# Analytic continuation and how it relates to the gamma function.

I am familiar with factorials, and I have read about the gamma function. From what I understand, the gamma function extends the concept of the factorial to complex numbers by nature of being an analytic continuation.

My question is: What is an analytic continuation? I would imagine that there are other functions that have analytic continuations. What are other examples of an analytic continuation?

Instead, to extend from a discrete function to a continuous function, the more appropriate terminology is to call it "interpolation". Once you have the gamma function that "interpolates" the factorial function to $[1,\infty)\subseteq\Bbb R$, then you can use analytic continuation to extend it to (most of) the rest of $\Bbb C$.
The infinite series expansion of a function often only converges for certain values of the variable $x$. So, simply put, analytic continuation attempts to define values for these functions in domains where their expansion diverges. You are basically attaching a holomorphic function to the original one. A popular example of analytic continuation is the Riemann Zeta function $\zeta(s)$ which is an analytic continuation of the following sum:$$\sum_{n=1}^\infty\frac 1{n^s}$$ Another well know example is the Complex Logarithm that extends the domain of $\ln|x|$ into $\mathbb{C}$.