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Fungrim entry: 348b26

j(τ)=2πiE6 ⁣(τ)E4 ⁣(τ)j(τ)j'(\tau) = -2 \pi i \frac{E_{6}\!\left(\tau\right)}{E_{4}\!\left(\tau\right)} j(\tau)
Assumptions:τH  and  E4 ⁣(τ)0\tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; E_{4}\!\left(\tau\right) \ne 0
TeX:
j'(\tau) = -2 \pi i \frac{E_{6}\!\left(\tau\right)}{E_{4}\!\left(\tau\right)} j(\tau)

\tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; E_{4}\!\left(\tau\right) \ne 0
Definitions:
Fungrim symbol Notation Short description
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
ModularJj(τ)j(\tau) Modular j-invariant
Piπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
EisensteinEEk ⁣(τ)E_{k}\!\left(\tau\right) Normalized Eisenstein series
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
Entry(ID("348b26"),
    Formula(Equal(ComplexDerivative(ModularJ(tau), For(tau, tau)), Mul(Mul(Neg(Mul(Mul(2, Pi), ConstI)), Div(EisensteinE(6, tau), EisensteinE(4, tau))), ModularJ(tau)))),
    Variables(tau),
    Assumptions(And(Element(tau, HH), NotEqual(EisensteinE(4, tau), 0))))

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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