First the background. I'm trying to solve for the roots of a rather messy complex equation. This is not the exact equation, but it's a decent (simpler) stand in:
Tan[x - I a] + I == I (x - I a)^(-1/2)
I can use FindRoot to solve for a particular root, i.e.
FindRoot[Tan[z] + I == (I) (z)^(-1/2), {z, Pi - .3 I}]
which gives {z -> 3.08945 - 0.465902 I}. The thing is, the real part of these equations (as well as the imaginary parts) alone have periodic solutions. For example, look graphically at this solution:
However, realizing that the real roots tend to be spaced roughly at integer factors of $\pi$ (i.e. consider Tan[x]==x^(-1/2)), using FindRoot[Tan[z] + I == (I) (z)^(-1/2), {z, 2 Pi - .3 I}] indeed gives the next root. However, in the actual equation some roots get spaced un-evenly, and it's possible to miss particular roots when using this guess method.
I recognize that another approach to doing this explicitly solves the real and imaginary parts of the equation as simultaneous roots to two real equations:
FindRoot[{Re[Tan[x + I y] + I] == Re[(I) (x + I y)^(-1/2)],
Im[Tan[x + I y] + I] == Im[(I) (x + I y)^(-1/2)]},
{{x, Pi}, {y, -.3}}]
and gives the same result, {x -> 3.08945, y -> -0.465902}.
Given this info, is there a way to use a RootSearch-type function to find all complex (simultaneous) roots to this (these) equation(s) over a particular range of real values without guessing each real part of root to be an integer value of $\pi$?
Edit As an added complication, I actually have two complex equations that I need to solve simultaneously, so if the solution could be generalizeable to allow for this sort of thing, that would be an added bonus.
Also Edit I've tried a few of the suggestions in the comments and answers below. Some work for the sample case I gave above, but for my actual problem they don't really work. So, I figured may as well give the whole crazy thing I'm trying to solve:
s1 = {(-2 ky Pi wavelength Cot[
1/2 (b ky + n Pi)] + (km - I nm) (4 Pi^2 +
ky^2 wavelength^2 -
2 ky (km - I nm) Pi wavelength Tan[
1/2 (b ky + n Pi)]))/((km - I nm) wavelength) == 0,
(-2 kx Pi wavelength Cot[
1/2 (a kx + m Pi)] + (km - I nm) (4 Pi^2 +
kx^2 wavelength^2 -
2 kx (km - I nm) Pi wavelength Tan[1/2 (a kx + m Pi)]))/((km - I nm) wavelength) == 0}
This equation needs to be solved for different values for the parameters in this equation a, b, wavelength, etc. Here's a sample:
wavelength = 26 10^-3;
nm = -100;
km = 5;
a = 13 10^-4;
b = 64 10^-5;
m = 1;
n = 0;
Now, if I try
Solve[{Sequence@@s1, Abs[kx] <= 10, Re[kx] >= 0, Abs[ky] <= 10, Re[ky] >= 0}]
I get an error "This system cannot be solved with the methods available to Solve.". However, trying Sequence@@N@s1 it suggests trying exact input. Similarly, Reduce does not work with N values, and with exact values takes impossibly long to solve.
So, I'm still looking for a fast numerical approach. Perhaps something like J.M. suggested in a comment could be generalized for more than two equations?
