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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\!\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\}
  • Note: because of the branch cut of the elliptic integral, only valid on part of the domain.
\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\}
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:
    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)))),
    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(" 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.

2019-09-22 15:43:45.410764 UTC