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WolframAlpha tells me that $I_0(x)K_1(x)+I_1(x)K_0(x)=\frac{1}{x}$ for any $x$, but I have no idea how to prove it.

Should I use the recurrence relations for Bessel functions? Or should I use the series expansion around 0?

Is this identity widely known? If so, I would like a reference. It greatly simplifies the expression that appears in my physics application.

Yuriy S
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    This is the Wronskian/Cross product for modified Bessel functions, see e.g. http://dlmf.nist.gov/10.28 for the general case – gammatester Feb 03 '16 at 14:08

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