elliptic (version 0.4-13)

view: Visualization of complex functions

Description

Visualization of complex functions using colourmaps and contours

Usage

view(x, y, z, scheme = 0, real.contour = TRUE, imag.contour = real.contour,
default = 0, col="black", r0=1, power=1, show.scheme=FALSE, ...)

Arguments

x

Vector showing real component of complex plane

y

Vector showing imaginary component of complex plane

z

Matrix of complex values to be visualized

scheme

Visualization scheme to be used. A numeric value is interpreted as one of the (numbered) provided schemes; see source code for details, as I add new schemes from time to time and the code would in any case dominate anything written here.

real.contour

Boolean with default TRUE meaning to draw contours of constant Re(z) and FALSE meaning not to draw them

imag.contour

Boolean with default TRUE meaning to draw contours of constant Im(z) and FALSE meaning not to draw them

default

Complex value to be assumed for colouration, if z takes NA or infinite values. Defaults to zero. Set to NA for no substitution (ie plot z as is); usually a bad idea.

col

Colour (sent to contour())

r0

If scheme=0, radius of Rieman sphere as used by Thaller

power

Defines a slight generalization of Thaller's scheme. Use high values to emphasize areas of high modulus (white) and low modulus (black); use low values to emphasize the argument over the whole of the function's domain

show.scheme

Boolean, with default FALSE meaning to perform as advertized and visualize a complex function; and TRUE meaning to return the function corresponding to the value of argument scheme.

...

Extra arguments passed to image() and contour()

Details

The examples given for different values of scheme are intended as examples only: the user is encouraged to experiment by passing homemade colour schemes (and indeed to pass such schemes to the author).

Scheme 0 implements the ideas of Thaller: the complex plane is mapped to the Riemann sphere, which is coded with the North pole white (indicating a pole) and the South Pole black (indicating a zero). The equator (that is, complex numbers of modulus r0) maps to colours of maximal saturation.

Function view() includes several tools that simplify the creation of suitable functions for passing to scheme.

These include: breakup{Breaks up a continuous map: function(x){ifelse(x>1/2,3/2-x,1/2-x)}} g{maps positive real to $[0,1]$: function(x){0.5+atan(x)/pi}} scale{scales range to $[0,1]$: function(x){(x-min(x))/(max(x)-min(x))}} wrap{wraps phase to $[0,1]$: function(x){1/2+x/(2*pi)}}

References

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

Examples

x <- seq(from=-4,to=4,len=55)
y <- x
z <- outer(x,1i*y,"+")
view(x,y,limit(1/z),scheme=2)
view(x,y,limit(1/z),scheme=18)


view(x,y,limit(1/z+1/(z-1-1i)^2),scheme=5)
view(x,y,limit(1/z+1/(z-1-1i)^2),scheme=17)

view(x,y,z^2,scheme=15,main="try finer resolution")
view(x,y,sn(z,m=1/2+0.3i),scheme=6,nlevels=33,drawlabels=FALSE)

view(x,y,limit(P(z,c(1+2.1i,1.3-3.2i))),scheme=2,nlevels=6,drawlabels=FALSE)
view(x,y,limit(Pdash(z,c(0,1))),scheme=6,nlevels=7,drawlabels=FALSE)
view(x,x,limit(zeta(z,c(1+1i,2-3i))),nlevels=6,scheme=4,col="white")

view(scheme=10, show.scheme=TRUE)