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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\!\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:νCandzCandRe ⁣(z)>0\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(z\right) > 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) > 0
Definitions:
Fungrim symbol Notation Short description
BesselJJν ⁣(z)J_{\nu}\!\left(z\right) Bessel function of the first kind
ConstPiπ\pi The constant pi (3.14...)
Integralabf ⁣(x)dx\int_{a}^{b} f\!\left(x\right) \, dx Integral
Sinsin ⁣(z)\sin\!\left(z\right) Sine
Expez{e}^{z} Exponential function
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
ReRe ⁣(z)\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))))

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

2019-08-25 15:30:03.056001 UTC