- #1

bjohnson2001

- 15

- 0

## Homework Statement

Let [itex]\Gamma[/itex] be the square whose sides have length 5, are parallel to the real and imaginary axis, and the center of the square is i. Compute the integral of the following function over [itex]\Gamma[/itex] in the counter-clockwise direction. You must use two different methods to solve the problem in order to receive full credit. Show all work.

[itex]\frac{e^{z}}{(z^{2} + 2*z + 1)}[/itex]

## Homework Equations

Residue for pole of order 2: [itex]Res(f(z),z_{0}) = \stackrel{lim}{_{z \rightarrow z_{0}}}\frac{d}{dz}(z-z_{0})^{2}*f(z)[/itex]

Residue Theorem: [itex]\oint_{C} f(z)dz = i2\pi\sum(Residues) [/itex]

## The Attempt at a Solution

When we rewrite the equation as: [itex]\frac{e^{z}}{(z+1)^{2}}[/itex] we can see that there is a pole of order 2 at z = -1. Since the singularities only have Real components, the function is meromorphic and the poles are isolated. The [itex]\Gamma[/itex] contour is analytic everywhere else on and inside [itex]\Gamma[/itex] so the poles are removable and we can use the Residue theorem to perform the integration.

First calculate the Residues: [itex]Res(f(z),z_{0}) = \stackrel{lim}{_{z \rightarrow -1}}\frac{d}{dz}((z+1)^{2}*\frac{e^{z}}{(z+1)^{2}})[/itex] = [itex]\frac{1}{e}[/itex]

Now we can use the Residue theorem to perform the integration:

[itex]\oint_{C} \frac{e^{z}}{(z+1)^{2}} = i2\pi\frac{1}{e} = i2.3115 [/itex]

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I am absolutely stuck on finding a second method to solve this problem! I thought maybe I could parametrize it but no symbolic integration exists for this function. I tried integration by parts, the int() function in MATLAB and my TI-89 all without success. The quad() function in MATLAB succeeded but unfortunately my prof is not allowing numeric integration methods..

Please help!