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Fungrim entry: fff8ff

n=0Un ⁣(x)znn!=ezx(cosh ⁣(zx21)+zxsinc ⁣(izx21))\sum_{n=0}^{\infty} U_{n}\!\left(x\right) \frac{{z}^{n}}{n !} = {e}^{z x} \left(\cosh\!\left(z \sqrt{{x}^{2} - 1}\right) + z x \operatorname{sinc}\!\left(i z \sqrt{{x}^{2} - 1}\right)\right)
Assumptions:xCandzCx \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}
TeX:
\sum_{n=0}^{\infty} U_{n}\!\left(x\right) \frac{{z}^{n}}{n !} = {e}^{z x} \left(\cosh\!\left(z \sqrt{{x}^{2} - 1}\right) + z x \operatorname{sinc}\!\left(i z \sqrt{{x}^{2} - 1}\right)\right)

x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
ChebyshevUUn ⁣(x)U_{n}\!\left(x\right) Chebyshev polynomial of the second kind
Powab{a}^{b} Power
Factorialn!n ! Factorial
Infinity\infty Positive infinity
Expez{e}^{z} Exponential function
Sqrtz\sqrt{z} Principal square root
ConstIii Imaginary unit
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("fff8ff"),
    Formula(Equal(Sum(Mul(ChebyshevU(n, x), Div(Pow(z, n), Factorial(n))), Tuple(n, 0, Infinity)), Mul(Exp(Mul(z, x)), Add(Cosh(Mul(z, Sqrt(Sub(Pow(x, 2), 1)))), Mul(Mul(z, x), Sinc(Mul(Mul(ConstI, z), Sqrt(Sub(Pow(x, 2), 1))))))))),
    Variables(x, z),
    Assumptions(And(Element(x, CC), Element(z, CC))))

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

2019-09-19 20:12:49.583742 UTC