Integration of Uniformly Convergent Series : Cauchy's Theorem and Morera's Theorem

Prove the following: Let be a sequence of functions continuous on a set containing the contour , and suppose that converges uniformly to on . Then the series converges to . Using this result and the Generalized Cauchy Integral formula for derivatives (see below), show the following: If all are analytic ...continues

Removable singularity

(See attached file for full problem description) --- Let have an isolated singularity at and suppose that is bounded in some punctured neighborhood of . Prove directly from the integral formula for the Laurent coefficients that for all j = 1,2,3,..., i.e. must have a removable singularity at . The integ ...continues

Cauchy Integral Formula

(See attached file for full problem description and embedded formulae) --- Why can (1) be regarded as a special case of (2)? (1) Cauchy's Integral formula (no need to prove): is a simple closed positively oriented contour. If is analytic in some simply connected domain D containing and if is any point inside , ...continues

Residues for integral a function

--- Using the method of residues, verify the following: --- (See attached file for full problem description)

Residues integral of trig function

Please see attached

Cauchy Integral Formula in an annulus

Please see attached.

Improper Integral

Please see attached

Complex numbers

What is the solution (-1+i squareroot of 3)exponent 9?

Vectors

Using the given vectors how do I find the specified dot product u=3i-8j;v=4i+9j find u.v

Vector subtraction

Given these vectors how do I find u - v? U=-8i-6j V=-10i+5j