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Fungrim entry: 99c077

Jn ⁣(z)=1π0πcos ⁣(ntzsin ⁣(t))dtJ_{n}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \cos\!\left(n t - z \sin\!\left(t\right)\right) \, dt
Assumptions:nZandzCn \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}
TeX:
J_{n}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \cos\!\left(n t - z \sin\!\left(t\right)\right) \, dt

n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}
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
ZZZ\mathbb{Z} Integers
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("99c077"),
    Formula(Equal(BesselJ(n, z), Mul(Div(1, ConstPi), Integral(Cos(Sub(Mul(n, t), Mul(z, Sin(t)))), Tuple(t, 0, ConstPi))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZ), Element(z, CC))))

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2019-08-21 11:44:15.926409 UTC