Fungrim entry: cac83e

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

Assumptions:

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

TeX:

J_{\nu}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \cos\!\left(\nu t - z \sin(t)\right) \, dt - \frac{\sin\!\left(\pi \nu\right)}{\pi} \int_{0}^{\infty} \exp\!\left(-z \sinh(t) - \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
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`Pi`	$\pi$	The constant pi (3.14...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Cos`	$\cos(z)$	Cosine
`Sin`	$\sin(z)$	Sine
`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("cac83e"),
    Formula(Equal(BesselJ(nu, z), Sub(Mul(Div(1, Pi), Integral(Cos(Sub(Mul(nu, t), Mul(z, Sin(t)))), For(t, 0, Pi))), Mul(Div(Sin(Mul(Pi, nu)), Pi), Integral(Exp(Sub(Neg(Mul(z, Sinh(t))), 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