Finding a Maclaurin representation of a complex function

Question

The problem I've found asks me to find the Maclaurin representation of \begin{equation} \log(1+2z) \end{equation} and thus find its radius of convergence, and then find 3 terms of it's Laurent expansion about singular point $z=0$

In my notes I've found Maclaurin expansions of other functions but never one involving the complex Log. Can anyone point me in the right direction?

Answer 1

You can consider its derivative: $\frac{\mathrm d}{\mathrm d z}\log(1+2z)=\frac{2}{1+2z}=\frac{2}{1-(-2z)}$. The MacLaurin series of this function can be obtained by noticing that this is the limit of the geometric series

$$2\sum_{k=0}^\infty (-2z)^k.$$

Now integrate each term to obtain the MacLaurin series of an antiderivative, which is the original function $\log(1+2z)$ plus a constant. The constant can be found by considering $f(0)=\log(1)=0$.

Answer 2

In the complex, the derivative of $\log z$ is also $\dfrac1z$. Hence the Tayor expansion is the same as in the reals.