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I am doing my PhD and I faced an integral of the CPV type integral of the form:

$$\int_{-1}^{1} \frac{\sqrt{1-t^2}\sin{kt}}{t-x}d t$$ or

$$\int_{-1}^{1} \frac{\sin{kt}}{(t-x)\sqrt{1-t^2}}dt$$

Is there a paper or book that gives the solution to these kind of integrals or the idea of how to solve them? There are some formulas but they do not contains the term [sin(t)]. Thank you

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It does not seem that an explicit closed form answer exists. So think what do you really want know about this integral. –  Alexandre Eremenko Dec 15 '12 at 2:41

2 Answers 2

I'll take your first integral, to give you an indication of what types of closed-form expressions you can expect:

$${\cal I}(x)={\cal P}\int_{-1}^{1}\frac{\sqrt{1-t^2}\sin kt}{t-x}dt$$

For $x=0$ this evaluates to

$${\cal I}(0)=\frac{\pi}{2}\left(J_{1}(k)[-2+\pi k H_{0}(k)]+kJ_{0}(k)[2-\pi H_{1}(k)]\right)$$

where $J_n$ is a Bessel function and $H_n$ a Struve function.

For nonzero $x$ the best I (read: Mathematica) can do is a power series expansion around $x=0$, containing all even powers $x^p$. Each term in this series contains the Bessel and Struve functions $J_0(k),J_1(k),H_0(k),H_1(k)$, multiplied by polynomials in $k$ of order $p$. Not particularly useful, but at least higher order Bessel/Struve functions do not appear.

The power series can be written in a compact form in terms of the generalized hypergeometric function ${\cal F}$ $\equiv$ $_{p}F_q$,

$${\cal I}(x)={\cal I}(0)+\sum_{n=1}^{\infty}x^{2n}(-1)^n \frac{\pi k^{2n-1}}{(2n-1)!} {\cal F}\left(-\frac{1}{2};n,n+\frac{1}{2};-\frac{1}{4}k^2\right)$$ The same applies to your second integral, same Bessel and Struve functions in different combinations, similar series of hypergeometric functions.

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Thanks a lot, it was helpful. –  Ahmed Jan 20 '13 at 18:16

Wikipedia claims that integrals of the form $\hat{I}_n(k)=\int_{-1}^1 \frac{T_n(t) e^{ikt}}{\sqrt{1-t^2}} dt$ are proportional to $J_n(k)$, where $T_n(t)$ is the Chebyshev polynomial of the first kind and $J_n(k)$ is the Bessel function of the first kind. Unfortunately, at the moment, I'm having trouble verifying this formula, though Wikipedia cites Erdélyi's Tables of integral transforms, which I've not checked myself.

However, if the above is true, then what you are computing is the Fourier (or rather the Fourier-sine) transform of the product of $P\frac{1}{t-x}$ and a linear combination of $I_n(t) = \frac{T_n(t)}{\sqrt{1-t^2}}$. On the other hand, it is well known that the Fourier transform of $P\frac{1}{x-t}$ is proportional to $e^{ikx}\operatorname{sign}(k)$, so the integral you are looking for is a convolution of the last expression with a linear combination of Bessel functions $J_n(k)$. For $x=0$, the convolution can be written in terms of indefinite integrals of Bessel functions, which are themselves linear combinations of Bessel functions. For $x\ne 0$, you'll need the indefinite integrals of $e^{ikx}J_n(k)$. And for these, I don't know if one can find closed form expressions. Though, this formula may be of help.

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Thank you Mr Igor, I appreciate your help. –  Ahmed Jan 20 '13 at 18:18

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