 Integral representation of the modified Bessel function of the second kind

ID: integral-representation-of-the-modified-bessel-function-of-the-second-kind

Codex  0 Created 2026-10-03 Updated 2026-10-14

For real

x > 0

and

ν

, one useful representation is

K_{ν} (x) = \frac{1}{2} \int_{- \infty}^{\infty} e^{- x c o s h t + ν t} d t .

It turns large-order asymptotics into a Laplace integral with a moving maximum.

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