Prove that the Bessel function $J_n(x)$ satisfied $\int x J_0^2(x) dx = \frac{x^2}{2}[J_0^2(x)+J_1^2(x)]$

Question

Prove that the Bessel function $J_n(x)$ satisfied $\int x J_0^2(x) dx = \frac{x^2}{2}[J_0^2(x)+J_1^2(x)]$

I really don't know where to even begin. Any hints or ideas is greatly appreciated. Thank you!

Answer 1

Well, all we need are two formulas as below: $$\frac{d}{dx}J_0(x)=-J_1(x),$$ $$\frac{d}{dx}J_1(x)=J_0(x)-x^{-1}J_1(x).$$

Now, let's use integration by parts:$$\int xJ_0^2(x)=\frac{1}{2}x^2J_0^2(x)+\int x^2J_0(x)J_1(x).$$

So, we just need to prove: $$\int x^2J_0(x)J_1(x)=\frac{x^2}{2}J_1^2(x).$$

And, this is almost obvious since:

$$\frac{d}{dx}(\frac{x^2}{2}J_1^2(x))=xJ_1^2(x)+x^2J_1(x)(J_0(x)-x^{-1}J_1(x)).$$

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Edit: The two formulas I mentioned can be found in most typical advanced engineering-mathematics books, for example by Erwin Kreyszig.