Fungrim entry: 38b4f3

\lambda'(\tau) = -\frac{4 i}{\pi} {\left(K\!\left(\lambda(\tau)\right)\right)}^{2} \left(\lambda(\tau) - 1\right) \lambda(\tau)

Assumptions:

\tau \in \left\{ {\tau}_{1} + n : {\tau}_{1} \in \operatorname{Interior}(\mathcal{F}_{\lambda}) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \right\}

References:

http://functions.wolfram.com/EllipticFunctions/ModularLambda/20/01/0001/ Note: because of the branch cut of the elliptic integral, only valid on part of the domain.

TeX:

\lambda'(\tau) = -\frac{4 i}{\pi} {\left(K\!\left(\lambda(\tau)\right)\right)}^{2} \left(\lambda(\tau) - 1\right) \lambda(\tau)

\tau \in \left\{ {\tau}_{1} + n : {\tau}_{1} \in \operatorname{Interior}(\mathcal{F}_{\lambda}) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \right\}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ModularLambda`	$\lambda(\tau)$	Modular lambda function
`ConstI`	$i$	Imaginary unit
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`EllipticK`	$K(m)$	Legendre complete elliptic integral of the first kind
`ModularLambdaFundamentalDomain`	$\mathcal{F}_{\lambda}$	Fundamental domain of the modular lambda function
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("38b4f3"),
    Formula(Equal(ComplexDerivative(ModularLambda(tau), For(tau, tau)), Mul(Mul(Mul(Neg(Div(Mul(4, ConstI), Pi)), Pow(EllipticK(ModularLambda(tau)), 2)), Sub(ModularLambda(tau), 1)), ModularLambda(tau)))),
    Variables(tau),
    Assumptions(Element(tau, Set(Add(Subscript(tau, 1), n), For(Tuple(Subscript(tau, 1), n)), And(Element(Subscript(tau, 1), Interior(ModularLambdaFundamentalDomain)), Element(n, ZZ))))),
    References("http://functions.wolfram.com/EllipticFunctions/ModularLambda/20/01/0001/ Note: because of the branch cut of the elliptic integral, only valid on part of the domain."))

Topics using this entry

Modular lambda function