Articles of laurent series

The field of Laurent series over $\mathbb{C}$ is quasi-finite

How can I prove that the field of Laurent series over $\mathbb{C}$ is quasi-finite, which means that it has a unique extension of degree $n$ for all $n \geqslant 1$ ? The article says that the extension of degree $n$ is $\mathbb{C}((T^{1/n}))$. I think that I understand why it is an extension of degree […]

Laurent series of $\frac{1-\cos(z)}{z^2}$

How do I calculate the Laurent series of $\frac{1-\cos(z)}{z^2}$? (I know the general formula as is shown here )

Help with this limit?

I am trying to focus on the limits of functions with similar series expansions and I stumbled on this. $$\lim_{x\to\infty}\left({\left(\frac{x^2+5}{x+5}\right)}^{1/2}\sin{\left({\left(\frac{x^3+5}{x+5}\right)}^{1/2}\right)}-(x-5)^{1/2}\sin{\left({\left(x^2-5x+25\right)}^{1/2}\right)}\right)=0$$ I heard the mean value is possible but the entire function is not bounded. I can take the taylor series at infinity however the terms would be undefined. I could use substitution with the taylor […]

Prove that $\zeta(4)=\pi^4/90$

I am asked to “use the calculus of residues” to prove that $$\displaystyle\sum\limits_{n=1}^{\infty} \frac{1}{n^4}=\frac{\pi^4}{90}$$ I think I can do this given the Laurent series for $\cot z$ centered at the origin, but I don’t know how to find the first few terms of the Laurent series (I can use Cauchy’s Integral Formula to find the […]

$e^{1/z}$ and Laurent expansion

$e^\frac1z$ is not holomorphic at $z=0$, but it is known that it can be expanded as $$e^\frac1z=1+\frac1z+\frac1{2!z^2}+\frac1{3!z^3}+\cdots$$ The coefficients of this Laurent expansion are computed the same way as Taylor’s. The question is how is that possible? If function is not holomoprhic at $z=0$, then it’s not true that it is holomophic at $|z|<R$ and […]

Classify the singularity – $\frac{1}{e^z-1}$

Definition : A isolated singularity is a pole of ordre $m$ if $f(z)= \sum_{k=-m}^{\infty} a_k (z-z_0)^k$, $a_m \not= 0$ I have to classify the singularity (removable, pole and essential) of $\frac{1}{e^z-1}$. I know that $e^z-1=0 \iff e^z=1 \iff z = 2\pi k i = z_k$ for each $k \in \mathbb{Z}$. In using what I found, […]

Prove Laurent Series Expansion is Unique

Suppose that $f$ is holomorphic on $A=\{r<|z|<R\}$, where $0\le r<R\le \infty$. Suppose that there are two series of complex numbers $(a_n)_{n\in{\mathbb Z}}$ and $(b_n)_{n\in\mathbb Z}$ such that $f(z)=\sum_{n=-\infty}^\infty a_n z^n=\sum_{n=-\infty}^\infty b_n z^n$ for $z\in A$. Show that $a_n=b_n$ for all $n\in\mathbb Z$. This means that the Laurent series expansion is unique. Hint: It suffices to […]

The degree of a polynomial which also has negative exponents.

In theory, we define the degree of a polynomial as the highest exponent it holds. However when there are negative and positive exponents are present in the function, I want to know the basis that we define the degree. Is the order of a polynomial degree expression defined by the highest magnitude of available exponents? […]

How to switch to a Laurent series' next convergence ring?

Given the Laurent series $\sum\limits_{k=-\infty}^\infty a_k^{(l)} z^k = f(z)|_{r_l<|z|<R_l}$ of a meromorphic function $f$ on $\mathbb C$ with convergence region $r_l< |z|< R_l$, one can use analytical continuation to obtain the values in the regions $r_m<|z|<R_m$ where either $r_m=R_l$ or $R_m=r_l$, i.e. switch between series converging in neighbouring rings the boundaries of which are touching […]

Laurent-series expansion of $1/(e^z-1)$

Find the Laurent series for the given function about the indicated point. Also, give the residue of the function at the point. $$ z\mapsto\frac{1}{e^z – 1} $$ The point is $z_0=0$ (four terms of laurent series). I have wrote $e^z -1$ as $z+z^2/2!+z^3/3!$…. Now i don’t know how to proceed with this further. Please answer […]