Bessel Function of the First Kind for Imaginary Argument

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

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Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Special Functions: $\text {II}$. Bessel functions: $5$