Calculating Antiderivative of Complex Integral

[rar0308]

Homework Statement

z is a complex variable.
What is antiderivative of $\frac{e^{-iz}}{z^2+(\mu r)^2}$?

Homework Equations

The Attempt at a Solution

To caluculate the Fourier transform encounterd in reading quantum phsycis i have to calculate this integral. I have little knowledge of complex analysis. How can I do this? And please recommend books in which this thing can be found.

 

[Whovian]

What are we antidifferentiating with respect to? I'm also assuming you mean AN antiderivative, or the general form of all antiderivatives.

 

[rar0308]

It's respect to z.

 

[fzero]

The integral you presumably want is related by a change of variables to the one computed here http://en.wikipedia.org/wiki/Residue_theorem#Example (it seems as clear as anything I would write). Give that a read through and post back with specific questions here. This is covered in almost every undergrad complex analysis text, as well as in most math methods texts.

 

[rar0308]

Very helpful
I read it.
Is $Res_{z=i}f(z)=\frac{e^{-t}}{2i}$ calculated by Cauchy's integral formula?
Why is the condition
$|e^{itz}| \leq 1$ required?

I'm reading quantum physics book. Fourier transforms appears in the book frequently. Which math book is good for learing introductory complex analysis so that there is no problem in doing quantum physics? Other than mathematical method books which is so brief .

 

[fzero]

Very helpful
I read it.
Is $Res_{z=i}f(z)=\frac{e^{-t}}{2i}$ calculated by Cauchy's integral formula?

Well, the residues should really be thought of as ingredients in the residue theorem, which is a generalization of the integral formula. However, in the case here, the poles in the function are sufficiently obvious that you can use the integral formula to compute the contour integral directly.

Why is the condition
$|e^{itz}| \leq 1$ required?

It's required so that

$$\int_{\mathrm{arc}}{|e^{itz}| \over |z^2+1|}\,|dz| \leq \int_{\mathrm{arc}}{1 \over |z^2+1|}\,|dz|$$

in the argument that the contribution to the integral from the arc vanishes.

I'm reading quantum physics book. Fourier transforms appears in the book frequently. Which math book is good for learing introductory complex analysis so that there is no problem in doing quantum physics? Other than mathematical method books which is so brief .

I haven't looked at it much myself, but the book that's always recommended for undergrads is Mary Boas "Mathematical Methods for the Physical Sciences." I've used Arfken's math methods book, but it's geared towards grad students and is more of a reference book.