Description Details Author(s) References Examples
A suite of elliptic and related functions including Weierstrass and Jacobi forms. Also includes various tools for manipulating and visualizing complex functions.
The DESCRIPTION file:
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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 AMS55 are implemented (for fun); the functions are named after
their equation numbers in AMS55; 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 numbertheoretic 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
NewtonRaphson iteration for complex functions
(?newton_raphson
).
Complex functions may be visualized using view()
; this is
customizable but has an extensive set of builtin colourmaps.
Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000000159820415>)
Maintainer: Robin K. S. Hankin <hankin.robin@gmail.com>
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, SpringerVerlag.
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, pp5157
E. L. Wachspress 2000. Evaluating Elliptic functions and their inverses. Computers and Mathematics with Applications, volume 29, pp131136
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, pp2126
S. Paszkowski 1997. Fast convergent quasipower series for some elementary and special functions. Computers and Mathematics with Applications, volume 33, number 1/2, pp181191
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, pp269275
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30  ## 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,23i))),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)

Attaching package: 'elliptic'
The following objects are masked from 'package:stats':
sd, sigma
The following object is masked from 'package:base':
is.primitive
[1] 22.974563.05323i
[1] 1 1 1 0 1 1 1 0 0 1
[1] 1+1.98i 22.03i
attr(,"class")
[1] "primitive"
[1] 1+0.05i 0+1.93i
attr(,"class")
[1] "primitive"
[1] 3.56164e178.881784e16i
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