# Fungrim entry: 38b4f3

$\lambda'(\tau) = -\frac{4 i}{\pi} {\left(K\!\left(\lambda\!\left(\tau\right)\right)\right)}^{2} \left(\lambda\!\left(\tau\right) - 1\right) \lambda\!\left(\tau\right)$
Assumptions:$\tau \in \left\{ {\tau}_{1} + n : {\tau}_{1} \in \operatorname{Interior}\!\left(\mathcal{F}_{\lambda}\right) \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}$
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\!\left(\tau\right)\right)\right)}^{2} \left(\lambda\!\left(\tau\right) - 1\right) \lambda\!\left(\tau\right)

\tau \in \left\{ {\tau}_{1} + n : {\tau}_{1} \in \operatorname{Interior}\!\left(\mathcal{F}_{\lambda}\right) \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}
Definitions:
Fungrim symbol Notation Short description
Derivative$\frac{d}{d z}\, f\!\left(z\right)$ Derivative
ModularLambda$\lambda\!\left(\tau\right)$ Modular lambda function
ConstI$i$ Imaginary unit
ConstPi$\pi$ The constant pi (3.14...)
Pow${a}^{b}$ Power
EllipticK$K\!\left(m\right)$ Complete elliptic integral of the first kind
SetBuilder$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$ Set comprehension
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(Derivative(ModularLambda(tau), tau, tau), Mul(Mul(Mul(Neg(Div(Mul(4, ConstI), ConstPi)), Pow(EllipticK(ModularLambda(tau)), 2)), Sub(ModularLambda(tau), 1)), ModularLambda(tau)))),
Variables(tau),
Assumptions(Element(tau, SetBuilder(Add(Subscript(tau, 1), n), 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

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

2019-09-22 15:43:45.410764 UTC