Let G be a region and suppose that f:G->C (C here is complex plane)is analytic and a in G such that |f(a)|=<|f(z)| for all z in G. Show that either f(a) = 0 or f is constant.

Let G be a region and suppose that f:G->C (C here is complex plane)is analytic and a in G such that |f(a)|=<|f(z)| for all z in G. Show that either f(a) = 0 or f is constant.

Complex Integration and Polynomials : If gamma(t) = Re^(it), 0 =< t =< 2 pi, show that the integral over gamma p'(z)/p(z) dz = 2 ( pi ) i n.

Let P(z) be polynomial of degree n and let R>0 be sufficiently large so that p never vanishes in { z: |z| >= R}. If gamma(t) = Re^(it), 0 =< t =< 2 pi, show that the integral over gamma p'(z)/p(z) dz = 2 ( pi ) i n.

Complex Analysis : Cauchy Theorem and Integral Formula- Let gamma be a closed rectifiable curve in C (Complex plane) and a is not an element of {gamma}. Show that for n>= 2 integral over gamma of ...

Let gamma be a closed rectifiable curve in C (Complex plane) and a is not an element of {gamma}. Show that for n>= 2 integral over gamma of (z-a)^-n dz = 0.

Power series representation of analytic functions (Complex integrals)

Evaluate the following integrals: a). integral over gamma of e^(iz) / z^2 dz, where gamma(t) = e^(it), 0= ...continues

Complex Integration using power series expansion of analytic functions

Evaluate the following integrals: (a) integral over gamma of (e^z - e^-z)/(z^n) dz, where n is positive integer and gamma(t) = e^(it), 0 =< t =< 2 pi (b) integral over gamma of (dz/(z^2 + 1) ) where gamma(t) = 2e^(it), 0 =< t =< 2pi ( Hint: expand (z^2 + 1)^-1 by means of partial fractions PLEASE USE POWER SERIE ...continues

Complex integration using series expansion of analytic functions

I want to check my answer: Evaluate the following integrals: integral over gamma for (sin z)/z dz, given that gamma(t) = e^(it) , 0= ...continues

Convergence of series (complex)

Find the radius of convergence of the series sum from n = 1 to infinity of n^3(z/3)^n. Does this series converge at any point on the boundary of the disk of convergence?

Analytic functions complex

Let f = u + iv be an analytic function on an open connected set G in C ( C = complex plane) where u and v are its real and imaginary parts. assume u(z) >= u(a) for some a in G and all z in G. Prove that f is constant.

Complex / Entire Function : Let f be an entire function such that |f(z)| =<10|z+1| for all |z|>100. Show that f is a linear function, f(z)= pz + q.

Let f be an entire function such that |f(z)| =<10|z+1| for all |z|>100 Show that f is a linear function, f(z)= pz + q.

Complex Cross Ratio : Evaluate cross ratio (infinity, 0, i, 1) give answer in the form a + ib where a,b in R.

Evaluate cross-ratio (infinity,0,i,1) give answer in the form a + ib where a,b in R.