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.

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Finding Laurent Series

This was an exam question and I was absolutely lost. Find positive constants $a,b$ such that the function $$\frac{1}{z^2+a} + \frac{1}{z+b}$$ has three Laurent series, one for each of the domains ...
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Laurent Series for $\frac{\cos(z)}{z^2}$ centered at $0$

How would you prove that this is entire if f(z) is (cos(z)-1) if z doesn't equal zero and is -1/2 if z equals zero?
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How do I find the Laurent series for $\frac{1}{z^2 - 4}$ at $z = 2$?

How do I find the Laurent series for $\frac{1}{z^2 - 4}$ at $z = 2$? Can anyone help me out with this? I used partial fraction decomposition and got the $\frac{1}{4}$ part of it, just don't ...
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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 ...
$e^{1/z}$ 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 ...