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# cauchy's residue theorem watch

1. Use the residue theorem to evaluate the following integral along the contour gamma which is the unit circle (positive orientation) centered at z=0 :

Integral over gamma (tan pi z) dz

I know that the only singularity inside gamma is at z=1/2. Then by cauchy's residue theorem the answer is 2pi i res(f,1/2). But i'm not sure how to find this residue since i dont know what type of singularity it is?
2. (Original post by Ishika)
Use the residue theorem to evaluate the following integral along the contour gamma which is the unit circle (positive orientation) centered at z=0 :

Integral over gamma (tan pi z) dz

I know that the only singularity inside gamma is at z=1/2. Then by cauchy's residue theorem the answer is 2pi i res(f,1/2). But i'm not sure how to find this residue since i dont know what type of singularity it is?
How about using the laurent series expansion for the function?
3. What is ?
4. (Original post by zcomputer5)
How about using the laurent series expansion for the function?
How do i find the laurent expansion of tan pi z about 1/2? I tried using tan(a+b) formula but it doesnt work..
5. First, make life a little easier by writing and rewriting in terms of . Then use the standard series expansion for sin to get the denominator.

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Updated: March 29, 2011
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