Find the Taylor series for $f(z)=e^z$ about $z_0=1+i$.

Question

Find the Taylor series for $f(z)=e^z$ about $z_0=1+i$.

I know that I want to use the geometric series for $e^z$ which goes $1+z+\frac{z^2}{2!}+\frac{z^3}{3!}…$, but this is centered around $z_0=0$. How would I go about changing this for $z_0=1+i$?

Answer 1

Since $e^z = e^{z_0} \cdot e^{z-z_0}$, we have

$$ e^z = e^{z_0} \cdot \sum_{k=0}^{\infty} \frac{(z-z_0)^k}{k!} = e^{1+i}\cdot\sum_{k=0}^{\infty} \frac{(z-1-i)^k}{k!}. $$