Exam May 2006Preben Alsholm, 2/12 2008computer:=ssystem("hostname"):
if computer=[0,"PC-PKA1"] then
libname:="C:/Documents and Settings/alsholm/Dokumenter/DMat/libDMat",libname
elif computer=[0,"pc-pka1"] then
libname:="J:/DMat/libDMat",libname
elif computer=[0,"alsholm-PC"] then
libname:="F:/DMat/libDMat",libname
else
print("Fremmed computer: Omdefin\303\251r selv libname.")
end if;with(DMat):with(plots):Problem 1Sum(z^j/(j+1)/(j+2),j=0..infinity);value(%);limit(%,z=1);Problem 2f:=z->(2*z-6)/(z+3);f(I);complexplot(5+4*exp(I*t),t=0..2*Pi,thickness=2,scaling=constrained,view=[0..10,-5..5]);conformal(5+z,z=0..4+2*Pi*I,coords=polar,view=[0..10,-5..5],color=[blue,black]):
conformal(z,z=0..10+2*Pi*I,coords=polar,view=[0..10,-5..5]):
p1:=display(%%,%,scaling=constrained):
conformal(f(5+z),z=0..4+2*Pi*I,coords=polar,color=[blue,black]):
conformal(f(z),z=-Pi/2*I..10+Pi/2*I,coords=polar):
p2:=display(%%,%,scaling=constrained):
display(Array([p1,p2]));f(z);The real axis is mapped onto itself since -3 is mapped to infinity andf(0),f(1);Using conformality we see that the imaginary axis is mapped onto the circle with diameter connecting the pointsf(0),limit(f(z),z=infinity);Again by conformality the circle C is mapped onto the circle with diameter connecting the pointsf(1),f(9);Sincef(1);is inside the image of the imaginary axis the image of the right half-plane is the interior of f(I). complexplot([f(t),f(I*t),f(5+4*exp(I*t))],t=-15..15,view=[-4..4,-4..4],thickness=3,scaling=constrained,color=[red,blue,maroon],legend=["f(R)","f(I)","f(C)"]);
g:=piecewise(5+r*cos(t)>=0,5+r*exp(I*t),undefined);animate(complexplot,[g,r=4..10,view=[0..12,-6..6]],t=0..2*Pi,frames=50,trace=50,scaling=constrained):
animate(complexplot,[f(g),r=4..10,view=[-2..2,-2..2]],t=0..2*Pi,frames=50,trace=50,scaling=constrained):
display(Array([%%,%]));finv:=unapply(solve(f(z)=w,z),w);animate(complexplot,[finv(r*exp(I*t)),t=0..2*Pi,scaling=constrained,view=[0..12,-6..6]],r=1..2,trace=25):
animate(complexplot,[finv(r*exp(I*t)),r=1..2,color=blue,scaling=constrained,view=[0..12,-6..6]],t=0..2*Pi,trace=25):
p1:=display(%%,%):
animate(complexplot,[r*exp(I*t),t=0..2*Pi,scaling=constrained],r=1..2,trace=25):
animate(complexplot,[r*exp(I*t),r=1..2,color=blue,scaling=constrained],t=0..2*Pi,trace=25):
p2:=display(%%,%):
display(Array([p1,p2]));p:=(n,fz,vw,nxy)->conformal(fz,z=1..2+2*Pi*I,vw,coords=polar,grid=[round(n),round(n)],nxy,scaling=constrained):
animate(p,[n,finv(z),-6*I..12+6*I,numxy=[50,50]],n=2..11,frames=10,trace =10):
animate(p,[n,z,-2-2*I..2+2*I,numxy=[50,50]],n=2..11,frames=10,trace =10):
display(Array([%%,%]));phi:=unapply(simplify(evalc(-1/ln(2)*ln(abs(f(x+I*y)))+2)),x,y);plot3d(phi(x,y),x=0..12,y=-6..6,view=1..2,axes=boxed);Problem 3F:=1/sin;G:=z->(z^2-Pi^2)/sin(z);singular(F(z),z);singular(G(z),z);series(F(z),z=0);series(G(z),z=0);Problem 4Int(x^(1/3)/(1+x^2),x=0..infinity): %=value(%);residue(z^(1/3)/(1+z^2),z=I);evalc(%);expand(2*Pi*I*%);Alternative computation of the integral:A:=Int(x^(1/3)/(1+x^2),x=0..infinity);simplify(IntegrationTools:-Change(A,x=t^3));The path:complexplot([5*exp(I*Pi/3*t/5),exp(I*Pi/3)*t,t],t=0..5,thickness=4);(1+exp(I*Pi/3))*A=2*Pi*I*residue(3*z^3/(1+z^6),z=exp(I*Pi/6));isolate(%,A);applyop(evalc,2,%);