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Fungrim entry: 38b4f3

λ(τ)=4iπ(K ⁣(λ(τ)))2(λ(τ)1)λ(τ)\lambda'(\tau) = -\frac{4 i}{\pi} {\left(K\!\left(\lambda(\tau)\right)\right)}^{2} \left(\lambda(\tau) - 1\right) \lambda(\tau)
Assumptions:τ{τ1+n:τ1Interior(Fλ)andnZ}\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
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
ModularLambdaλ(τ)\lambda(\tau) Modular lambda function
ConstIii Imaginary unit
Piπ\pi The constant pi (3.14...)
Powab{a}^{b} Power
EllipticKK(m)K(m) Complete elliptic integral of the first kind
ModularLambdaFundamentalDomainFλ\mathcal{F}_{\lambda} Fundamental domain of the modular lambda function
ZZZ\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."))

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

2020-01-31 18:09:28.494564 UTC