Complex functions and Taylor series

Question

Find the Taylor series arround $z_0=0$ write radius of convergence

a) $f(z)=\cosh(z)$

b) $f(z)=\log(z+1)$

I don't know how it works with the complex functions. Could you show me the workflow? I will be so grateful!

Answer 1

For example

$$\frac1{1+z}=\sum_{n=0}^\infty(-1)^nz^n\;,\;\;|z|<1\implies \text{Log}\,z=\int\frac{dz}{1+z}=\sum_{n=0}^\infty(-1)^n\int z^n\,dz$$

as termwise integration is allowed within the convergence interval, so...

Answer 2

Actually the Taylor series is meant for complex functions (polynomials have a much more consistent theory, with the fundamental theorem of algebra). The radius of convergence is the radius to the first pole you encounter. Logarithm $\log(z+1)$ has a problem at $z=-1$, which should hint at how far the series converges. For $\cosh z$, you should remember it is a whole function (like $\exp$, all trigs and so on), so there are no poles and the radius of convergence is infinite.