# Taylor series and Laurent series

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.