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.
Best Answer
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.