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#1 2012-10-25 20:52:56

zetafunc.
Guest

Is this contour integration correct?

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?

#2 2012-10-25 22:50:28

scientia
Full Member

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Re: Is this contour integration correct?

What is C?

#3 2012-10-25 22:58:05

zetafunc.
Guest

Re: Is this contour integration correct?

C is some simple closed curve about 0.

#4 2012-10-25 23:04:59

scientia
Full Member

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Re: Is this contour integration correct?

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

#5 2012-10-26 00:08:39

zetafunc.
Guest

Re: Is this contour integration correct?

Why is it holomorphic on C?

#6 2012-10-26 02:42:19

scientia
Full Member

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Re: Is this contour integration correct?

Let
so
where
and
.









Since the partial derivatives are continuous for all
and satisfy the Cauchy–Riemann equations
and
,
is holomorphic on the complex plane.

#7 2012-10-26 03:03:37

zetafunc.
Guest

Re: Is this contour integration correct?

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?

#8 2012-10-26 03:10:09

zetafunc.
Guest

Re: Is this contour integration correct?

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?

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