Laurent series with nonzero negative infinity term

Question

I was told that the most general case of a Laurent series has its negative index at negative infinity instead of at -m for some integer m. Can someone give an example of a function with this form? Thank you in advance!

Answer 1

One example is

$$e^{1/z} = \sum_{n = -\infty}^\infty a_n z^n,\quad z \neq 0$$

where $a_n = 0$ if $n > 0$ and $a_n = \frac{1}{(-n)!}$ if $n \le 0$.

Answer 2

For example, $e^{1/x}$ centered at $x = 0$.

Answer 3

An example: $$e^{\frac{1}{x}}= \sum_{n=0}\frac{1}{n!x^n}$$