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at the moment I am trying to reproduce the results of a paper.

There, it turns out that a specific physical problem is mapped onto an integral to be calculated:

$$I(\Theta; a, b) = 2\cdot\int_0^\infty \frac{a-u^2}{\left( u^2+\frac{a^2}{b-a}\right) \left(u^2+\frac{b^2}{b-a} \right) \sqrt{a - u^2 -\frac{a}{1-a/b}\sin^2(\Theta) } }\mathrm {d}u \equiv \int_0^\infty f(u)du$$

where I took the liberty to replace $\epsilon_d\rightarrow a$ and $\epsilon_m \rightarrow -b$ in contrast to the paper and one can assume that both

$a,b > 0$ and $b > a$.

Somehow, Mathematica manages to be able to calculate the numerical values of this integral with some warning messages due to the pole at $$u_p = \sqrt{a -\frac{a}{1-a/b}\sin^2(\Theta)}$$

Also, Mathematica can calculate the indefinite integral, both for $\Theta = 0$ (which is a special case to compare results) and in the general case. Nevertheless, I am not able to use the result since it is indefinite in all cases for $u\rightarrow \infty$.

So, I am asking for some advice to calculate the integral at hand with respect to given constant $a$ and $b$.
The special case of $\Theta = 0$ might already be worth to take a look since it is much easier to calculate it than for the general case.

In the meantime I tried something like a Cauchy principal value integration around $u_p$ using the parametrization $u_\delta (\varphi) = u_p -\delta e^{\mathrm{i}\varphi}$ along a half circle $C_\delta$ interpreting $u^2$ as $\bar{u}u$. Then,
$$I_\delta = \int_0^\pi f(u_\delta(\varphi))\delta d\varphi$$ is the integral around $C_\delta$ which turned out to vanish for $\delta\rightarrow 0$ such that the whole integral should be given in terms of a principal value one. Noteworthy, I am not sure if my result is correct.

Please, if my question is not stated correctly, or anything is obvious don't hesitate to give me some advice.

Thank you in advance.
Sincerely

Robert

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I don't see a question mark! :-). Can you please explicitly state what you are looking for? –  Aryabhata Jan 6 '11 at 18:34
    
@Moron: Thank you to point out that my question is not really clear. I want to calculate the given integral with respect to the constant $a$ and $b$. Today, I tried to calculate the contribution at the pole via Cauchy principal value integration but my vanishing result seams erroneous. I think I am just missing some basic "singular" integral skills to calculate the integral efficiently and I thought this might be just the right place to ask :) PS.: I will update the question tomorrow due to my further calculations. Sincerely –  Robert Filter Jan 6 '11 at 21:36
    
I suggest you update the question with your previous comment. Not everyone bothers to read the comments. –  Aryabhata Jan 6 '11 at 21:46
    
@Moron: Thank you for the push :) I hope it is much more convenient now. Greets –  Robert Filter Jan 6 '11 at 22:30

1 Answer 1

up vote 2 down vote accepted

I am answering my own question here since I hope that it can safe some time for somebody.

To compare my results with that of the paper I calculated a complex quantity called reflection coefficient $r = |r|e^{\mathrm{i}\varphi}$. The absolute value $|r|$ tells you roughly how much of some field gets back-reflected in some domain at the boundary to another one, depending on several parameters. Its phase $\varphi$ is also of great importance since it will be useful for effects like interference for linear phenomena.

My results were simply different from those presented in the given paper since I did not care for the convention used for the calculation of the phase. Taking the modulus of $-\varphi$ with respect to $\pi$, I was able to reconstruct all given data.

I apologize for any inconvenience. Furthermore, an analytic calculation of the integral at hand is only feasible for the special case of $\Theta = 0$ (using Mathematica) since the results blow up quickly.
Greetings

Robert

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