Checking if a function is analytic or entire

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To begin, I think I might be using holomorphic and analytic (incorrectly) interchangeably. When I think of analytic I think of something being able to be represented locally by a power series. holomorphic however makes me think of a function being complex differentiable locally. I am a bit confused because they seem to be used interchangeably in complex analysis and that one implies the other.


Now my question, what is the best way to check that a function is analytic?/ I know this is an incredibly basic question and I look for answers but I just don't seem to find, for example, just a theorem I can apply. I know I have the C-R equations but if they are satisfied that only gaurantees the function is holomorphic at that point. To be analytic at a point, it should be complex differentiable in an entire open set containing the point.


Are there any useful little tricks to check for this? Or really good theorems to apply? And similarly with a function being entire (holomorphic everywhere?) is it just kind of an issue of analyzing the function and figuring it out (and plus I know say, polys are entire, cosz, sinz, and e^z are entire so combinations of them will be) or are there any hard and fast theorems I can apply that simplify the task?
 

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