JacobiElliptic API
Index
JacobiElliptic.CarlsonAlg.amJacobiElliptic.CarlsonAlg.ellipkeJacobiElliptic.FukushimaAlg.EJacobiElliptic.FukushimaAlg.FJacobiElliptic.FukushimaAlg.JJacobiElliptic.FukushimaAlg.KJacobiElliptic.FukushimaAlg.PiJacobiElliptic.FukushimaAlg.acnJacobiElliptic.FukushimaAlg.asnJacobiElliptic.FukushimaAlg.cnJacobiElliptic.FukushimaAlg.dnJacobiElliptic.FukushimaAlg.sn
Elliptic Integrals
JacobiElliptic.FukushimaAlg.K — Function$K(m) = \int_0^{\pi/2}\frac{d\theta}{\sqrt{1-k^2\sin(\theta)^2}}.$
Returns the complete elliptic integral of the first kind.
Arguments
m: Elliptic modulus
JacobiElliptic.FukushimaAlg.E — Function$E(m) = \int_0^{\pi/2}\sqrt{1-k^2\sin(\theta)^2}d\theta.$
Returns the complete elliptic integral of the second kind.
Arguments
m: Elliptic modulus
$E(\varphi|\, m) = \int_0^{\varphi}d\theta \sqrt{1-m\sin(\theta)^2}.$
Returns the incomplete elliptic integral of the second kind.
Arguments
φ: Amplitudem: Elliptic modulus
JacobiElliptic.FukushimaAlg.F — Function$F(\varphi |\, m) = \int_0^{\varphi}\frac{d\theta}{\sqrt{1-m\sin(\theta)^2}}.$
Returns the incomplete elliptic integral of the first kind.
Arguments
φ: Amplitudem: Elliptic modulus
JacobiElliptic.FukushimaAlg.Pi — Function$\Pi(n|\,m)=\int_{0}^{1 }{\frac{1}{1-nt^{2}}}{\frac{dt}{\sqrt{\left(1-mt^{2}\right)\left(1-t^{2}\right)}}}.$
Returns the complete elliptic integral of the third kind.
Arguments
n: Characteristicm: Elliptic modulus
$\Pi (n;\varphi \,|\,m)=\int _{0}^{\sin \varphi }{\frac {1}{1-nt^{2}}}{\frac {dt}{\sqrt {\left(1-mt^{2}\right)\left(1-t^{2}\right)}}}.$
Returns the incomplete elliptic integral of the third kind.
Arguments
n: Characteristicφ: Amplitudem: Elliptic modulus
JacobiElliptic.FukushimaAlg.J — Function$J (n;\varphi \,|\,m)=\frac{\Pi(n;\varphi|\, m) - F(\varphi|\,m)}{n}.$
Returns the associate incomplete elliptic integral of the third kind.
Arguments
n: Characteristicφ: Amplitudem: Elliptic modulus
$J(n;\varphi \,|\,m)=\frac{\Pi(n;\pi/2|\, m) - K(m)}{n}.$
Returns the associate complete elliptic integral of the third kind.
Arguments
n: Characteristicm: Elliptic modulus
Jacobi Elliptic Functions
JacobiElliptic.FukushimaAlg.sn — Function$\text{sn}(u|\,m)=\sin\,\text{am}(u,m)$, where $\text{am}(u|\,m)=F^{-1}(u|\,m)$ is the Jacobi amplitude.
Returns the Jacobi Elliptic sn.
Arguments
u: Amplitudem: Elliptic modulus
JacobiElliptic.FukushimaAlg.cn — Function$\text{cn}(u|\,m)=\cos\,\text{am}(u,m)$, where $\text{am}(u|\,m)=F^{-1}(u|\,m)$ is the Jacobi amplitude.
Returns the Jacobi Elliptic cn.
Arguments
u: Amplitudem: Elliptic modulus
JacobiElliptic.FukushimaAlg.dn — Function$\text{dn}(u|\,m)=\frac{d}{d u}\,\text{am}(u,m)$, where $\text{am}(u|\,m)=F^{-1}(u|\,m)$ is the Jacobi amplitude.
Returns the Jacobi Elliptic dn.
Arguments
u: Amplitudem: Elliptic modulus
JacobiElliptic.FukushimaAlg.asn — Function$\text{asn}(u|\,m)=\text{sn}^{-1}(u,m)$.
Returns the inverse Jacobi Elliptic sn.
Arguments
u: Amplitudem: Elliptic modulus
JacobiElliptic.FukushimaAlg.acn — Function$\text{acn}(u|\,m)=\text{cn}^{-1}(u,m)$.
Returns the inverse Jacobi Elliptic cn.
Arguments
u: Amplitudem: Elliptic modulus
JacobiElliptic.CarlsonAlg.am — Functionam(u::Real, m::Real, [tol::Real=eps(Float64)])Returns amplitude, φ, such that u = F(φ | m)
Landen sequence with convergence to tol used if √(tol) ≤ m ≤ 1 - √(tol)
JacobiElliptic.CarlsonAlg.ellipke — Functionellipke(m::Real) returns (K(m), E(m)) for scalar 0 ≤ m ≤ 1