I have a integral function like below. $ J_0 $ is Bessel function and c is a constant.
$$ \int \left( J_0\left(c*x \right) \right)^2 dx $$
In the Mathematica software, it gives me the equation as the result.
$$ x * HypergeometricPFQ[\{ 1/2, 1/2 \}, \{1, 1, 3/2\}, -c^2 x^2] $$
This software does not present any steps for derivation.
Could you help me confirm these equation?
Here is the code
Integrate[BesselJ[0, c*x]^2, x]
Thank you
You have $$J_0(x)=\sum_{n=0}^\infty (-1)^n \frac{x^{2 n} }{2^{2 n}\,\Gamma (n+1)^2 }$$ $$\big[J_0(x)\big]^2=\frac 1 {\sqrt \pi}\sum_{n=0}^\infty (-1)^n\frac{ \Gamma \left(n+\frac{1}{2} \right)}{ \Gamma (n+1)^3}x^{2n}$$ $$\int \big[J_0(x)\big]^2\,dx=\frac 1 {\sqrt \pi}\sum_{n=0}^\infty (-1)^n\frac{ \Gamma \left(n+\frac{1}{2}\right)}{(2 n+1) \Gamma (n+1)^3} x^{2 n+1}$$ $$\int \big[J_0(x)\big]^2\,dx=x \, _2F_3\left(\frac{1}{2},\frac{1}{2};1,1,\frac{3}{2};-x^2\right)$$