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Fungrim entry: 026e44

0eatθ3 ⁣(x,ibt)dt=πabcosh ⁣((2x1)πab)sinh ⁣(πab)\int_{0}^{\infty} {e}^{-a t} \theta_{3}\!\left(x , i b t\right) \, dt = \sqrt{\frac{\pi}{a b}} \frac{\cosh\!\left(\left(2 x - 1\right) \sqrt{\frac{\pi a}{b}}\right)}{\sinh\!\left(\sqrt{\frac{\pi a}{b}}\right)}
Assumptions:aCandRe(a)>0andbCandRe(b)>0andx[0,1]a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(a) > 0 \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(b) > 0 \,\mathbin{\operatorname{and}}\, x \in \left[0, 1\right]
References:
  • https://doi.org/10.1016/0022-0728(88)87001-3
TeX:
\int_{0}^{\infty} {e}^{-a t} \theta_{3}\!\left(x , i b t\right) \, dt = \sqrt{\frac{\pi}{a b}} \frac{\cosh\!\left(\left(2 x - 1\right) \sqrt{\frac{\pi a}{b}}\right)}{\sinh\!\left(\sqrt{\frac{\pi a}{b}}\right)}

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(a) > 0 \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(b) > 0 \,\mathbin{\operatorname{and}}\, x \in \left[0, 1\right]
Definitions:
Fungrim symbol Notation Short description
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Expez{e}^{z} Exponential function
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
ConstIii Imaginary unit
Infinity\infty Positive infinity
Sqrtz\sqrt{z} Principal square root
Piπ\pi The constant pi (3.14...)
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
ClosedInterval[a,b]\left[a, b\right] Closed interval
Source code for this entry:
Entry(ID("026e44"),
    Formula(Equal(Integral(Mul(Exp(Mul(Neg(a), t)), JacobiTheta(3, x, Mul(Mul(ConstI, b), t))), For(t, 0, Infinity)), Mul(Sqrt(Div(Pi, Mul(a, b))), Div(Cosh(Mul(Sub(Mul(2, x), 1), Sqrt(Div(Mul(Pi, a), b)))), Sinh(Sqrt(Div(Mul(Pi, a), b))))))),
    Variables(a, b, x),
    Assumptions(And(Element(a, CC), Greater(Re(a), 0), Element(b, CC), Greater(Re(b), 0), Element(x, ClosedInterval(0, 1)))),
    References("https://doi.org/10.1016/0022-0728(88)87001-3"))

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2019-11-19 15:10:20.037976 UTC