Stereographic projection

1. Let z and z' be points in C with corresponding points on the unit sphere Z and Z' by stereographic projection. Let N be the north pole N(0,0,1). a) Show that z and z' are diametrically opposite on the unit sphere iff z(z bar)'=-1 ps. here z bar means conjugate of z b) Show that the triangles Nz'z and NZZ' are similar. The order of the vertices is important and is as given. Use this to derive the formula for the euclidean distance in R^3 d(Z,Z') = (2|z-z'|)/(sqrt(1+|z|^2)x sqrt(1+|z'|^2)) c) Show that the stereographic projection preserves angles by looking at two lines l1 and l2 through the point z in the complex plane and their images of the Riemann sphere, which are two arcs thru t... click for more

Conformal mappings

Let O be the upper half of the unit disc D. Find a conformal mapping f: O->D that maps {-1,0,1} to {-1,-i,1}. Find z in O with f(z)=0

Find all cube roots of –8, in rectangular coordinates

Find all cube roots of the number -8 and state the final answer in rectangular coordinates.

Cross ratio

a) Let z1 and z2 be two points on a circle C. Let z3 and z4 be symmetric with respect to the circle. Show that the cross ratio (z1,z2,z3,z4) has absolute value 1. b)Let ad-bc=1, c not zero and consider T(z)=(az+b)/(cz+d). Show that it increases lengths and areas inside the circle|cz+d|=1 and decreases lengths and areas outside the circle|cz+d|=1

Contour integrals

Prove that integral(o to pi/2) of sin^2n theta d(theta)=pi(1x3x5x...2n-1)/2(2x4x6x...2n)