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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?

Thanks in advance!

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Really you don't do analytic continuation to extend the factorial fuction to the gamma function. Analytic continuation is when you have a function defined over a region, like a whole interval of the real line, not a discrete set of points (even if an infinite set) like the factorial function.

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 $\Bbb C$.

Anayltic continuation can be understood by what happens with polynomials. Two polynomials that are equal on an interval are equal everywhere. Functions that can be expanded in power series (which are like infinite degree polynomials) have the same uniqueness property. That uniqueness property allows us to extend them uniquely (if at all).

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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 $\mathbb{C}$.

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