how to solve such transcendental equation?

(rhoL/rhoA*L)*(1-(ka/ks)^2+(ky*L/(ks*L))^2)*(ky*L) = 1/tan(ky*L);

ka=f/C1, ks=f/C2

rhoL,rhoA,L,C1 and C2 are constants

f has a range from (0,10000).

ky can be real, imaginary or complex. how to solve for the complete set of ky?

[Matt Fig]

What values do the constants have?

[Rui]

you can simplify the equation as f = @(x) afa*x^3-1/tan(x). adjust afa to make the solution contain a imaginary part

[Matt Fig]

Have a look at the comments in my answer.

Answer 1

[Matt Fig]

There are several ways to go about this. One is to use FZERO. Since you didn't actually give the values for the constants, I'll show you a different example.

Solve this equation for x on [0 10]: exp(-x)*x^2 = sin(x)/x

% First write it as a vectorized anonymous function.
% Solving the equation for zero.
f = @(x) exp(-x).*x.^2 - sin(x)./x;  % Notice the dots (.)
% Now we plot it to get an idea of where the zeros are.
x = 0:.01:10;  
plot(x,f(x))  % Look for the zeros - looks like 3.
% Now find the roots.
cnt = 1;
for ii = [2,6.5,9.5]  % This vector has the guesses.
   rt(cnt) = fzero(f,ii); % Pass each guess to FZERO.
   cnt = cnt + 1;
end

Let's look at the roots:

rt  % Display the roots.  Match our guesses? (Yes)

Now just do the same with your equation....

[Rui]

I hope things would be as easy as you listed there. The problem is the roots can be real, imaginary or complex and It's not possible to guess the approximate roots. I tried eval(solve()), it's not only slow but also gives merely the first root.

[Matt Fig]

In that case, have a look at this function: NEWTZERO.

You still have not given enough information for me to copy/paste and see if I can get results, but let's give it a try with your generic function:

afa = 3i+4;
f = @(x) afa*x.^3-1./tan(x);
newtzero(f)

ans =

     0.6379 - 0.0954i
    -0.6379 + 0.0954i

Also, you can use the solve function if you want more digits:

syms x
solve((3i+4)*x.^3-1./tan(x))

[Walter Roberson]

It is difficult to provide rigorous solving methods without knowing the ranges of each of the constants.

I could do a piecewise analysis of all of the combinations of possibilities and maybe come up with a complete method of finding all of the roots without any advanced constraints being known, but I don't think that would be a productive use of my time.