Understanding the Rieman Surface of the Complex Logarithm

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I was reading Complex Logarithm from Wikipedia. I got stuck on a line. I am quoting "Another way to resolve the indeterminacy is to view the logarithm as a function whose domain is not a region in the complex plane, but a Riemann surface that covers the punctured complex plane in an infinite-to-1 way". I understood everything before this line.

I could not understand this picture as well.

enter image description here https://en.wikipedia.org/wiki/Complex_logarithm--is the link.

Can anyone make me understand by giving simple example? I really can not make progress on it.

Can anyone please help me?