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I understand that a Taylor series of a real or complex function $f(x)$ about a point $x=a$ is given by

$\displaystyle\sum_{n=0}^{\infty}{\frac{f^{(n)}(a)(x-a)^n}{n!}}$

where $f^{(n)}(a)$ is the $n$-th derivative of $f(x)$ evaluated at $a$.

On the other hand, a Laurent series of a complex function $f(z)$ about a point $z=c$ is given by

$f(z)=\displaystyle\sum_{n=-\infty}^{\infty}{a_n(z-c)^n}$

where $\displaystyle a_n=\frac{1}{2\pi i}\oint_{\gamma}{\frac{f(z)dz}{(z-c)^{n+1}}}$.

My question is, are there instances when a Taylor series and a Laurent series of the same function about the same point ever equal? I have made a few searches on Wolfram Alpha, and sometimes when I ask for a Laurent series for a function about a point, they give me the Taylor series at that point. I am therefore left to believe that a Taylor series and a Laurent series are equal sometimes(?)

Thank you for your feedback.

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Laurent series is unique. If a function has a Taylor series and a Laurent series about a point then the two coincide. Which means the function is analytic at that point.

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The essential difference between Taylor and Laurent series at a point is their ranges of expansions. When you expand a function by Laurent series at a point, that point is not included in the range of expansion, while for Taylor series, it is included. In other words, the range for Laurent series is ring, for Taylor is circle.

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