Simple case of analytic continuation

I have yet to formally study complex analysis, yet the topic of analytic continuation, specifically with respect to the Riemann Zeta function, is fascinating.

My question is, what are some relatively simple cases of analytic continuation?

I ask because I am looking for a basic example to study and in the case of Riemann Zeta, the derivation seems rather complicated.

Is the continuation of $$f(z)=\sqrt{z}$$ a relatively simple case of analytic continuation? Based on this Wolfram MathWorld page, I would think so; however, the article goes on to say that the continued power series of the function can become a multivalued function as shown in the following graphic. Must an analytic continuation necessarily yield a multivalued function?

Thank you in advance.

The analytic function given by the sum of the series $$f(z):=\sum_{n=0}^{+\infty}z^n$$ and the function $$g(z)=\frac1{1-z}$$ agree on the open disk $$\{|z|<1\}$$ (which is the domain of $$f$$), but the latter is analytic on $$\Bbb C\setminus\{1\}$$, thus $$g$$ extend the function $$f$$ on this bigger domain.