Complex Integration

Let be the unit circle, followed in the positive direction. Evaluate the integral Please show all steps to the solution.

Complex Analysis

See attached Let be the unit circle, followed in the positive direction,Evaluate the integral

Complex Variables

Please show all steps to solution (see attached) Let be the curve parameterized by ζ(t) = for Evaluate the integral dz

Complex Variables

Please show all steps to solution. Suppose that P is a polynomial of degree n≥1.Show that P has exactly n roots in C Hint: Fundamental Theorem of Algebra

Complex Variables

Please show all steps to solution. Suppose that P is a polynomial with no roots on the contour Y .Show that the number of roots of P in the region enclosed by Y is given by

Complex Variables : The Minimum Principle

Please see the attached file for the fully formatted problem. The Minimum Principle: Let f be analytic in a bounded region D and continuous and nonzero on bar-D. Show that |f(z)| attains its minimum on the boundary of D. Hint: Consider the function g(z) =1/f(z).

Isolated singularities

Please show all steps to solution. See attached Classify with proof all the isolated singularities in C of ...

Complex Variables

Please show all steps to solution. Find the Laurent series....

Proof

Prove that an entire function that omits the value 0 and 1 is a constant.

Maclaurin series

Please show all steps to solution. Find the Maclaurin series for...and describe the region of analyticity.