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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.


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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
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.


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