Vrftsjtryk

So the problem states:

Say $f(z) := log z$ is the principal branch of the logarithm (the primitive of $1/z$ on the region $Bbb Csetminus (-infty,0]$). 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

$$a_0 = log sqrt{2} + ifrac{3pi}{4},,,text{and},,,a_n = (-1)^{n+1}frac{e^{-3pi in/4}}{n2^n/2}$$

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} + ifrac{3pi}{4} $ and determine the radius of convergence.

The function $g(z):={1over z}$ 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 $Omegasubsetdot{mathbb C}$, 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
$${rm Log}(-1+i)={1over2}log2+{3piover4}>i .$$
Standing at the point $p:=-1+iindot{mathbb C}$ 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 ${1over2}log2+{3piover4}>i$ 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 $Omega'subset DcapOmega$ 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.

The first part of your question is really asking how to find the real and imaginary parts of the logarithm. So just write it out!
$$
e^{x+ i y} = -1 + i
$$
Now use Euler's identity to get simultaneous equations for $x$ and $y$:
$$
e^x cos y = -1 ~,~ e^x sin y = 1
$$
Hopefully you can solve these (the solution for $y$ is only unique because we choose a particular branch of the logarithm).

As for the second part, what do you know about the radius of convergence of the Taylor series for a holomorphic function?

The Maple command $$convert(ln(z), FPS, z = -1+I) $$ produces $$ ln left( -1+i right) +sum _{k=0}^{infty } left( -frac {
left( 1/2+1/2,i right)^k}{2,k+2}-frac {i left( 1/2+1/2,i
right)^k}{2,k+2} right) left( z+1-i right) ^{k+1}
.$$ Next,
$$normal(-(1/2+1/2*I)^k/(2*k+2)-I*(1/2+1/2*I)^k/(2*k+2)) $$ outputs $$frac{left( -1/2-1/2, i right) left( 1/2+1/2,i right)^k} {k+1}.
$$