Laurent Series

Find a Laurent series which converges for |z-1|>1 for f(z) = (z-1)^2/(z(z-2)) ... {see attachment for complete question}

Residue Theorem and L'Hopital's Rule

Please evaluate the attached by means of the residue theorem.

Quotient Roots Contour Integral Proof

Suppose that p(z) and q(z) are polynomials with complex coefficients with the property that deg q(z)>=degp(z) + 2. If C is a positively oriented simple closed contour containing all the roots of q(z) on its interior, then prove that: the contour integral about C of (p(z)/q(z))dz=0.

Isolated Zeros Proof

Let f be analytic on a domain D. Prove that if f is not identically zero, then the zeros of f in D are isolated. (That is, prove that if f is not identically zero and if z(0) is a point in D with f(z(0))=0, then there exists e>0 such that f(z)=/0 for all z in the region 0<|z-z(0)|

Analytic Zeros Proof

Let f be analytic on a domain D. Prove that if f(z(0))=0 and if f is not identically zero, then z(0) is a zero of f of some finite order m.

Analyticity question

Suppose f is analytic on the disk |z|<1 and that f(0)=0. Let g(z)=f(z)/z. Then g is analytic on the region 0<|z|<1. How can you define g(0) to make g an analytic function on all of |z|<1? Briefly explain why the choice makes g analytic at 0.

Definite integral

Evaluate the integral from 0 to infinity of X^2/(X^4+2X^2+1) dx

Integral

Evaluate the integral from 0 to inf. of [cos(3x)-cos(5x)]/x^2 dx

Integral of a hard polynomial

Suppose that p(z) and q(z) are polynomials with a complex coefficients with the property that deg q(z) is greater than or equal to deg p(z)+2. If C is a positively oriented simple closed contour containing all of the roots of q(z) on its interior, then prove that integral C of p(z)/q(z) dz = 0

Uning the Euler's identity to calculate the root of i

Calculate the square root of i using a direct method and show that using Euler's formula yield the same result.