Integral of the Von karman equation
What is the result of this integral, and how can I proceed:
$$ \int_{-\infty}^{\infty}{c_{1} \over\left(1 + c_{2}\,x^{2}\right)^{5/6}}\, \cos\left(x\tau\right)\,{\rm d}x\,,\qquad c_{1}, c_{2}\mbox{: positive constants.} $$
I see in one book that the result contains the ${\tt 2_{\rm nd}\ \mbox{type Bessel function}}$.
Solution 1:
Related problems (I). Follow the steps
1) the integrand is even so you can integrate on the interval $(0,\infty)$
2) expand the function $\cos(x)$ in terms of its Taylor series
3) use the beta function to evaluate the integral and then resum.
Added: Here is a closed form for a more general integral
$$ c_1\int_{-\infty}^{\infty}{\frac{\cos\left(x\tau\right)}{\left(1 + c_{2}\,x^{2}\right)^{\alpha}}}\, \,{\rm d}x = \frac{ c_{{1}}{c_{{2}}}^{-1/4-1/2\,\alpha}{\pi}^{3/2} \,{\tau}^{\alpha-1/2}{2}^{1/2-\alpha} }{\Gamma \left(\alpha\right) \cos \left( \pi \,\alpha \right) } {{\rm I}_{1/2-\alpha}\left({\frac { {\tau}}{\sqrt {c_{{2}}}}}\right)} $$
in terms of the modified Bessel function of the first kind. Pay attention for what values of $\alpha$ the above formula make sense.
Solution 2:
$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ $\ds{\int_{-\infty}^{\infty}{c_{1} \over\pars{1 + c_{2}\,x^{2}}^{5/6}}\, \cos\pars{x\tau}\,\dd x\,,\qquad c_{1}, c_{2}\mbox{: positive constants.}}$
\begin{align}&\int_{-\infty}^{\infty}{c_{1} \over\pars{1 + c_{2}\,x^{2}}^{5/6}}\, \cos\pars{x\tau}\,\dd x =2c_{1}\int_{0}^{\infty}{\cos\pars{x\verts{\tau}} \over\pars{1 + c_{2}\,x^{2}}^{5/6}}\, \,\dd x \\[3mm]&={2c_{1} \over c_{2}^{5/6}}\int_{0}^{\infty} {\cos\pars{x\color{#c00000}{\verts{\tau}}}\over \bracks{x^{2} + \pars{\color{#c00000}{c_{2}^{-1/2}}}^{2}}^{\color{#c00000}{1/3} + 1/2}}\,\dd x \end{align}
With the Bessel Function Identity ${\bf 9.6.25}$: $$ {\rm K}_{\nu}\pars{xz} ={\Gamma\pars{\nu + 1/2}\pars{2z}^{\nu} \over \root{\pi}x^{\nu}} \int_{0}^{\infty}{\cos\pars{xt} \over \pars{t^{2} + z^{2}}^{\nu + 1/2}}\,\dd t $$ we'll have \begin{align} &\int_{-\infty}^{\infty}{\cos\pars{x\tau} \over\pars{1 + c_{2}\,x^{2}}^{5/6}}\, \,\dd x ={\root{\pi}\verts{\tau}^{1/3} \over \Gamma\pars{1/3 + 1/2}\pars{2/\root{c_{2}}}^{1/3}}\, {\rm K}_{1/3}\pars{\verts{\tau}\,{1 \over \root{c_{2}}}} \end{align}
\begin{align}&\color{#66f}{\large% \int_{-\infty}^{\infty}{c_{1} \over\pars{1 + c_{2}\,x^{2}}^{5/6}}\, \cos\pars{x\tau}\,\dd x} \\[3mm]&=\color{#66f}{\large{c_{1}\root{\pi}\over \Gamma\pars{5/6}}\,\pars{\verts{\tau}\root{c_{2}} \over 2}^{1/3} \,{\rm K}_{1/3}\pars{\verts{\tau} \over \root{c_{2}}}} \end{align}