I'm not an expert with this type of functions and its transformations, but I would like to know how to work with them when they are in an integral. First of all I know the definition of Bessel functions and Legendre functions. And the relation between them, with the generating function:

$$e^{tx}J_0\left ( t\sqrt{1-x^2} \right )=\sum \frac {P_n(x)}{n!}t^n $$

The integrals that I don't know how to start to solve are:

$$\int_{0}^{\pi }e^{r\cos}~J_0(r\sin\theta )\sin\theta ~d\theta $$ And $$\int_{-1}^{1 }e^{ax}~J_0\left ( a\sqrt{1-x^2} \right )~P_m(x)~dx $$

If someone can explain how to work with integrals and summations in this cases I'd be really grateful. Thanks!!


1 Answer 1



Try using that $$\int_{-1}^{1}P_m(x)P_n(x) dx=\frac{2}{2n+1}\delta_{mn}$$

Where $\delta_{mn}$ represents the Kronecker delta alongwith the Generating functions you have at hand.

Also with the generating function you have, try substituting $x=\cos \theta $ in the first integral, and then use that $$\int_{-1}^{1}P_n(x)dx=0$$ $\forall$ $n\ge 1$ which is pretty evident from the property of Legendre polynomials that $$P_n(-x)=(-1)^nP_n(x)$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.