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eXorikos

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## Homework Statement

Calculate the residu for the singularity of [tex] \frac{z \sin{z}}{\left( z - \pi \right)^3} [/tex]

## Homework Equations

[tex]R \left( a_0 \right) = \frac{1}{2 \pi i} \oint \frac{z \sin{z}}{\left( z - \pi \right)^3} dz[/tex]

## The Attempt at a Solution

[tex]\pi [/tex] is an essential singularity so the residue cannot be calculated using the limit.

I have tried calculating the integral around the path from 3-i to 4-i to 4+i to 3+i to 3-i. In principle this should work, but Maple gives me a wrong answer if I try to evaluate.

[tex]\frac{1}{2 \pi i} \left( \int^4_3 \frac{\left( x - i \right) \sin{x - i}}{\left( x - i - \pi \right)^3} dx + \int^1_{-1} \frac{\left( 4 + iy \right) \sin{4 + iy}}{\left( 4 + iy - \pi \right)^3} dy + \int^3_4 \frac{\left( x + i \right) \sin{x + i}}{\left( x + i - \pi \right)^3} + \int^{-1}_1 \frac{\left( 3 + iy \right) \sin{3 + iy}}{\left( 3 + iy - \pi \right)^3} dy \right)[/tex]

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