PortraitsWithPoles.m

%% Phase portraits for functions with poles
% Nick Trefethen, March 2013

%%
% (Chebfun2 example complex/PortraitsWithPoles.m)
% [Tags: #phase portraits, #poles, #Chebfun2, #SMASH]

function PortraitsWithPoles

%% Phase portraits for analytic functions
% Here is a phase portrait [1] of the function $f(z) = z^2(z^3-1) $ on the
% square $[-1.5,1.5,-1.5,1.5]$:
ff = @(z) z.^2.*(z.^3-1);
d = 1.5*[-1 1 -1 1];
f = chebfun2(ff,d);
plot(f), axis(d), axis square on

%% Phase portraits for functions with poles
% Now suppose the zeros of $f$ at the origin were poles instead of zeros. Then
% Chebfun2 couldn't represent $f$, so we couldn't do a phase portrait directly.
% However, for a phase portrait, all that matters is the phase!  So we should be
% able to replace $f$ by another function with the same phase but smooth
% modulus, like $f/(1+|f|^2)$. I've implemented this with a function called
% "smash" (picking a name reminiscent of a command in TeX).

function g = smash(f)
  absf = abs(f);
  g = f./(1+absf.^2);      % smooth function with same phase as f
  g(isnan(g)) = 0;         % give 0 rather than NaN at poles
end

%%
% It works!  Notice that in the middle, yellow is clockwise of red rather than
% counterclockwise, indicating poles rather than zeros.
ff = @(z) z.^(-2).*(z.^3-1);
g = chebfun2(@(z) smash(ff(z)),d);
plot(g), axis(d), axis square on

%%
% For another example, here is $\tan((3+3i)z)$.  Compare Figure 3.20 of [1].
hh = @(z) tan((3+3i)*z);
h = chebfun2(@(z) smash(hh(z)),d);
plot(h), axis(d), axis square on


%%
% These experiments are promising.  In fact, I am not sure how robust the
% technique is; some careful exploration may be needed, and one might want to
% loosen the Chebfun2 convergence criterion, since the aim here is only to
% draw a plot in any case.  Also, the "smash" process as I've defined it isn't
% scale-invariant; instead of $1$ in the denominator, there should be a number
% representing a typical magnitude of $f$.

%% 
% Perhaps it is worth emphasizing that although it is appealing to be able to
% use Chebfun2 to draw phase portraits for a wider range of functions than one
% might have imagined, ultimately there is no good reason for the phase
% portraitist to use Chebfun2.  A phase portrait is just a pixel-by-pixel
% image; it gets no benefit from the global high-accuracy representations of
% Chebfun2.

%% References
%
% 1. E. Wegert, Visual Complex Functions: An Introduction with Phase Portraits,
%    Birkhauser/Springer, Basel, 2012.

end