Fungrim entry: cac83e

J_{\nu}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \cos\!\left(\nu t - z \sin\!\left(t\right)\right) \, dt - \frac{\sin\!\left(\pi \nu\right)}{\pi} \int_{0}^{\infty} \exp\!\left(-z \sinh\!\left(t\right) - \nu t\right) \, dt

Assumptions:

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(z\right) \gt 0

TeX:

J_{\nu}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \cos\!\left(\nu t - z \sin\!\left(t\right)\right) \, dt - \frac{\sin\!\left(\pi \nu\right)}{\pi} \int_{0}^{\infty} \exp\!\left(-z \sinh\!\left(t\right) - \nu t\right) \, dt

\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(z\right) \gt 0

Definitions:

Fungrim symbol	Notation	Short description
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`ConstPi`	$\pi$	The constant pi (3.14...)
`Sin`	$\sin\!\left(z\right)$	Sine
`Exp`	${e}^{z}$	Exponential function
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part

Source code for this entry:

Entry(ID("cac83e"),
    Formula(Equal(BesselJ(nu, z), Sub(Mul(Div(1, ConstPi), Integral(Cos(Sub(Mul(nu, t), Mul(z, Sin(t)))), Tuple(t, 0, ConstPi))), Mul(Div(Sin(Mul(ConstPi, nu)), ConstPi), Integral(Exp(Sub(Neg(Mul(z, Sinh(t))), Mul(nu, t))), Tuple(t, 0, Infinity)))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, CC), Greater(Re(z), 0))))

Topics using this entry

Bessel functions