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Calculate $\displaystyle \int_\Gamma \frac 1{z^4 + 81}$ where $\Gamma: |z+i| = \frac 34$

Can somebody help me with this question please or give me a hint on how to get started, as I have never seen a question with gamma like this and I have no idea how to start.

Thanks

Ant
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bws
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1 Answers1

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Solve for $z^4 = -81$, you will get four roots and write it as $\frac{1}{z^4 + 81}=\frac{1}{(z-a)(z-b)(z-c)(z-d)}$ where a,b,c,d are roots of the above $z^4 = -81$. Then use partial fractions $\frac{1}{(z-a)(z-b)(z-c)(z-d)} = \frac{A}{z-a}+\frac{B}{z-b}+\frac{C}{z-c}+\frac{D}{z-d}$. Find the value of A,B,C,D and then substitute in the above equation and then integrate using Cauchys integral formula.

Sam Christopher
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  • What do you need partial fractions for? You just need to calculate the residue :-) – Ant May 23 '15 at 14:31
  • are the roots 3/√(2) + [3/√(2)]i, -3/√(2) + [3/√(2)]i, -3/√(2) - [3/√(2)]i, 3/√(2) - [3/√(2)]i ? – bws May 23 '15 at 14:54
  • I do not understand how to do partial fractions with this question – bws May 23 '15 at 14:57
  • and none of the roots are in gamma are they? i'm confused... – bws May 23 '15 at 15:04