Integral $\int_0^\infty \frac{\sqrt{\sqrt{\alpha^2+x^2}-\alpha}\,\exp\big({-\beta\sqrt{\alpha^2+x^2}\big)}}{\sqrt{\alpha^2+x^2}}\sin (\gamma x)\,dx$

I am having trouble showing this equality is true$$
\int_0^\infty \frac{\sqrt{\sqrt{\alpha^2+x^2}-\alpha}\,\exp\big({-\beta\sqrt{\alpha^2+x^2}\big)}}{\sqrt{\alpha^2+x^2}}\sin (\gamma x)\,dx=\sqrt\frac{\pi}{2}\frac{\gamma \exp\big(-\alpha\sqrt{\gamma^2+\beta^2}\big)}{\sqrt{\beta^2+\gamma^2}\sqrt{\beta+\sqrt{\beta^2+\gamma^2}}},
$$
$$
\mathcal{Re}(\alpha,\beta,\gamma> 0).
$$
I do not know how to approach it because of all the square root functions.

It seems if $x=\pm i\alpha \ $ we may have some convergence problems because of the denominator. Perhaps there are ways to solve this using complex methods involving the branch cut from the square root singularity. I just do not know what to choose $f(z)$ for a suitable complex function to represent the integrand.

I also tried differentiating under the integral signs w.r.t $\alpha,\beta,\gamma$ but it did not simplify anything. Thanks. How can we calculate this integral?

Answer

Replace $\alpha$, $\beta$ and $\gamma$ with $a$, $b$ and $c$ respectively.

With the substitution $x=a\sinh t$, the integral can be written as:
$$\begin{aligned}
I & = \sqrt{2a}\int_0^{\infty} e^{-ab\cosh t}\sin(ac\sinh t)\sinh \left(\frac{t}{2}\right)\,dt \\
&=-\sqrt{2a}\Im\left(\int_0^{\infty} e^{-ab\cosh t}\cos\left(ac\sinh t+\frac{it}{2}\right)\,dt \right)
\end{aligned}$$

Thanks to sir O.L. for evaluating the final integral here: Integral: $\int_0^{\infty} e^{-ab\cosh x}\cos\left(ac\sinh(x)+\frac{ix}{2}\right)\,dx$

The result is hence,
$$\begin{aligned}
I & = -\sqrt{2a}\Im\left(e^{-\frac{i}{2}\arctan\frac{c}{b}}\sqrt{\frac{\pi}{2a\sqrt{b^2+c^2}}}e^{-a\sqrt{b^2+c^2}}\right) \\
&=\sqrt{2a}\sqrt{\frac{\pi}{2a\sqrt{b^2+c^2}}}e^{-a\sqrt{b^2+c^2}} \sin\left(\frac{1}{2}\arctan\frac{c}{b}\right) \\
&=\sqrt{\frac{\pi}{2}}\sqrt{\frac{1}{\sqrt{b^2+c^2}}}e^{-a\sqrt{b^2+c^2}}\frac{c}{\sqrt{\left(\sqrt{ b^2+c^2}+b \right)\sqrt{b^2+c^2}}} \\
&=\boxed{\sqrt{\dfrac{\pi}{2}}\dfrac{c\exp\left(-a\sqrt{b^2+c^2}\right)}{\sqrt{b^2+c^2}\sqrt{\sqrt{b^2+c^2}+b}}}
\end{aligned}$$
I used the following to simplify the above expression
$$\begin{aligned}
\sin\left(\frac{1}{2}\arctan\frac{c}{b}\right) &=\sqrt{\frac{1-\cos\left(\arctan\frac{c}{b}\right)}{2}}\\
&= \frac{1}{\sqrt{2}}\sqrt{1-\frac{b}{\sqrt{b^2+c^2}}}\\
&= \frac{1}{\sqrt{2}}\sqrt{\frac{\sqrt{b^2+c^2}-b}{\sqrt{b^2+c^2}}}=\frac{1}{\sqrt{2}}\frac{c}{\sqrt{\left(\sqrt{ b^2+c^2}+b \right)\sqrt{b^2+c^2}}}
\end{aligned}$$

Attribution
Source : Link , Question Author : Jeff Faraci , Answer Author : Community

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