# Bessel function with complex argument and index

let be a Bessel function $J_{u} (x)$

if both the index 'u' and the argument 'x' are complex

$$J_{ia}(ib)$$

for real 'a' and 'b' what is then the name for this function ??

if only the index is complex $J_{ia}(b)$ or the argumentis complex $J_{a}(ib)$

what is the name of this function ??

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For real $\alpha$ there is "modified Bessel function" $I_\alpha(x) = i^{-\alpha} J_\alpha(ix)$ –  Gerald Edgar Dec 18 '12 at 13:09