How to evaluate the integral $\int_{1}^{\infty} x^{-5/3} \cos\left((x-1) \tau\right) dx$

Question

I would like to evaluate the following integral: $$\int_{1}^{\infty} x^{-5/3} \cos\left((x-1) \tau\right) dx$$ I get the Integral by Maple and it gives the Lommel function. After that, I will search an asymptotic as $\tau$ goes to $\infty$.

Answer 1

Related problems (I). Here is how you can evaluate the integral. Making the change of variables $u=x-1$ gives

$$I = \int_{0}^{\infty}\frac{\cos(\tau u)}{(1+u)^{5/3}}du. $$

Using the Taylor series of $\cos(\tau u)$ we get

$$ I = \sum_{k=0}^{\infty}\frac{(-1)^k \tau^{2k} }{(2k)!} \int_{0}^{\infty}\frac{u^{2k}}{(1+u)^{5/3}}du. $$

Making the change of variables $1+u=\frac{1}{t}$ casts the above integral in terms of the $\beta$ function and then resum the series and the result will follow in terms of Lommel function.

$$I= \frac{3}{2}-{\frac {27}{8}}\,{x}^{2}+{\frac {27}{8}}\,{x}^{\frac{7}{6}}s_{\frac{11}{6},\frac{1}{2}}( x ). $$

Another form for the solution can be obtained in terms of the hypergeometric function

$$ I = \frac{3}{2} {}_1F_2\left( 1;\frac{1}{6},\frac{2}{3}; -\frac{x^4}{4} \right) . $$