Fungrim home page

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\!\left(\tau\right)\right)\right)}^{2} \left(\lambda\!\left(\tau\right) - 1\right) \lambda\!\left(\tau\right)
Assumptions:τ{τ1+n:τ1Interior ⁣(Fλ)andnZ}\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
Derivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Derivative
ModularLambdaλ ⁣(τ)\lambda\!\left(\tau\right) Modular lambda function
ConstIii Imaginary unit
ConstPiπ\pi The constant pi (3.14...)
Powab{a}^{b} Power
EllipticKK ⁣(m)K\!\left(m\right) Complete elliptic integral of the first kind
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
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(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