Complex integration

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

Power series

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

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(z_i), i=1,2,..n

Holomorphic functions

Let f(z) be holomorphic on |z|<1 and |f(z)|<1/(1-|z|) for |z|<1. Show that the Taylor coefficients an of f(z) satisfy:|an| less tha([(n+1)(1+1/n)]^n less than e(n+1)