Fungrim entry: c29d6f

K_{\nu}\!\left(z\right) = \int_{0}^{\infty} \exp\!\left(-z \cosh(t)\right) \cosh\!\left(\nu t\right) \, dt

Assumptions:

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0

TeX:

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

Definitions:

Fungrim symbol	Notation	Short description
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified Bessel function of the second kind
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Exp`	${e}^{z}$	Exponential function
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("c29d6f"),
    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))))

Topics using this entry

Bessel functions