Hi. How do I find the power series for tanh(z) (in the complex plane, if that matters, which it doesn't).
I know the power series for sinh(z) and cosh(z) (same as sin(t), cos(t), but without the sign changes):
Now what? You can't divide series (or, at a minimum, I don't know how to divide series). So what's next?
You can use the same derivation as any other Taylor series:
You have to evaluate as well as the first, second, third, etc. derivatives at the point .
You should find that when you substitute the values into your Taylor series formula that you can simplify using Bessel numbers.
An 'a little unconventional' approach: the function is the solution of the following ODE...
If You suppose that is analytic in [and that is true...] is...
All right!... now from (1) and (2) You can derive the as follows...
... and so one...