Bessel Function of the First Kind for Imaginary Argument

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Theorem

Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$.


Then:

$\map {J_n } {i x} = i^{-n} \, \map {I_n} x$

where:

$i$ denotes the imaginary unit
$\map {I_n} x$ denotes the modified Bessel function of the first kind of order $n$.


Proof


Sources