Laurent Series

Find the Laurent series about all singular points of {see attachment} Thanks.

Mapping (Quarter-Plane; Half-Line)

Find the image of the quarter-plane {see attachment} under the mapping {see attachment}. Show graphs (shaded regions) in the w-plane and identify the images of the half-lines {see attachment}.

Computing an Integral Using Residues

Use residue theory to compute the integral: (see attachment)

Classifying Isolated Singularities

Classify all the isolated singularities of the following functions (classify as removable, pole of order m, or essential). Explain the reasoning for each classification. *See attachment for functions*

Laurent Series

Find a Laurent series which converges for ... {see attachment for complete question}

Residue Theorem; l'Hopital's Rule

Please evaluate the attached by means of the residue theorem - thanks!

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 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 anaalytic 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.