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Fungrim entry: e2288d

θ2 ⁣(0,1+yi)=1+i2yθ3 ⁣(0,1+iy)\theta_{2}\!\left(0 , 1 + y i\right) = \frac{1 + i}{\sqrt{2 y}} \theta_{3}\!\left(0 , 1 + \frac{i}{y}\right)
Assumptions:y∈(0,∞)y \in \left(0, \infty\right)
TeX:
\theta_{2}\!\left(0 , 1 + y i\right) = \frac{1 + i}{\sqrt{2 y}} \theta_{3}\!\left(0 , 1 + \frac{i}{y}\right)

y \in \left(0, \infty\right)
Definitions:
Fungrim symbol Notation Short description
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
ConstIii Imaginary unit
Sqrtz\sqrt{z} Principal square root
OpenInterval(a,b)\left(a, b\right) Open interval
Infinity∞\infty Positive infinity
Source code for this entry:
Entry(ID("e2288d"),
    Formula(Equal(JacobiTheta(2, 0, Add(1, Mul(y, ConstI))), Mul(Div(Add(1, ConstI), Sqrt(Mul(2, y))), JacobiTheta(3, 0, Add(1, Div(ConstI, y)))))),
    Variables(y),
    Assumptions(Element(y, OpenInterval(0, Infinity))))

Topics using this entry

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

2021-03-15 19:12:00.328586 UTC