Fungrim entry: 348b26

j'(\tau) = -2 \pi i \frac{E_{6}\!\left(\tau\right)}{E_{4}\!\left(\tau\right)} j(\tau)

Assumptions:

\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
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ModularJ`	$j(\tau)$	Modular j-invariant
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`HH`	$\mathbb{H}$	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))))

Topics using this entry

Modular j-invariant