K_{ν} (x) = \frac{1}{2} \int_{- \infty}^{\infty} exp {- x cosh t + ν t} d t = \frac{1}{2} \int_{- \infty}^{\infty} e^{ν h (t)} d t, h (t) = t - \frac{x}{ν} cosh t .

The unique stationary point is the strict maximum

t_{0} = arsinh \frac{ν}{x}, ν h (t_{0}) = ν arsinh \frac{ν}{x} - ν^{2} + x^{2}, ν h^{''} (t_{0}) = - ν^{2} + x^{2} .

K_{ν} (x) \sim \frac{1}{2} (\frac{2 π}{ν ^{2} + x ^{2}})^{1/2} exp {ν arsinh \frac{ν}{x} - ν^{2} + x^{2}} .

For fixed

x > 0

ν \to + \infty

arsinh \frac{ν}{x} = lo g \frac{2 ν}{x} + O (ν^{- 2}), ν^{2} + x^{2} = ν + O (ν^{- 1}) .

K_{ν} (x) \sim (\frac{π}{2 ν})^{1/2} (\frac{2 ν}{e x})^{ν} .

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