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Fungrim entry: 8a857c

0eatθ1 ⁣(x,ibt)dt=πabsinh ⁣(2xπab)cosh ⁣(πab)\int_{0}^{\infty} {e}^{-a t} \theta_{1}\!\left(x , i b t\right) \, dt = \sqrt{\frac{\pi}{a b}} \frac{\sinh\!\left(2 x \sqrt{\frac{\pi a}{b}}\right)}{\cosh\!\left(\sqrt{\frac{\pi a}{b}}\right)}
Assumptions:aC  and  Re(a)>0  and  bC  and  Re(b)>0  and  x[12,12]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[-\frac{1}{2}, \frac{1}{2}\right]
References:
  • https://doi.org/10.1016/0022-0728(88)87001-3
TeX:
\int_{0}^{\infty} {e}^{-a t} \theta_{1}\!\left(x , i b t\right) \, dt = \sqrt{\frac{\pi}{a b}} \frac{\sinh\!\left(2 x \sqrt{\frac{\pi a}{b}}\right)}{\cosh\!\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[-\frac{1}{2}, \frac{1}{2}\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("8a857c"),
    Formula(Equal(Integral(Mul(Exp(Mul(Neg(a), t)), JacobiTheta(1, x, Mul(Mul(ConstI, b), t))), For(t, 0, Infinity)), Mul(Sqrt(Div(Pi, Mul(a, b))), Div(Sinh(Mul(Mul(2, x), Sqrt(Div(Mul(Pi, a), b)))), Cosh(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(Neg(Div(1, 2)), Div(1, 2))))),
    References("https://doi.org/10.1016/0022-0728(88)87001-3"))

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2021-03-15 19:12:00.328586 UTC