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Fungrim entry: c29d6f

Kν ⁣(z)=0exp ⁣(zcosh(t))cosh ⁣(νt)dtK_{\nu}\!\left(z\right) = \int_{0}^{\infty} \exp\!\left(-z \cosh(t)\right) \cosh\!\left(\nu t\right) \, dt
Assumptions:νC  and  zC  and  Re(z)>0\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0
K_{\nu}\!\left(z\right) = \int_{0}^{\infty} \exp\!\left(-z \cosh(t)\right) \cosh\!\left(\nu t\right) \, dt

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0
Fungrim symbol Notation Short description
BesselKKν ⁣(z)K_{\nu}\!\left(z\right) Modified Bessel function of the second kind
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Expez{e}^{z} Exponential function
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
Source code for this entry:
    Formula(Equal(BesselK(nu, z), Integral(Mul(Exp(Neg(Mul(z, Cosh(t)))), Cosh(Mul(nu, t))), For(t, 0, Infinity)))),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, CC), Greater(Re(z), 0))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC