- #1

brianhawaiian

- 12

- 0

## Homework Statement

Calculate the residue at each isolated singularity in the complex plane

e^(1/z)

## Homework Equations

#1 Simple pole at z0 then,

Res[f(z), z0] = lim (z - z0)(f(z)) as z goes to z0.

#2 Double pole at z0 then,

Res[f(z), z0] = lim d/dz [(z - z0)^2*f(z)] as z goes to z0.

#3 If f(z) and g(z) are analytic at z0, and if g(z) has a simple zero at z0 then

Res[f(z)/g(z), z0] = f(z0)/g'(z0)

#4 If g(z) is analytic and has a simple zero at z0, then

Res[1/g(z), z0] = 1/g'(z0).

## The Attempt at a Solution

The problem occurs when z = 0 so looking at

Res[e^(1/z), 0], Using #1, #3, #4 don't help the problem. So using #2

lim as z goes to z0 [d/dz z^2 * e^(1/z)], there's still a problem... I'm completely lost at this point.

## Homework Statement

Evaluate the following integral, using the residue theorem

Integral |z| = 1 (sin(z)/z^2)dz

## Homework Equations

See Above

## The Attempt at a Solution

How would I start this? z = 0 gives a problem, would I take the integral first and then evaluate?