The relationship between social class and spending on welfare and education. Using Chi-squared test to study existence of relationship. For specific problem solved in this posting, please see the posted problem.

Please see attached file with two questions with charts. Thank you! Here we examine the relationship between social class and spending on welfare and education. Lower Class Working Class Middle Class Upper Class Spending on welfare Too little 18 59 73 4 About right 12 109 132 10 Too much 16 142 121 12 Total ...continues

Contour Integral Problem

Please see attached file. R is a closed region for real valued functions. Does this imply it is rectangular? It is based on Green's Theorem, so I think it is considered rectangular.

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.

Prove that the function is analytic.

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.