1. Complex Power Series

Can anyone help with this question?

Find the power series expansions of
(a) log z about z = 1;
(b) 1/(1 + z) about z = −5.
What is the radius of convergence in each case?

Thanks for any help

2. Originally Posted by alawrie
Can anyone help with this question?

Find the power series expansions of
(a) log z about z = 1;
(b) 1/(1 + z) about z = −5.
What is the radius of convergence in each case?

Thanks for any help
For (a), where are you stuck? What have you tried?

(b) Note that $\displaystyle \frac{1}{1 + z} = \frac{1}{(z + 5) -4}$.

Region I: $\displaystyle 0 < \frac{|z + 5|}{4} < 1$.

$\displaystyle \frac{1}{(z + 5) -4} = \frac{-1}{4 - (z + 5)} = -\frac{1}{4} \left( \frac{1}{1 - \left(\frac{z+5}{4}\right)} \right)$ and you can get a series using the sum of an infinite geometric series, noting that $\displaystyle r = \frac{z + 5}{4}$.

Region II: $\displaystyle \frac{|z + 5|}{4} >1$.

$\displaystyle \frac{1}{(z + 5) -4} = \frac{1}{z + 5} \left( \frac{1}{1 - \left[ \frac{4}{z + 5}\right]} \right)$ and you can get a series using the sum of an infinite geometric series, noting that $\displaystyle r = \frac{4}{z + 5}$.