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Fungrim entry: 37e644

drdτrθj(s) ⁣(z,τ)=1(4πi)rθj(2r+s) ⁣(z,τ)\frac{d^{r}}{{d \tau}^{r}} \theta^{(s)}_{j}\!\left(z , \tau\right) = \frac{1}{{\left(4 \pi i\right)}^{r}} \theta^{(2 r + s)}_{j}\!\left(z , \tau\right)
Assumptions:j{1,2,3,4}andzCandτHandrZ0sZ0j \in \left\{1, 2, 3, 4\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0} \in s \in \mathbb{Z}_{\ge 0}
TeX:
\frac{d^{r}}{{d \tau}^{r}} \theta^{(s)}_{j}\!\left(z , \tau\right) = \frac{1}{{\left(4 \pi i\right)}^{r}} \theta^{(2 r + s)}_{j}\!\left(z , \tau\right)

j \in \left\{1, 2, 3, 4\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0} \in s \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Powab{a}^{b} Power
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("37e644"),
    Formula(Equal(ComplexDerivative(JacobiTheta(j, z, tau, s), tau, tau, r), Mul(Div(1, Pow(Mul(Mul(4, ConstPi), ConstI), r)), JacobiTheta(j, z, tau, Add(Mul(2, r), s))))),
    Variables(j, z, tau, r, s),
    Assumptions(And(Element(j, Set(1, 2, 3, 4)), Element(z, CC), Element(tau, HH), Element(r, ZZGreaterEqual(0), Element(s, ZZGreaterEqual(0))))))

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

2019-09-16 21:17:18.797188 UTC