Calculating the integral of some bessel function

Question

How can I calculate$$\int \frac {x\left(J_{ \frac 34}(\frac {x^2}{2})-J_{- \frac 54}(\frac {x^2}{2})\right)}{2J_{- \frac 14}(\frac {x^2}{2})}dx~?$$

Answer 1

You have some recurrence formulas that you can maybe use in this case: $$J_{n+1}(x)-J_{n-1}(x)=-2J'_n(x)$$ Since $n+1=\frac 3 4\implies n=-\frac 14$ And $$n-1=-\frac 5 4$$ So you have : $$J_{3/4}(x)-J_{-5/4}(x)=-2J'_{-1/4}(x)$$ And for the integral: $$ \begin{align} I=&\int \frac {x\left(J_{ \frac 34}(\frac {x^2}{2})-J_{- \frac 54}(\frac {x^2}{2})\right)}{2J_{- \frac 14}(\frac {x^2}{2})}dx \\ I=&-\int \frac x{J_{- \frac 14}(\frac {x^2}{2})} \frac {dJ_{-\frac 14}(x^2/2)}{d \frac {x^2}2}dx\\ I=&-\int \frac {dJ_{- \frac 14}(\frac {x^2}{2})} {J_{-\frac 14}(\frac {x^2}{2})}\\ I=&-\ln \left ( J_{- \frac 14}(\frac {x^2}{2}) \right ) \end{align} $$