So the problem states:
Say $f(z) := \log z$ is the principal branch of the logarithm (the primitive of $1/z$ on the region $\Bbb C\setminus (-\infty,0]$). Show that the Taylor series of $f(z)$ about $z_0 = -1 + i$ takes the form
$$\log z = \sum_{n=0}^{\infty} a_n(z-(-1+i))^n $$
with$$a_0 = \log \sqrt{2} + i\frac{3\pi}{4}\,\,\,\text{and}\,\,\,a_n = (-1)^{n+1}\frac{e^{-3\pi in/4}}{n2^n/2}$$
Determine the radius of convergence of this series. Explain why the series does not represent $f(z)$ in its entire disk of convergence.
My main concern here is how to show $\log(-1+i) = \log \sqrt{2} + i\frac{3\pi}{4} $ and determine the radius of convergence.
Best Answer
The function $g(z):={1\over z}$ is analytic in $\dot{\mathbb C}:={\mathbb C}\setminus\{0\}$, but has no primitive defined in all of $\dot{\mathbb C}$. The function $g$ however has primitives in suitable subdomains $\Omega\subset\dot{\mathbb C}$, the most famous one being the principal value $${\rm Log}(z):=\log|z|+ i\>{\rm Arg}(z)\ ,$$ which is defined on $\Omega:={\mathbb C}\setminus\{≤0\ {\rm real\ axis}\}$. In particular $${\rm Log}(-1+i)={1\over2}\log2+{3\pi\over4}\>i\ .$$ Standing at the point $p:=-1+i\in\dot{\mathbb C}$ we see that the function $g$ is analytic in a disk $D$ of radius $\sqrt{2}$ around $p$. Therefore $g$ has primitives which are analytic in $D$, whence have a power series development with center $p$ and convergence radius $\sqrt{2}$. These primitives are equal up to an additive constant, and one of them has the value ${1\over2}\log2+{3\pi\over4}\>i$ at $p$. Since ${\rm Log}$ is a primitive of $g$ in the neighborhood of $p$ having exactly this value at $p$ the corresponding series represents ${\rm Log}$ in any domain $\Omega'\subset D\cap\Omega$ containing the point $p$. The points of $\Omega$ lying below the negative real axis do not belong to such an $\Omega'$. Therefore the obtained series does not represent ${\rm Log}$ there.