Is this contour integration correct?

zetafunc. · 2012-10-24 21:52:56

I was trying to find

, where z is complex and n is a constant.

Let z = ln(t), then dz = (1/t)dt. This transforms the integral to

Is this correct? And what would the graph of this look like?

scientia · 2012-10-24 23:50:28

What is C?

zetafunc. · 2012-10-24 23:58:05

C is some simple closed curve about 0.

scientia · 2012-10-25 00:04:59

Then shouldn't it be 0 by Cauchy's integral theorem since is holomorphic on ?

zetafunc. · 2012-10-25 01:08:39

Why is it holomorphic on C?

scientia · 2012-10-25 03:42:19

Let so where and .

Since the partial derivatives are continuous for all

and satisfy the CauchyRiemann equations and , is holomorphic on the complex plane.

zetafunc. · 2012-10-25 04:03:37

Thanks for that, I did not know of this method. Is that the only way to show a function like this is holomorphic on C? Would I have to do this before I evaluate any contour integral?

zetafunc. · 2012-10-25 04:10:09

Oh wait, I just noticed...

and since

for n ≠ 1, z ∈ Z, then every term reduces to zero, so the contour integral is zero... right?

Math Is Fun Forum

#1 2012-10-24 21:52:56

Is this contour integration correct?

#2 2012-10-24 23:50:28

Re: Is this contour integration correct?

#3 2012-10-24 23:58:05

Re: Is this contour integration correct?

#4 2012-10-25 00:04:59

Re: Is this contour integration correct?

#5 2012-10-25 01:08:39

Re: Is this contour integration correct?

#6 2012-10-25 03:42:19

Re: Is this contour integration correct?

#7 2012-10-25 04:03:37

Re: Is this contour integration correct?

#8 2012-10-25 04:10:09

Re: Is this contour integration correct?

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