elliptic-package: Weierstrass and Jacobi Elliptic Functions

In elliptic: Weierstrass and Jacobi Elliptic Functions

Description

A suite of elliptic and related functions including Weierstrass and Jacobi forms. Also includes various tools for manipulating and visualizing complex functions.

Details

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The primary function in package elliptic is P(): this calculates the Weierstrass P function, and may take named arguments that specify either the invariants g or half periods Omega. The derivative is given by function Pdash and the Weierstrass sigma and zeta functions are given by functions sigma() and zeta() respectively; these are documented in ?P. Jacobi forms are documented under ?sn and modular forms under ?J.

Notation follows Abramowitz and Stegun (1965) where possible, although there only real invariants are considered; ?e1e2e3 and ?parameters give a more detailed discussion. Various equations from AMS-55 are implemented (for fun); the functions are named after their equation numbers in AMS-55; all references are to this work unless otherwise indicated.

The package uses Jacobi's theta functions (?theta and ?theta.neville) where possible: they converge very quickly.

Various number-theoretic functions that are required for (eg) converting a period pair to primitive form (?as.primitive) are implemented; see ?divisor for a list.

The package also provides some tools for numerical verification of complex analysis such as contour integration (?myintegrate) and Newton-Raphson iteration for complex functions (?newton_raphson).

Complex functions may be visualized using view(); this is customizable but has an extensive set of built-in colourmaps.

Author(s)

Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000-0001-5982-0415>)

Maintainer: Robin K. S. Hankin <[e-mail]>

References

• R. K. S. Hankin. Introducing Elliptic, an R package for Elliptic and Modular Functions. Journal of Statistical Software, Volume 15, Issue 7. February 2006.

• M. Abramowitz and I. A. Stegun 1965. Handbook of Mathematical Functions. New York, Dover.

• K. Chandrasekharan 1985. Elliptic functions, Springer-Verlag.

• E. T. Whittaker and G. N. Watson 1952. A Course of Modern Analysis, Cambridge University Press (fourth edition)

• G. H. Hardy and E. M. Wright 1985. An introduction to the theory of numbers, Oxford University Press (fifth edition)

• S. D. Panteliou and A. D. Dimarogonas and I. N .Katz 1996. Direct and inverse interpolation for Jacobian elliptic functions, zeta function of Jacobi and complete elliptic integrals of the second kind. Computers and Mathematics with Applications, volume 32, number 8, pp51-57

• E. L. Wachspress 2000. Evaluating Elliptic functions and their inverses. Computers and Mathematics with Applications, volume 29, pp131-136

• D. G. Vyridis and S. D. Panteliou and I. N. Katz 1999. An inverse convergence approach for arguments of Jacobian elliptic functions. Computers and Mathematics with Applications, volume 37, pp21-26

• S. Paszkowski 1997. Fast convergent quasipower series for some elementary and special functions. Computers and Mathematics with Applications, volume 33, number 1/2, pp181-191

• B. Thaller 1998. Visualization of complex functions, The Mathematica Journal, 7(2):163–180

• J. Kotus and M. Urb\'anski 2003. Hausdorff dimension and Hausdorff measures of Julia sets of elliptic functions. Bulletin of the London Mathematical Society, volume 35, pp269-275

Examples

     ## Example 8, p666, RHS:
 P(z=0.07 + 0.1i, g=c(10,2)) 

     ## Now a nice little plot of the zeta function:
 x <- seq(from=-4,to=4,len=100)
 z <- outer(x,1i*x,"+")
 par(pty="s")
 view(x,x,limit(zeta(z,c(1+1i,2-3i))),nlevels=3,scheme=1)
 view(x,x,P(z*3,params=equianharmonic()),real=FALSE)

     ## Some number theory:
 mobius(1:10)
 plot(divisor(1:300,k=1),type="s",xlab="n",ylab="divisor(n,1)")

    ## Primitive periods:
 as.primitive(c(3+4.01i , 7+10i))
 as.primitive(c(3+4.01i , 7+10i),n=10)   # Note difference

    ## Now some contour integration:
 f <- function(z){1/z}
 u <- function(x){exp(2i*pi*x)}
 udash <- function(x){2i*pi*exp(2i*pi*x)}
 integrate.contour(f,u,udash) - 2*pi*1i


 x <- seq(from=-10,to=10,len=200)
 z <- outer(x,1i*x,"+")
 view(x,x,P(z,params=lemniscatic()),real=FALSE)
 view(x,x,P(z,params=pseudolemniscatic()),real=FALSE)
 view(x,x,P(z,params=equianharmonic()),real=FALSE)

Example output

Attaching package: 'elliptic'

The following objects are masked from 'package:stats':

    sd, sigma

The following object is masked from 'package:base':

    is.primitive

[1] -22.9745-63.05323i
 [1]  1 -1 -1  0 -1  1 -1  0  0  1
[1]  1+1.98i -2-2.03i
attr(,"class")
[1] "primitive"
[1] 1+0.05i 0+1.93i
attr(,"class")
[1] "primitive"
[1] -3.56164e-17-8.881784e-16i

elliptic documentation built on May 2, 2019, 9:37 a.m.