I have to solve a kind of integral problem analytically : $$ \int_{-1}^{1}\frac{e^{-ik\sqrt{1-x}}}{\sqrt{1-x}}P_n(x)\text{d}x $$ where $P_n(x)$ is Legendre polynomial, $i$ is imaginary unit and $k$ is a constant. Given $n$, Mathematica can help me get the very answer.

But I want to make sure whether there are some other methods that can help get the general answer, i.e., not given $n$ before. Thank you in advance.


We have: $$\begin{eqnarray*} \int_{-1}^{1}\frac{e^{-ik\sqrt{1-x}}}{\sqrt{1-x}}\,P_n(x)\,dx &=& 2\sqrt{2}\int_{0}^{1}e^{-ik\sqrt{2}x} P_n(1-2x^2)\,dx\\&=&\sqrt{2}\int_{0}^{1}\frac{e^{-ik\sqrt{2x}}}{\sqrt{x}}P_n(1-2x)\,dx\end{eqnarray*}$$ hence we are essentially decomposing $\frac{e^{-ik\sqrt{2x}}}{\sqrt{x}}$ in terms of the shifted Legendre polynomials.

In order to compute the last integral, we can expand $\frac{e^{-ik\sqrt{2x}}}{\sqrt{x}}$ as a Taylor series, then exploit:

$$\begin{eqnarray*}\int_{0}^{1}x^{\alpha}P_n(2x-1)\,dx &=& \frac{1}{n!}\int_{0}^{1}x^{\alpha}\frac{d^n}{dx^n}(x^2-x)^n\,dx=\frac{\alpha!^2}{(\alpha+n+1)!(\alpha-n)!} \end{eqnarray*}$$ that follows from integration by parts.


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