Avoiding Poles in the Complex Plane

I have functions defined by nonlinear differential equations. I know the
initial conditions at exactly one point, z0, of the complex plane. The
complex plane is covered with an infinite lattice of simple poles, whose
position is unknown. The goal is to generate a contour map of the
absolute value of the functions over a portion of the complex plane
which includes some of the poles.

To find the value at the point z1, I integrate along the line

z0 + (z1-z0)t,

or t going from 0 to 1. This works fine except when the line passes too
close to a pole causing the numerical integrate to fail. This eliminates
a wedge of the plane behind the pole from the plot.

I need some way to dynamically adjust the trajectory so as to reach the
point z but miss any poles that would disrupt the integration.

As a concrete example consider

c'[z] = -s[z]^6,. c[0] = 1,
s'[z] = c[z]^6, s[0] = 0.

these functions generate a lattice of poles with a seven fold symmetry.
I would like to integrate along some path

z[t] = u[t] + i v[t], z[0] = z0, z[1] = z1

such that a function similar to

|z - z1| [a + (1-a)(|c[z]| + |s[z]|)],

remains below some bound. Clearly, a set of straight lines can be found
by stopping the integration and inserting a dog leg when necessary. What
would be nice is differential formulation that would allow an
uninterrupted integration from z0 to z1.
.

Relevant Pages

  • Re: Avoiding Poles in the Complex Plane
    ... initial conditions at exactly one point, z0, of the complex plane. ... complex plane is covered with an infinite lattice of simple poles, ... which includes some of the poles. ... by stopping the integration and inserting a dog leg when necessary. ...
    (sci.math.num-analysis)
  • Re: Writing our own low level m-file
    ... all the poles in the complex plane by your ... favorite method and using the residue theorem. ... with little success as far as I could ...
    (comp.soft-sys.matlab)
  • David Ullrich : Please if you would help with a previous question
    ... function on the complex plane with simple poles at n = 1,2,3,... ... P_nwill be partial sums of the last infinite sum where |z| < n-1. ...
    (sci.math)
  • Meromorphic function, Weierstrass product
    ... Find a meromorphic function on the complex plane having simple poles at ... My first reaction to find a meromorphic function would be to take 1/f. ... function that has simple zeroes at i, 2i, 3i, ... ...
    (sci.math)
  • Re: Numerical integration with many points
    ... steps and use Boole's rule to perform the integration. ... I would recommend VERY STRONGLY analytically ... continuing into the complex plane, ... choose the exponentially decreasing direction, ...
    (sci.math.num-analysis)
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