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.

1,190 questions
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Laurent series expansion in powers of z

$f(z) = \frac {e^{z^{2}}}{z^3}$, valid for $|z| >0$ The way I am thinking about this problem is to factor out the $e^{z^{2}}$ term as I am use to Laurent series of the form $\frac{1}{1-w}$. I am ...
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why there isn't laurent series at a non isolated singularity

I am studying the laurent series expansion around singularities. I don't understand why at a non-isolated singularity the laurent series expansion doesn't exist, whereas we define the laurent series ...
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Finding the singularities of a complex function.

I need to find and classify the singularities of the function: $f(z) = \frac{z^2+1} {z^4-2}$. I'm aware that I'm going to have to first find the Laurent series corresponding to this function. I ...
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Evaluating the Laurent series of $\frac{1}{\exp (z) -1 }$

I am trying to find the Laurent series of the function $f(z) =\frac{1}{\exp (z) -1 }$ above but I don't know how to find the term $$a_{-1} = \frac{1}{2\pi i}\int f(\xi ) \,d\xi$$ I tried ...
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Laurent Series of $\frac{e^z}{z(1+z^2)}$ [at z=0]

How to get the Laurent Series of $$\frac{e^z}{z(1+z^2)}$$ at $z=0$? I know the answer is $$\frac{1}{z} + 1 - \frac{1}{2}z - \frac{5}{6}z^2 + \frac{13}{24}z^3 + \frac{101}{120}z^4 + O(z^5)$$ But I ...
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Show that the singular point of the following functions is a pole and verify using Laurent’s expansion

Given that I have $\frac{1-e^{2z}}{z^3}$ and $\frac{e^{2z}}{(1-z)^2}$, how would I go about verifying it in each case using Laurent's expansion? I can find the residues and I know the singular points ...
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Clasifying singularities.

I would like to clasify the singularities of the function $$f(z) = \frac{1}{z^2+2z+1}$$ I tried to make the Laurent expansion on $z = -1$, but at this point the function is not holomorphic and I ...
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Obtaining Laurent series of 1/f(z) given the laurent series of f(z).

Suppose I have a complex function ($f_{}^{}(z_{}^{})$), analytic in the annular region $\mathcal{C} = \{z : 0\leq r_{}^{}<|z_{}^{}-z_{0}^{}|<R_{}^{} \leq \infty\}$ with the Laurent series ...
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Calculate residue in every pole of $\frac{z^2}{(z-2)^2(\cos{z}-1)^3}$

The problem is as simple as the title suggests. Of course, the formula for the residue of an order $n$ pole involving $n-1$ derivatives could be applied, but the computation will be extremely long, as ...
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Finding the Laurent series of the following function

I need to find the laurent series and the residue of the following complex function $$f(z)=(z+1)^2e^{3/z^2}$$ at $z=0$. Since $e^z=\sum z^n/n!$, then e^{3/z^2}=\sum_{n=0}^\infty \frac{3^n/n!}{z^{...