Harmonic Functions

a)Let a be less than b and set M(z)=(z-ia)/(z-ib). Define the lines L1={z:F(z)=b}, L2={z:F(z)=a} and L3={z:R(z)=0}. The three lines split the complex plane into 6 regions. Determine the image of them in the complex plane. b) Let log be principal branch of the logarithm. Show that log(M(z)) is defined for all z in C with the ...continues

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 ...continues

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.

Circles and Cross Ratios

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 outsid ...continues

Contour Integrals

Prove that integral (0 to pi/2) of sin^2n theta d(theta)=pi(1x3x5x...2n-1)/2(2x4x6x...2n)

Complex Integration : Holomorphisms

Let f(z) be holomorphic on the unit disc and f(0)=1. By working with 1/2ipi(integral over unit circle of [2+,-(z+1/z)]f(z) dz/z) prove that a)2/pi(integral(0 -2pi) of f(e^itheta)cos^2theta/2 d(theta))=2 + f'(0) b)2/pi(integral(0-2pi) of f(e^itheta)sin^2theta/2 d(theta))=2-f'(0)

Holomorphic Functions

If f(z) is holomorphic on |z|<1, f(0)=1, and for all |z|<=1 we have R(f(z))>=0, then show that -2<=R(f'(0))<=2 keywords: holomorphisms

Power Series and Holomorphic Functions

Let f(z) be holomorphic in the region |z|<=R with power series expansion f(z)=sum(n=0 to infinity) a_nz^n. Let the partial sum of the series be defined as s_N(z)=sum(n=0 to N) a_nz^n Show that for |z|less than R we have s_n(z)= 1/i2pi(integral over |w|=R of f(w)[(w^N+1 - z^N+1)/(w-z)]dw/w^N+1)

Polynomials with Complex Roots and Holomorphic Functions

Let C be a circle enclosing the distinct points z1,z2,...zn. Let p(z)=(z-z1)(z-z2)...(z-zn) be the polynomial of degree n with roots at these points. Let f(z) be holomorphic in a disc that includes C. Show that P(z)=1/i2pi(integral over C of (f(w)/p(w)[(p(w)-p(z)/w-z)]dw) is a polynomial of degree n-1, with the property P(z_i)=f ...continues