Power Series Expansion about Point

[curtdbz]

So this is a REALLY elementary question but I can't seem to find the answer on the net, or maybe I did but just keep skipping over it some how. (by the way, this is with respect to complex numbers $$z \in C$$ which is used in Complex Analysis, thus why I chose this forum). I know what it means when someone gives a function and says "Expand into a power series about $$z$$", but what does it mean when they say to expand about, say, $$z = 4 + i$$ for a function like $$(e^{z} - 1)^{-1}$$.

I don't need to know the answer, just the general formula for something like that. Again, elementary I know. Thanks! Any help appreciated.

 

[Mute]

In a regular power series expansion, when you expand the function f(x) about a point a, you get the expansion

$$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n.$$
(when it exists and all that jazz).

If f is a function of a complex variable z, then for analytic functions you get the same form of the series expansion,

$$f(z) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(z-a)^n.$$
where both z and a may be complex numbers. In your example, a = 4-i and f(z) = 1/(exp(z)-1).