# Questions tagged [laurent-series]

The Laurent series is a generalisation of the power series which allows negative indices and is essential for investigating the behaviour of functions near poles.

1,136 questions
32 views

### Expansion of the form $\sum_{n=0}^\infty c_n z^n$ [on hold]

write down an expansion of the form $\sum_{n=0}^\infty c_n z^n$ for $\frac{(a+iz)}{a-iz}$
66 views

### What is the Laurent series of $z+(1/z)$? [on hold]

What is the Laurent series of $z+(1/z)$? Is it already in the form of its series?
43 views

### Principal Part of Laurent Series Converges in Punctured Disc

I'm trying to work through the following problem: Prove that if the holomorphic function $f$ has an isolated singularity at $z_{0}$, then the principal part of the Laurent series of $f$ at $z_{0}$ ...
39 views

33 views

### Determining the type of singularities

Determine the type of singularities of $$f(z)=\frac{1}{(z-1)\cot(\pi/z)}\tag{1}$$ We first rewrite the function: $$f(z)=\frac{1}{(z-1)\cot(\pi/z)}=\frac{\sin(\pi/z)}{(z-1)\cos(\pi/z)} \tag{2}$$ ...
29 views

### Why is $z_0=0$ not an essential singularity of $\tan(1/z)$ What is this singularity?
Since for any $\epsilon>0~\exists |z|=|\frac{2}{(2k+1)\pi}|<\epsilon$ for some $k$ large enough such that $\tan(1/z)=\pm\infty$ there cannot be a Laurent series at $0$. Does there need to be a ...