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I'm physics student and not very good at proof. My mathematical physics textbook states the orthogonality of Bessel functions, http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

basically same as the equation (53) in the linked page.

However, textbook says that there is a condition that $\nu>-1$ but doens't explain why. Wikipedia also states the same equation and condition but doesn't explain why. I can guess that it's related to the convergence of the integral but I'm not sure how to show it.

Can somebody help me?

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1 Answer 1

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Use the limiting form $J_{\nu}(az) \sim \left(\frac{az}{2}\right)^{\nu+1}/\Gamma(\nu+1).$ Then for $z$ near $0$ the integrand behaves like

$$z J_{\nu}(az)J_{\nu}(bz) \sim f(\nu,a,b) \left(\frac{z}{2}\right)^{2\nu+1}$$

where $f(\nu,a,b)$ is independent of $z$.

Therefore the integral converges if $2\nu + 1 > -1 $ or $\nu >-1.$

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