Functions to compute the Jacobi elliptic functions, which are doubly periodic generalizations of trigonometric and hyperbolic functions.
Jacobi Elliptic SN, CN and DN Given:
$$u = \int_0^\phi\frac{d\psi}{\sqrt{1-k^2\text{sin}^2\psi}}$$
jacobi_sn(k, u): \(\text{sin}\psi\)jacobi_cn(k, u): \(\text{cos}\psi\)jacobi_dn(k, u): \(\sqrt{(1 - k^2\text{sin}^2\psi)}\)jacobi_cd(k, u): \(\frac{cn(k, u)}{dn(k, u)}\)jacobi_cs(k, u): \(\frac{cn(k, u)}{sn(k, u)}\)jacobi_dc(k, u): \(\frac{dn(k, u)}{cn(k, u)}\)jacobi_ds(k, u): \(\frac{dn(k, u)}{sn(k, u)}\)jacobi_nc(k, u): \(\frac{1}{cn(k, u)}\)jacobi_nd(k, u): \(\frac{1}{dn(k, u)}\)jacobi_ns(k, u): \(\frac{1}{sn(k, u)}\)jacobi_sc(k, u): \(\frac{sn(k, u)}{cn(k, u)}\)jacobi_sd(k, u): \(\frac{sn(k, u)}{dn(k, u)}\)
Usage
jacobi_elliptic(k, u)
jacobi_cd(k, u)
jacobi_cn(k, u)
jacobi_cs(k, u)
jacobi_dc(k, u)
jacobi_dn(k, u)
jacobi_ds(k, u)
jacobi_nc(k, u)
jacobi_nd(k, u)
jacobi_ns(k, u)
jacobi_sc(k, u)
jacobi_sd(k, u)
jacobi_sn(k, u)Value
For jacobi_elliptic, a list containing the values of the Jacobi elliptic functions: sn, cn, dn. For individual functions, a single numeric value is returned.
See also
Boost Documentation for more details on the mathematical background.
Examples
# All three principal Jacobi Elliptic Functions at once
k <- 0.5
u <- 2
jacobi_elliptic(k, u)
#> $sn
#> [1] 0.9628982
#>
#> $cn
#> [1] -0.269865
#>
#> $dn
#> [1] 0.8764741
#>
# Individual Jacobi Elliptic Functions
jacobi_cd(k, u)
#> [1] -0.3078984
jacobi_cn(k, u)
#> [1] -0.269865
jacobi_cs(k, u)
#> [1] -0.2802632
jacobi_dc(k, u)
#> [1] -3.247825
jacobi_dn(k, u)
#> [1] 0.8764741
jacobi_ds(k, u)
#> [1] 0.9102458
jacobi_nc(k, u)
#> [1] -3.705557
jacobi_nd(k, u)
#> [1] 1.140935
jacobi_ns(k, u)
#> [1] 1.038531
jacobi_sc(k, u)
#> [1] -3.568074
jacobi_sd(k, u)
#> [1] 1.098604
jacobi_sn(k, u)
#> [1] 0.9628982