Contour Integrals and Green's Theorem

Please see attached file. Complex Analysis Problem Problem. Show that if C is a positively oriented simple closed contour, then the area of the region enclosed by C can be written .…. Suggestion: You can use the form: R is a closed region for real valued functions. Does this imply it is rectangular? It is based on ...continues

An application of Cauchy's theorem to a non-simple curve

Please see attached file.

Some simple applications of the Cauchy-Goursat theorem

Use the Cauchy theorem to show that the integral around the unit circle |z|=1, traversed in either direction, is zero for each of the following functions: 1) f(z)=z exp(-z) 2) f(z)=tan(z) 3) f(z)=Log(z+2) The attached file contains this question written more clearly with correct mathematical notation.

Analytic Functions

Let f:from C to C be analytic. Define g:from C to C by g(x)= ~(f(~z))^2. Show that g is analytic. (Note: "~" here represents an over-bar., i.e., one over the whole set of parentheses and the other just over letter "z" ).

An example using Cauchy's theorem

Let f be entire. Evaluate the integral from zero to 2 pi of f(z_0+re^(i theta)) e^(ik theta), where z_0 is a constant and k is a constant greater than or equal to 1.

The function f(z) is entire and Im f <=0. Prove that f is a constant.

The function f(z) is analytic over the whole complex plane and Im f <= 0. Prove that f is a constant.

Show that the integral of the analytic function is independent of radius.

Let f be analytic on │z│> 1. Show that if r > 1, then the integral of f over C(0,r) is independent of r.

Prove that a power series converges absolutely everywhere or nowhere on its circle of convergence.

Prove that a power series converges absolutely everywhere or nowhere on its circle of convergence.

Analytic functions. Either f or g is zero.

Let G be a region and let f and g be analytic functions on G such that f(z)g(z)=0 for all z in G. Show that either f=0 or g=0.

Cauchy integral formula

Show that the Cauchy Integral Formula follows from Cauchy's Theorem.