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Fungrim entry: 7ae3ed

Iν ⁣(z)=1π0πexp ⁣(zcos ⁣(t))cos ⁣(νt)dtsin ⁣(πν)π0exp ⁣(zcosh ⁣(t)νt)dtI_{\nu}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \exp\!\left(z \cos\!\left(t\right)\right) \cos\!\left(\nu t\right) \, dt - \frac{\sin\!\left(\pi \nu\right)}{\pi} \int_{0}^{\infty} \exp\!\left(-z \cosh\!\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:
I_{\nu}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \exp\!\left(z \cos\!\left(t\right)\right) \cos\!\left(\nu t\right) \, dt - \frac{\sin\!\left(\pi \nu\right)}{\pi} \int_{0}^{\infty} \exp\!\left(-z \cosh\!\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
BesselIIν ⁣(z)I_{\nu}\!\left(z\right) Modified 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
Expez{e}^{z} Exponential function
Sinsin ⁣(z)\sin\!\left(z\right) Sine
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("7ae3ed"),
    Formula(Equal(BesselI(nu, z), Sub(Mul(Div(1, ConstPi), Integral(Mul(Exp(Mul(z, Cos(t))), Cos(Mul(nu, t))), Tuple(t, 0, ConstPi))), Mul(Div(Sin(Mul(ConstPi, nu)), ConstPi), Integral(Exp(Sub(Neg(Mul(z, Cosh(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.

2019-08-21 11:44:15.926409 UTC