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**zetafunc.****Guest**

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****Banned**- Registered: 2009-11-13
- Posts: 224

What is *C*?

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**zetafunc.****Guest**

C is some simple closed curve about 0.

**scientia****Banned**- Registered: 2009-11-13
- Posts: 224

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

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**zetafunc.****Guest**

Why is it holomorphic on C?

**scientia****Banned**- Registered: 2009-11-13
- Posts: 224

Let so where and .

Since the partial derivatives are continuous for all

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

**zetafunc.****Guest**

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.****Guest**

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?Pages: **1**