Fungrim home page

Fungrim entry: cac83e

Jν ⁣(z)=1π0πcos ⁣(νtzsin(t))dtsin ⁣(πν)π0exp ⁣(zsinh(t)νt)dtJ_{\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:νCandzCandRe(z)>0\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
BesselJJν ⁣(z)J_{\nu}\!\left(z\right) Bessel function of the first kind
Piπ\pi The constant pi (3.14...)
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Sinsin(z)\sin(z) Sine
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:
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

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

2019-11-19 15:10:20.037976 UTC