1tMIVtdIiEjOzckJCIrKUg+MSwjISIqJCEyJCoqKVwnM21adjghIzs3JCQiMVV4IWYkKWZvLSMhIzokITJiJVxGciYzYkwiISM7NyQkIjFsTVhzJVsmUT8hIzokISRLIiEiIzckJCIyUFFTTShHPyk0IyEjOyQhMlt5X046T087IiEjOzckJCIxWE5oJz1ObDojISM6JCEkMyIhIiM3JCQiMTorb1ZyZiM9IyEjOiQhMmkkeSlcdUVVKyIhIzs3JCQiKzZuJT5FIyEiKiQhMV83Jyk0TyRHdikhIzs3JCQiMWAqb25iJiplRiMhIzokITBfRGFtLEtgKSEjOjckJCIyOSVSWyRcRigpRyMhIzskISMlKSEiIzckJCIyMCM+LyFIXFhPIyEjOyQhMXYlSCllQ3l6cCEjOzckJCIxXyssRSMqXFRDISM6JCEiJyEiIjckJCIxOjlrZCgzNlkjISM6JCExdSpIVFhkPV0mISM7NyQkIitCVEY4RCEiKiQhMW10NjMtVWpYISM7NyQkIjJkUiw9YU9TYiMhIzskITFNQTdGQ0QqKVIhIzs3JCQiMnZnJmVVPFAqZSMhIzskISNPISIjNyQkIjI4I3BPODtQT0UhIzskITJvTSp5WFdcdkIhIzw3JCQiMic0NWpgaEEicCMhIzskISM3ISIjNyQkIjInNDVqYGhBInAjISM7JCEwJm9KL3M7JCpcISM7NyQkIjInNDVqYGhBInAjISM7JCIjNyEiIzckJCIxLXhzNHh2V0UhIzokIjJCTnZGQkNXTSMhIzw3JCQiMnZnJmVVPFAqZSMhIzskIiNPISIjNyQkIitCVEY4RCEiKiQiMW10NjMtVWpYISM7NyQkIjFfKyxFIypcVEMhIzokIiInISIiNyQkIjJzKlt2SlYjeUwjISM7JCIxMS9RJTRBYW4oISM7NyQkIjI5JVJbJFxGKClHIyEjOyQiIyUpISIjNyQkIis2biU+RSMhIiokIjFfNycpNE8kR3YpISM7NyQkIjFYTmgnPU5sOiMhIzokIiQzIiEiIzckJCIxVEgqPjxkPzIjISM6JCIyZD5LQDpKOEUiISM7NyQkIjFsTVhzJVsmUT8hIzokIiRLIiEiIzckJCIrKUg+MSwjISIqJCIxKiopXCczbVp2OCEjOjckJCIyYlREKj1ZMFc+ISM7JCIkYyIhIiM3JCQiMmhZK1tqOXgjPiEjOyQiMCI+Ij03byJSOyEjOTckJCIybiF5JCpvKDN1Jz0hIzskIiM9ISIiNyQkIjJpKVwoUkM7NCQ9ISM7JCIyMyRvQHdQZ3I+ISM7NyQkIjJgODhmOF9ZIT0hIzskIiQvIyEiIzckJCIrJyk9SGY8ISIqJCIxPGFmMndRXEEhIzo3JCQiMjM5QmQvKkdhPCEjOyQiJEcjISIjNyQkIjJDIzQ/LkNnYTwhIzskIjJEQThpN3lXRyMhIzs3JCQiMi90al0yaGRyIiEjOyQiJF8jISIjNyQkIjJNa0xRIWZdPTwhIzskIjJiSW9TdVoqZUQhIzs3JCQiMk9JIlJBRDsmbyIhIzskIiR3IyEiIzckJCIyJClIW1cxYykqbyIhIzskIjJsT1RhSzFqI0chIzs3JCQiMjgnPWd1KFw0bSIhIzskIiNJISIiN2duNyQkIjFSYWRLYkJBWSEjOiQiI0khIiI3JCQiMk4nNC1scndGWSEjOyQiMiVRNFAiKXkhMyFIISM7NyQkIjF4ZnkleUEhKWYlISM6JCIkdyMhIiM3JCQiMCwpcDxzPC1ZISM5JCIxJzNoNFtXX28jISM6NyQkIjFETjZLVVVuWCEjOiQiJF8jISIjNyQkIjF6JXBJVFwpcFghIzokIjA8NC5FOmhaIyEjOTckJCIxJTNhOUUnKilHWCEjOiQiJEcjISIjNyQkIjF5NFo/QjxIWCEjOiQiMiVcXE1BIWZcRiMhIzs3JCQiK0BNKlFfJSEiKiQiMUYpUmxnKFFcQSEjOjckJCIyMHlZZlFQSVwlISM7JCIyKCpHXklNbCVwPyEjOzckJCIxIj5xNzxMJnlXISM6JCIkLyMhIiM3JCQiMWkhKjRwSyIqXFchIzokIjJtMXY3JWVrcT0hIzs3JCQiMnYmPUNRbHg6VyEjOyQiIz0hIiI3JCQiMSZ6bCl5OUcnUiUhIzokIjIsYW8lRy8nPW8iISM7NyQkIjF3KFwjKW9JIlJWISM6JCIkYyIhIiM3JCQiMiZwKSpmMnlPSVYhIzskIjIuSU8iW0whW10iISM7NyQkIis0Z2NzVSEiKiQiMjFXWCMzbVp2OCEjOzckJCIyWClvQXJhS2NVISM7JCIyRjQzN2QzYkwiISM7NyQkIjFpZHVNb2pXVSEjOiQiJEsiISIjNyQkIjAjPXpMQylcPSUhIzkkIjJVI1FtYGhqajYhIzs3JCQiMVpAaD8sbEVUISM6JCIkMyIhIiM3JCQiMXNmbGoiKWUrVCEjOiQiMjBWP2B1RVUrIiEjOzckJCIrKGZRNy0lISIqJCIxeSU0Q2lMR3YpISM7NyQkIjEyYXNdKCpHMlMhIzokIjFMU3NwOz9MJikhIzs3JCQiMWUoKWY4eVglKlIhIzokIiMlKSEiIzckJCIybWJ5cyxPJz1SISM7JCIxPitVaEN5enAhIzs3JCQiMjlkQDYzJ29UUSEjOyQiIichIiI3JCQiMWAnSCZcbDJBUSEjOiQiMVsleVdYZD1dJiEjOzckJCIrJT02KnBQISIqJCIxPC5AMC1ValghIzs3JCQiMSdcNWB3WyJIUCEjOiQiMVVQMEVDRCopUiEjOzckJCIxQyM0WWM4UXAkISM6JCIjTyEiIzckJCIyLiQqZlBwOG9rJCEjOyQiMmoreVhXJVx2QiEjPDckJCIyKFwwYGAiZj5mJCEjOyQiIzchIiM3JCQiMihcMGBgImY+ZiQhIzskIjJsNCopKSk9bkoqXCEjPTckJCIyKFwwYGAiZj5mJCEjOyQhIzchIiM3JCQiMnYnM2QoZkYlUU8hIzskITJBW0VjQkNXTSMhIzw3JCQiMUMjNFljOFFwJCEjOiQhI08hIiM3JCQiKyU9NipwUCEiKiQhMTwuQDAtVWpYISM7NyQkIjI5ZEA2MydvVFEhIzskISInISIiNyQkIjIkM2dBdTRPWFIhIzskITBaIyl6MkFhbighIzo3JCQiMWUoKWY4eVglKlIhIzokISMlKSEiIzckJCIrKGZRNy0lISIqJCExeCU0Q2lMR3YpISM7NyQkIjFaQGg/LGxFVCEjOiQhJDMiISIjNyQkIjFlO0NOIkc2QCUhIzokITJfeUFAOko4RSIhIzs3JCQiMWlkdU1valdVISM6JCEkSyIhIiM3JCQiKzRnY3NVISIqJCEyM1dYIzNtWnY4ISM7NyQkIjF3KFwjKW9JIlJWISM6JCEkYyIhIiM3JCQiMXghUUNucWFOJSEjOiQhMS0oUj83byJSOyEjOjckJCIydiY9Q1FseDpXISM7JCEjPSEiIjckJCIyMEMnSGohcEFYJSEjOyQhMUxlWndQZ3I+ISM6NyQkIjEiPnE3PEwmeVchIzokISQvIyEiIzckJCIrQE0qUV8lISIqJCEyayMpUmxnKFFcQSEjOzckJCIxJTNhOUUnKilHWCEjOiQhJEcjISIjNyQkIjBhaVIhSGVHWCEjOSQhMSpcXmo3eVdHIyEjOjckJCIxRE42S1VVblghIzokISRfIyEiIzckJCIwODRMU3pZYyUhIzkkITJiblxTdVoqZUQhIzs3JCQiMXhmeSV5QSEpZiUhIzokISR3IyEiIzckJCIydiJ6blUjSExmJSEjOyQhMihvISlIRGpJRUchIzs3JCQiMVJhZEtiQkFZISM6JCEjSSEiIi0lJkNPTE9SRzYmJSRSR0JHJCIjNSEiIiQiIiEhIiIkIiIhISIiLSUlVklFV0c2JDskIiIhISIiJCIrM2A9JEcnISIqOyQhI0khIiIkIiNJISIiLSYlJl9BWElTRzYjIiIiNiYtJSZDT0xPUkc2JiUkUkdCRyQiIiEhIiIkIiIhISIiJCIiISEiIi0lKkxJTkVTVFlMRUc2IyIiIS0lKlRISUNLTkVTU0c2IyIiIS0lLVRSQU5TUEFSRU5DWUc2IyQiIiEhIiItJiUmX0FYSVNHNiMiIiM2Ji0lJkNPTE9SRzYmJSRSR0JHJCIiISEiIiQiIiEhIiIkIiIhISIiLSUqTElORVNUWUxFRzYjIiIhLSUqVEhJQ0tORVNTRzYjIiIhLSUtVFJBTlNQQVJFTkNZRzYjJCIiISEiIi0lK0FYRVNMQUJFTFNHNiQtSSNtaUc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IjY1USJ1NiIvJSdmYW1pbHlHUSE2Ii8lJXNpemVHUSMxMDYiLyUlYm9sZEdRJmZhbHNlNiIvJSdpdGFsaWNHUSV0cnVlNiIvJSp1bmRlcmxpbmVHUSZmYWxzZTYiLyUqc3Vic2NyaXB0R1EmZmFsc2U2Ii8lLHN1cGVyc2NyaXB0R1EmZmFsc2U2Ii8lK2ZvcmVncm91bmRHUShbMCwwLDBdNiIvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XTYiLyUnb3BhcXVlR1EmZmFsc2U2Ii8lK2V4ZWN1dGFibGVHUSZmYWxzZTYiLyUpcmVhZG9ubHlHUSZmYWxzZTYiLyUpY29tcG9zZWRHUSZmYWxzZTYiLyUqY29udmVydGVkR1EmZmFsc2U2Ii8lK2ltc2VsZWN0ZWRHUSZmYWxzZTYiLyUscGxhY2Vob2xkZXJHUSZmYWxzZTYiLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lLG1hdGh2YXJpYW50R1EnaXRhbGljNiItSSNtaUc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IjY1USJ2NiIvJSdmYW1pbHlHUSE2Ii8lJXNpemVHUSMxMDYiLyUlYm9sZEdRJmZhbHNlNiIvJSdpdGFsaWNHUSV0cnVlNiIvJSp1bmRlcmxpbmVHUSZmYWxzZTYiLyUqc3Vic2NyaXB0R1EmZmFsc2U2Ii8lLHN1cGVyc2NyaXB0R1EmZmFsc2U2Ii8lK2ZvcmVncm91bmRHUShbMCwwLDBdNiIvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XTYiLyUnb3BhcXVlR1EmZmFsc2U2Ii8lK2V4ZWN1dGFibGVHUSZmYWxzZTYiLyUpcmVhZG9ubHlHUSZmYWxzZTYiLyUpY29tcG9zZWRHUSZmYWxzZTYiLyUqY29udmVydGVkR1EmZmFsc2U2Ii8lK2ltc2VsZWN0ZWRHUSZmYWxzZTYiLyUscGxhY2Vob2xkZXJHUSZmYWxzZTYiLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lLG1hdGh2YXJpYW50R1EnaXRhbGljNiItJSlfVklTSUJMRUc2IyIiIi0lJVJPT1RHNictJSlCT1VORFNfWEc2IyQiJF0mISIiLSUpQk9VTkRTX1lHNiMkIiRTIiEiIi0lLUJPVU5EU19XSURUSEc2IyQiJStNISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJTVQISIiLSUpQ0hJTERSRU5HNiItJStBTk5PVEFUSU9ORzYnLSUpQk9VTkRTX1hHNiMkIiIhISIiLSUpQk9VTkRTX1lHNiMkIiIhISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlK1MhIiItJS5CT1VORFNfSEVJR0hURzYjJCIlK1MhIiItJSlDSElMRFJFTkc2Ig==Ig==

display(Ax,Ix);

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If we take Iy in [pi, 3 pi], there are more than 2 components in Iy Iy:=implicitplot([Rytilde(u,v,beta,theta)=0,u=beta+Pi/2,u=beta+Pi/2-2*Pi,u=beta+Pi/2+2*Pi,u=beta-Pi/2,u=beta-Pi/2+2*Pi],u=Pi..3*Pi,v=-vmax..vmax,color=blue,linestyle=dash);

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A better choice might be the interval [beta, beta + pi] Ay:=implicitplot([u=beta+Pi/2,u=beta+3*Pi/2],u=beta..beta+2*Pi,v=-vmax..vmax,color=blue,view=[0..beta+2*Pi, -vmax..vmax],linestyle=dash);

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Ig==

Iy:=implicitplot([Rytilde(u,v,beta,theta)=0],u=beta..beta+2*Pi,v=-vmax..vmax,color=blue,view=[0..beta+2*Pi, -vmax..vmax]#,linestyle=dash );

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display(Ay,Iy);

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All four branches together, and their asymptotes display(Ax,Ay,Ix,Iy,view=[min(Pi/2,Pi-theta)-1..max(Pi+2*beta-theta,beta+3*Pi/2)+1,-2..2]);

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A picture also showing the translates of (Ix+) by 2 beta, and the translate of (Iy+) by -2beta Tx:=implicitplot([Rxtilde(u-2*beta,v,theta)=0],u=2*beta..Pi+2*beta,v=-vmax..vmax,view=[0..2*Pi+2*beta,-vmax..vmax],color=red,linestyle=dash);

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

Ty:=implicitplot([Rytilde(u+2*beta,v,beta,theta)=0],u=Pi-beta..2*Pi-beta,v=-vmax..vmax,view=[0..2*Pi+2*beta,-vmax..vmax],color=blue,linestyle=dash);

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display(Tx,Ty,Ix,Iy,view=[min(Pi/2,Pi-theta)-1..max(Pi+2*beta-theta,beta+3*Pi/2)+1,-2..2]);

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Another example, xhete theta >pi/2 and beta-theta<pi/2 beta:=3*Pi/4: epsilon:=Pi/6: delta:=3*Pi/4: theta:=2*Pi/3: evalf(theta-Pi/2); evalf(beta-theta-Pi/2);

JCIrZXgpZkImISM1

JCErUnAqKjM4ISIq

Are our conditions fulfilled? test1:= evalb(evalf(theta-beta)<0) and (evalf(theta)>0) and (evalf(beta-Pi)<0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZ0ZXN0MUdGKEkldHJ1ZUdGJjcjRi4=

test2:= evalb(evalf(beta-epsilon-delta+Pi)>0) and evalb(evalf(theta-beta+epsilon)>0) and evalb(evalf(delta-theta)>0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZ0ZXN0MkdGKEkldHJ1ZUdGJjcjRi4=

vmax:=3: The curve Ix, with its asymptotes (first the asymptotes) Ax:=implicitplot([u=Pi/2,u=3*Pi/2],u=0..2*Pi,v=-vmax..vmax,view=[0..2*Pi,-vmax..vmax],color=red,linestyle=dash): Ix:=implicitplot([Rxtilde(u,v,theta)=0],u=0..2*Pi,v=-vmax..vmax,view=[0..2*Pi,-vmax..vmax],color=red): display(Ax,Ix);

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Ay:=implicitplot([u=beta+Pi/2,u=beta+3*Pi/2],u=beta..beta+2*Pi,v=-vmax..vmax,color=blue,view=[0..beta+2*Pi, -vmax..vmax],linestyle=dash): Iy:=implicitplot([Rytilde(u,v,beta,theta)=0],u=beta..beta+2*Pi,v=-vmax..vmax,color=blue,view=[0..beta+2*Pi, -vmax..vmax]#,linestyle=dash ): display(Ay,Iy);

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All four branches together, and their asymptotes display(Ax,Ay,Ix,Iy,view=[min(Pi/2,Pi-theta)-1..max(Pi+2*beta-theta,beta+3*Pi/2)+1,-2..2]);

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A picture also showing the translates of (Ix+) by 2 beta, and the translate of (Iy+) by -2beta Tx:=implicitplot([Rxtilde(u-2*beta,v,theta)=0],u=2*beta..Pi+2*beta,v=-vmax..vmax,view=[0..2*Pi+2*beta,-vmax..vmax],color=red,linestyle=dash): Ty:=implicitplot([Rytilde(u+2*beta,v,beta,theta)=0],u=Pi-beta..2*Pi-beta,v=-vmax..vmax,view=[0..2*Pi+2*beta,-vmax..vmax],color=blue,linestyle=dash): display(Tx,Ty,Ix,Iy,view=[min(Pi/2,Pi-theta)-1..max(Pi+2*beta-theta,beta+3*Pi/2)+1,-2..2]);

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Another example, where now theta<pi/2 but beta-thet>pi/2 beta:=7*Pi/8: epsilon:=2*Pi/3: delta:=Pi/3: theta:=Pi/4: evalf(theta-Pi/2); evalf(beta-theta-Pi/2);

JCErTjspUiZ5ISM1

JCIrPTMqcCNSISM1

Are our conditions fulfilled? test1:= evalb(evalf(theta-beta)<0) and (evalf(theta)>0) and (evalf(beta-Pi)<0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZ0ZXN0MUdGKEkldHJ1ZUdGJjcjRi4=

test2:= evalb(evalf(beta-epsilon-delta+Pi)>0) and evalb(evalf(theta-beta+epsilon)>0) and evalb(evalf(delta-theta)>0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZ0ZXN0MkdGKEkldHJ1ZUdGJjcjRi4=

The curve Ix, with its asymptotes (first the asymptotes) Ax:=implicitplot([u=Pi/2,u=3*Pi/2],u=0..2*Pi,v=-vmax..vmax,view=[0..2*Pi,-vmax..vmax],color=red,linestyle=dash): Ix:=implicitplot([Rxtilde(u,v,theta)=0],u=0..2*Pi,v=-vmax..vmax,view=[0..2*Pi,-vmax..vmax],color=red): display(Ax,Ix);

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Ay:=implicitplot([u=beta+Pi/2,u=beta+3*Pi/2],u=beta..beta+2*Pi,v=-vmax..vmax,color=blue,view=[0..beta+2*Pi, -vmax..vmax],linestyle=dash): Iy:=implicitplot([Rytilde(u,v,beta,theta)=0],u=beta..beta+2*Pi,v=-vmax..vmax,color=blue,view=[0..beta+2*Pi, -vmax..vmax]#,linestyle=dash ): display(Ay,Iy);

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All four branches together, and their asymptotes display(Ax,Ay,Ix,Iy,view=[min(Pi/2,Pi-theta)-1..max(Pi+2*beta-theta,beta+3*Pi/2)+1,-2..2]);

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A picture also showing the translates of (Ix+) by 2 beta, and the translate of (Iy+) by -2beta Tx:=implicitplot([Rxtilde(u-2*beta,v,theta)=0],u=2*beta..Pi+2*beta,v=-vmax..vmax,view=[0..2*Pi+2*beta,-vmax..vmax],color=red,linestyle=dash): Ty:=implicitplot([Rytilde(u+2*beta,v,beta,theta)=0],u=Pi-beta..2*Pi-beta,v=-vmax..vmax,view=[0..2*Pi+2*beta,-vmax..vmax],color=blue,linestyle=dash): display(Tx,Ty,Ix,Iy,view=[min(Pi/2,Pi-theta)-1..max(Pi+2*beta-theta,beta+3*Pi/2)+1,-2..2]);

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Un un peu de trigo expand(cos(u)*cos(t-b)-cos(u-b)*cos(t));

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factor(%);

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combine(%,trig);

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combine(cos(u)*sin(t)-cos(t)*sin(u),trig);

LUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJkkidUdGJyEiIkkidEdGJyIiIg==

with(SolveTools[Inequality]);

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The basic conditions that hold throughout the paper (divided by pi) conds:={0<t,0<d,0<e,0<b,b<1, d<1,e<1, t<b,d-1<b-e, b-e<t, t<d};

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basic:=LinearMultivariateSystem(conds,[t,b,d,e])

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nops(basic);

IiIk

op(1,basic);

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op(2,basic);

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op(3,basic);

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La premi\303\250re r\303\251ponse est suspecte. On essaye avec solve. Vu de loin c'est mieux basicsolve:=solve(conds,[t,b,d,e]);

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op(1,basicsolve);op(2,basicsolve);op(3,basicsolve);

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NykySSJ0RzYiIiIiMiIiIUYkMkYkSSJiR0YlMkYqRiYvSSJkR0YlRioySSJlR0YlRiYyLCZGKkYmRiQhIiJGLw==

NyoySSJ0RzYiIiIiMiIiIUYkMkYkSSJiR0YlMkYqRiYyRipJImRHRiUyRi1GJjJJImVHRiUsKEYtISIiRipGJkYmRiYyLCZGKkYmRiRGMkYw

solve(conds union {2*d-t=b},[t,b,d,e]);

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solve(conds union {2*d-t=1},[t,b,d,e]);

Warning, computation interrupted

solve(conds union {2*d-t=b+2},[t,b,d,e]);

NyI=

solve(conds union {2*d-t=+3*b+1},[t,b,d,e]);

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solve(conds union {2*b-2*d-t=1},[t,b,d,e]);

NyM3KTJJInRHNiIjIiIiIiIkMiIiIUkidEc2IjJJImJHNiIiIiIyLCYjIiIiIiIjIiIiKiYjIiIkIiIjIiIiSSJ0RzYiIiIiIiIiSSJiRzYiL0kiZEc2IiwoSSJiRzYiIiIiKiYjIiIiIiIjIiIiSSJ0RzYiIiIiISIiIyIiIiIiIyEiIjJJImVHNiIiIiIyLCZJImJHNiIiIiJJInRHNiIhIiJJImVHNiI=

solve(conds union {2*d-t=2*b+1},[t,b,d,e]);

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solve(conds union {1+2*b-2*d-t=1*b},[t,b,d,e]);

Warning, computation interrupted

solve(conds union {2*b-2*d-t=-1},[t,b,d,e]);

NyM3KTJJInRHNiIiIiIyIiIhSSJ0RzYiMkkidEc2IkkiYkc2IjJJImJHNiIsJiMiIiIiIiMiIiIqJiMiIiIiIiMiIiJJInRHNiIiIiIiIiIvSSJkRzYiLCgqJiMiIiIiIiMiIiJJInRHNiIiIiIhIiJJImJHNiIiIiIjIiIiIiIjIiIiMkkiZUc2IiwmIyIiIiIiIyIiIiomIyIiIiIiIyIiIkkidEc2IiIiIiIiIjIsJkkiYkc2IiIiIkkidEc2IiEiIkkiZUc2Ig==

solve(conds union {2*b-2*d-t=-3},[t,b,d,e]);

NyI=

solve(conds union {2*b-2*e-t=-1+2*b,2*d-t=b+1},[t,b,d,e]);

NyU3KDJJInRHNiIjIiIiIiIkMiIiIUYlMkYlSSJiR0YmMkYtLCYjRigiIiNGKComRjBGKEYlRihGKC9JImRHRiYsKComRjBGKEYtRihGKEYyRihGMEYoL0kiZUdGJiwmRjBGKEYyISIiNycvRiVGJzJGLSNGMUYpMkYnRi0vRjQsJkY2RihGPkYoL0Y4Ric3KDJGJUYwMkYnRiVGLDJGLSwmRiVGOkYoRihGM0Y3

solve(conds union {2*b-2*e-t=-1+1*b,2*d-t=1},[t,b,d,e]);

NyM3KDJJInRHNiIiIiIyIiIhSSJ0RzYiMkkidEc2IkkiYkc2IjJJImJHNiIiIiIvSSJkRzYiLCYjIiIiIiIjIiIiKiYjIiIiIiIjIiIiSSJ0RzYiIiIiIiIiL0kiZUc2IiwoKiYjIiIiIiIjIiIiSSJiRzYiIiIiIiIiKiYjIiIiIiIjIiIiSSJ0RzYiIiIiISIiIyIiIiIiIyIiIg==

solve(conds union {2*b-2*e-t=-1+2*b,2*d-t=b+1},[t,b,d,e]);

NyU3KDJJInRHNiIjIiIiIiIkMiIiIUYlMkYlSSJiR0YmMkYtLCYjRigiIiNGKComRjBGKEYlRihGKC9JImRHRiYsKComRjBGKEYtRihGKEYyRihGMEYoL0kiZUdGJiwmRjBGKEYyISIiNycvRiVGJzJGLSNGMUYpMkYnRi0vRjQsJkY2RihGPkYoL0Y4Ric3KDJGJUYwMkYnRiVGLDJGLSwmRiVGOkYoRihGM0Y3

e1:=0: K:=0: solve(conds union {1+2*b-2*e-t=e1*b+K},[t,b,d,e]);

NyM3KTJJInRHNiIiIiIyIiIhSSJ0RzYiMkkidEc2IkkiYkc2IjJJInRHNiJJImRHNiIySSJiRzYiLCYjIiIiIiIjIiIiKiYjIiIiIiIjIiIiSSJ0RzYiIiIiIiIiMkkiZEc2IiwmIyIiIiIiIyIiIiomIyIiIiIiIyIiIkkidEc2IiIiIiIiIi9JImVHNiIsKCMiIiIiIiMiIiJJImJHNiIiIiIqJiMiIiIiIiMiIiJJInRHNiIiIiIhIiI=

e2:=0: K:=-0: solve(conds union {-1+2*d-t=e2*b+K},[t,b,d,e]);

NyU3KTJJInRHNiIiIiIyIiIhRiUyRiVJImJHRiYyRissJiNGJyIiI0YnKiZGLkYnRiVGJ0YnL0kiZEdGJkYtMkkiZUdGJiwoRi5GJ0YrRidGMCEiIjIsJkYrRidGJUY2RjQ3KEYkRigvRitGLUYxMkY0RicyLCZGLkYnRjBGNkY0NylGJEYoMkYrRicyRi1GK0YxRjtGNw==

m1:=-7: solve(conds union {1+2*b-2*e-t=m1*b},[t,b,d,e]);

NyM3KTJJInRHNiIjIiIiIiIpMiIiIUkidEc2IjJJInRHNiJJImJHNiIySSJ0RzYiSSJkRzYiMkkiYkc2IiwmKiYjIiIiIiIqIiIiSSJ0RzYiIiIiIiIiIyIiIiIiKiIiIjJJImRHNiIsKComIyIiKCIiIyIiIkkiYkc2IiIiIiEiIiomIyIiIiIiIyIiIkkidEc2IiIiIiIiIiMiIiIiIiMiIiIvSSJlRzYiLCgqJiMiIioiIiMiIiJJImJHNiIiIiIiIiIqJiMiIiIiIiMiIiJJInRHNiIiIiIhIiIjIiIiIiIjIiIi

r1:=0: solve(conds union {1+2*b-2*e-t=(2*r1+1)*b-1},[t,b,d,e]);

NyI=

r2:=1: solve(conds union {-1+2*d-t=2*r2*b-1},[t,b,d,e]);

NyM3KTJJInRHNiIjIiIjIiIkMiIiIUkidEc2IjJJInRHNiJJImJHNiIySSJiRzYiLCYqJiMiIiIiIiMiIiJJInRHNiIiIiIhIiIiIiIiIiIvSSJkRzYiLCZJImJHNiIiIiIqJiMiIiIiIiMiIiJJInRHNiIiIiIiIiIySSJlRzYiLCYqJiMiIiIiIiMiIiJJInRHNiIiIiIhIiIiIiIiIiIyLCZJImJHNiIiIiJJInRHNiIhIiJJImVHNiI=

e2:=1: K:=-2: solve(conds union {-1+2*d-t=e2*b+K},[t,b,d,e]);

NyI=

m2:=1:eps2:=1: solve(conds union {-1+2*d-t=m2*b-eps2},[t,b,d,e]);

NyM3KTJJInRHNiIiIiIyIiIhSSJ0RzYiMkkidEc2IkkiYkc2IjJJImJHNiIiIiIvSSJkRzYiLCYqJiMiIiIiIiMiIiJJInRHNiIiIiIiIiIqJiMiIiIiIiMiIiJJImJHNiIiIiIiIiIySSJlRzYiIiIiMiwmSSJiRzYiIiIiSSJ0RzYiISIiSSJlRzYi

r1:=0: r2:=-1: e1:=0: e2:=1: solve(conds union {1+2*b-2*e-t=(2*r1+e1)*b,-1+2*d-t=(2*r2+e2)*b},[t,b,d,e]);

NyU3KDJJInRHNiIjIiIiIiIkMiIiIUYlMkYlSSJiR0YmMkYtLCYqJiNGKCIiI0YoRiVGKEYoRjFGKC9JImRHRiYsKEYwRihGMUYoKiZGMUYoRi1GKCEiIi9JImVHRiYsKEYxRihGLUYoRjBGNzcnL0YlRicyRi0jRjJGKTJGJ0YtL0Y0LCZGPkYoRjZGNy9GOSwmRidGKEYtRig3KDJGJUYxMkYnRiVGLDJGLSwmRiVGN0YoRihGM0Y4

1,1 eps1:=1: eps2:=1: e1:=1: e2:=0: solve(conds union {1+2*b-2*e-t=(e1)*b-eps1,-1+2*d-t=(e2)*b-eps2},[t,b,d,e]);

NyI=

eps1:=1: eps2:=1: e1:=0: e2:=1: solve(conds union {1+2*b-2*e-t=(e1)*b-eps1,-1+2*d-t=(e2)*b-eps2},[t,b,d,e]);

NyI=

0,1 eps1:=0: eps2:=1: e1:=0: e2:=1: solve(conds union {1+2*b-2*e-t=(e1)*b-eps1,-1+2*d-t=(e2)*b-eps2},[t,b,d,e]);

NyM3KDJJInRHNiIiIiIyIiIhSSJ0RzYiMkkidEc2IkkiYkc2IjJJImJHNiIsJiomIyIiIiIiIyIiIkkidEc2IiIiIiIiIiMiIiIiIiMiIiIvSSJkRzYiLCYqJiMiIiIiIiMiIiJJInRHNiIiIiIiIiIqJiMiIiIiIiMiIiJJImJHNiIiIiIiIiIvSSJlRzYiLCgjIiIiIiIjIiIiSSJiRzYiIiIiKiYjIiIiIiIjIiIiSSJ0RzYiIiIiISIi

eps1:=0: eps2:=1: e1:=1: e2:=0: solve(conds union {1+2*b-2*e-t=(e1)*b-eps1,-1+2*d-t=(e2)*b-eps2},[t,b,d,e]);

NyI=

eps1:=0: eps2:=1: e1:=0: e2:=0: solve(conds union {1+2*b-2*e-t=(e1)*b-eps1,-1+2*d-t=(e2)*b-eps2},[t,b,d,e]);

NyI=

eps1:=0: eps2:=1: e1:=1: e2:=1: solve(conds union {1+2*b-2*e-t=(e1)*b-eps1,-1+2*d-t=(e2)*b-eps2},[t,b,d,e]);

NyM3KDJJInRHNiIiIiIyIiIhSSJ0RzYiMkkidEc2IkkiYkc2IjJJImJHNiIiIiIvSSJkRzYiLCYqJiMiIiIiIiMiIiJJInRHNiIiIiIiIiIqJiMiIiIiIiMiIiJJImJHNiIiIiIiIiIvSSJlRzYiLCgjIiIiIiIjIiIiKiYjIiIiIiIjIiIiSSJiRzYiIiIiIiIiKiYjIiIiIiIjIiIiSSJ0RzYiIiIiISIi

1,0 eps1:=1: eps2:=0: e1:=0: e2:=1: solve(conds union {1+2*b-2*e-t=(e1)*b-eps1,-1+2*d-t=(e2)*b-eps2},[t,b,d,e]);

NyI=

eps1:=1: eps2:=0: e1:=0: e2:=0: solve(conds union {1+2*b-2*e-t=(e1)*b-eps1,-1+2*d-t=(e2)*b-eps2},[t,b,d,e]);

NyI=

eps1:=1: eps2:=0: e1:=1: e2:=1: solve(conds union {1+2*b-2*e-t=(e1)*b-eps1,-1+2*d-t=(e2)*b-eps2},[t,b,d,e]);

NyI=

eps1:=1: eps2:=0: e1:=1: e2:=0: solve(conds union {1+2*b-2*e-t=(e1)*b-eps1,-1+2*d-t=(e2)*b-eps2},[t,b,d,e]);

NyI=

b:=1/2: d:=5/12: e:=5/6: t:=1/3: map(evalb, conds union {1+2*b-2*e-t=(2*r1+e1)*b,-1+2*d-t=(2*r2+e2)*b});

PCNJJXRydWVHJSpwcm90ZWN0ZWRH

e1:=0: e2:=1: for r1 from -4 to 5 do for r2 from -5 to r1-1 do if solve(conds union {1+2*b-2*e-t=(2*r1+e1)*b,-1+2*d-t=(2*r2+e2)*b},[t,b,d,e])=[] then print (r1,r2): else print(OK): fi: od: od:

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GAMMA(1/2);

KiQpSSNQaUclKnByb3RlY3RlZEcjIiIiIiIjRic=

Pole r\303\251el sur R poleR:=LinearMultivariateSystem(conds union{2*b=2*e+t},[t,b,d,e]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZwb2xlUkdGKDwkNyY8JDIiIiFJInRHRigyRjMiIiI8JDJJImJHRihGNTJGM0Y4PCQxRjhJImRHRigyRjxGNTwjL0kiZUdGKCwmKiYjRjUiIiNGNUYzRjUhIiJGOEY1NyZGMEY2PCQyRjxGODJGM0Y8Rj43I0Yu

nops(poleR);

IiIj

poleR[1];poleR[2];

NyY8JDIiIiFJInRHNiIyRiYiIiI8JDJJImJHRidGKTJGJkYsPCQxRixJImRHRicyRjBGKTwjL0kiZUdGJywmKiYjRikiIiNGKUYmRikhIiJGLEYp

NyY8JDIiIiFJInRHNiIyRiYiIiI8JDJJImJHRidGKTJGJkYsPCQySSJkR0YnRiwyRiZGMDwjL0kiZUdGJywmKiYjRikiIiNGKUYmRikhIiJGLEYp

poleRsolve:=solve(conds union{2*b=2*e+t},[t,b,d,e]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SStwb2xlUnNvbHZlR0YoNyM3KTJJInRHRigiIiIyIiIhRjEyRjFJImJHRigyRjFJImRHRigyRjZGMjJGOEYyL0kiZUdGKCwmKiYjRjIiIiNGMkYxRjIhIiJGNkYyNyNGLg==

Le 4e exemple LinearMultivariateSystem(conds union{e+d+b=2},[t,b,d,e]): nops(%);

IiNH

racine double \303\240 P, revient \303\240 dire s1= -e^{2i beta} conddouble:=LinearMultivariateSystem(conds union{e+d+b=2,2*e+t=2},[t,b,d,e]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SStjb25kZG91YmxlR0YoPCY3JTwkSSJiR0YoSSJ0R0YoPCMvSSJkR0YoLCgqJiMiIiIiIiNGOUYyRjlGOUY5RjlGMSEiIjwjL0kiZUdGKCwmRjlGOUY3Rjs3JjwkMUY4RjIyRjIjRjoiIiQ8JDFGMUZEMkYyRjFGM0Y8NyY8JDIiIiFGMjJGMkY4PCRGRzJGOEYxRjNGPDcmPCRGS0ZDPCQyRkRGMTJGMUY/RjNGPDcjRi4=

nops(%);

IiIl

op(1,conddouble);op(2,conddouble);op(3,conddouble);op(4,conddouble);

NyU8JEkiYkc2IkkidEdGJTwjL0kiZEdGJSwoKiYjIiIiIiIjRi1GJkYtRi1GLUYtRiQhIiI8Iy9JImVHRiUsJkYtRi1GK0Yv

NyY8JDEjIiIiIiIjSSJ0RzYiMkYoI0YnIiIkPCQxSSJiR0YpRisyRihGLzwjL0kiZEdGKSwoKiZGJUYmRihGJkYmRiZGJkYvISIiPCMvSSJlR0YpLCZGJkYmRjVGNg==

NyY8JDIiIiFJInRHNiIyRiYjIiIiIiIjPCQxSSJiR0YnI0YrIiIkMkYpRi48Iy9JImRHRicsKComRilGKkYmRipGKkYqRipGLiEiIjwjL0kiZUdGJywmRipGKkY2Rjc=

NyY8JDIiIiFJInRHNiIyRiYjIiIjIiIkPCQyRilJImJHRicyRi4sJiIiIkYxKiYjRjFGKkYxRiZGMSEiIjwjL0kiZEdGJywoRjJGMUYxRjFGLkY0PCMvSSJlR0YnRjA=

solve(conds union{e+d+b=2,2*e+t=2},[t,b,d,e]);

NyU3KDJJInRHNiIjIiIiIiIjMiIiIUYlMkkiYkdGJiwmKiZGJ0YoRiVGKCEiIkYoRigyRidGLS9JImRHRiYsKEYoRihGL0YoRi1GMC9JImVHRiZGLjcnL0YlRicyRi0jIiIkIiIlRjEvRjMsJiMiIiZGPEYoRi1GMC9GNkY6NygyRiUjRilGOzJGJ0YlMkYlRi1GLEYyRjU=

Un exemple map(evalb,subs(t=1/3,b=3/4, e=5/6,d=5/12,conds union{e+d+b=2,2*e+t=2}));

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Des cas o\303\271 w(y(s1)) est racine de S (pas forc\303\251ment tous !). Pour le moment on n'en a pas trouv\303\251. cond2:=LinearMultivariateSystem(conds union{e+d+b=2,2*e+t>2*b, 1< 2*e+t, 2*e+t<1+2*b,b<1/2},[t,b,d,e]): nops(cond2);

IiIh

cond2solve:=solve(conds union{e+d+b=2,2*e+t>2*b, 1< 2*e+t, 2*e+t<1+2*b,b<1/2},[t,b,d,e]): nops(cond2solve);

IiIh

cond2:=LinearMultivariateSystem(conds union{e+d+b=2,2*e+t>2*b, 2< 2*e+t, 2*e+t<2+2*b,b<1/2},[t,b,d,e]): nops(cond2);

IiIh

solve(conds union{e+d+b=2,2*e+t>2*b, 2< 2*e+t, 2*e+t<2+2*b,b<1/2},[t,b,d,e]);

NyI=

apres un long moment, message d'erreur cond2:=LinearMultivariateSystem(conds union{2*b<2*e+t, 1< 2*e+t, 2*e+t<1+2*b,b<1/2},[t,b,d,e]):

Error, (in solver) invalid input: LinearMultivariateSystemInternal expects its 1st argument, eqns, to be of type ({list, set})({`<`, `<=`, `=`}), but received {-1+2*b-t = 0, b-d <> 0}

cond2:={0<t,t<b,b<1,b>1/2,b<t+e,e+b+t<2,e<1,e+b>1};

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZjb25kMkdGKDwqMiIiIUkidEdGKDIiIiIsJkkiZUdGKEYzSSJiR0YoRjMyI0YzIiIjRjYyRjZGMzJGNiwmRjFGM0Y1RjMyRjVGMzJGMUY2MiwoRjVGM0Y2RjNGMUYzRjk3I0Yu

solve(cond2 union {2<2*e+t,2*e+t<b+2},[t,b,d,e]);

Nyc3KTJJInRHNiIjIiIiIiIjMiIiIUYlMkkiYkdGJiwmRihGKEYlISIiMkYnRi0vSSJkR0YmRjIySSJlR0YmRigyLCYqJkYnRihGJUYoRi9GKEYoRjQ3KEYkRiovRi1GLkYxRjNGNTcpRiRGKjJGLUY2MkYuRi1GMTJGNCwoRi1GL0YlRi9GKUYoRjU3KC9GJUYnMkYtIyIiJCIiJUYwRjEyRjQsJkYtRi8jRkNGKUYoMkZCRjQ3KTJGJSNGKUZDMkYnRiUyRiVGLUY7RjFGPUY1

op(1,%);

NykySSJ0RzYiIyIiIiIiIzIiIiFGJDJJImJHRiUsJkYnRidGJCEiIjJGJkYsL0kiZEdGJUYxMkkiZUdGJUYnMiwmKiZGJkYnRiRGJ0YuRidGJ0Yz

Exemple pour le cas interm\303\251diaire map(evalb,subs(d=2-e-b,t=1/4, b=2/3,e=8/9, cond2 union {2<2*e+t,2*e+t<b+2}));

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subs(d=2-e-b,t=1/4, b=2/3,e=8/9, d);

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subs(d=2-e-b,t=1/4, b=2/3,e=8/9, cos((b-2*e-t+1)*Pi));evalf(%);

LUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJComIyIjOCIjTyIiIkkjUGlHRiVGLiEiIg==

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subs(b=2/3,cos(Pi*(1-b)));evalf(%);

LUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJComIyIiIiIiJEYsSSNQaUdGJUYsRiw=

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Exemple pour une racine double \303\240 P solve(cond2 union {2=2*e+t},[t,b,e]);

NyU3JzJJInRHNiIjIiIiIiIjMiIiIUYlMkkiYkdGJiwmKiZGJ0YoRiVGKCEiIkYoRigyRidGLS9JImVHRiZGLjcmL0YlRicyRi0jIiIkIiIlRjEvRjNGNzcnMkYlI0YpRjgyRidGJTJGJUYtRixGMg==

subs(d=2-e-b,t=1/2, b=2/3,e=3/4, d);

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map(evalb,subs(d=2-e-b,t=1/2, b=2/3,e=3/4,cond2 union {2=2*e+t} ));

PCNJJXRydWVHJSpwcm90ZWN0ZWRH

Cas standard solve(cond2 union {2*b>2*e+t},[t,b,e]);

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

op(1,%);

NygySSJ0RzYiIyIiIyIiJDIiIiFGJDJJImJHRiUsJiomIyIiIkYnRjBGJEYwRjBGL0YwMiwmKiYjRjAiIiVGMEYkRjBGMEYvRjBGLDJJImVHRiUsJkYuISIiRixGMDIsJkYsRjlGMEYwRjc=

subs(d=2-e-b,t=1/4, b=2/3,e=1/2, d);

IyIiJiIiJw==

map(evalb,subs(d=2-e-b,t=1/4, b=2/3,e=1/2, cond2 union {2*b>2*e+t}));

PCNJJXRydWVHJSpwcm90ZWN0ZWRH

Evitons aussi la racine double map(evalb,subs(d=2-e-b,t=1/4, b=2/3,e=1/2, {2=2*e+t}));

PCNJJmZhbHNlRyUqcHJvdGVjdGVkRw==

solve(cond2 union {2*b+2*e+t<2},[t,b,e]);

NyI=

11./36; 23./72;

JCIrY2JiYkkhIzU=

JCIrV1dXJT4kISM1

11./40;

JCIrKysrXUYhIzU=

subs(t=1/4,b=5/9,cond2 union {2*b+2*e+t<2});

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solve(%);

PCQyIyIjNiIjT0kiZUc2IjJGJyMiI0IiI3M=

map(evalb,subs(t=1/4,b=5/9,e=31/100, {t < 1/2, 0 < t, b < (1/4)*t+1/2, 1/2 < b, e < 1-(1/2)*t-b, b-t < e}));;

PCNJJXRydWVHJSpwcm90ZWN0ZWRH

map(evalb,subs(t=1/4,b=5/9,e=31/100,cond2 union {2*b+2*e+t<2} ));;

PCNJJXRydWVHJSpwcm90ZWN0ZWRH

subs(t=1/4,b=5/9,e=31/100,1-2*e-t);

IyIjOCIkKyI=

1/9.;

JCIrNjY2NjYhIzU=

subs(t=1/4,b=5/9,e=31/100,2-e-b);

IyIlQDUiJCsq

Valeurs du papier conds union { d+e+b=2}

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map(evalb,subs(b=2/3,d=5/6,e=1/2,t=1/4,conds union {2*e+t<2, d+e+b=2} ));;

PCNJJXRydWVHJSpwcm90ZWN0ZWRH

map(evalb,subs(b=2/3,d=4/9,e=8/9,t=1/4,conds union {2*e+t>2, d+e+b=2} ));;

PCNJJXRydWVHJSpwcm90ZWN0ZWRH

map(evalb,subs(b=2/3,d=7/12,e=3/4,t=1/2,conds union {2*e+t=2, d+e+b=2} ));;

PCNJJXRydWVHJSpwcm90ZWN0ZWRH