Some questions for entire function

[emptyboat]

The definition of entire function in complex analysis is
'If a complex function is analytic at all finite points of the complex plane , then it is said to be entire, sometimes also called "integral" (Knopp 1996, p. 112).'

I have questions about the terms 'finite points'.
Why are this terms here?
And If the terms are omitted, the function has another name or meaning?

 

[g_edgar]

The definition of entire function in complex analysis is
'If a complex function is analytic at all finite points of the complex plane , then it is said to be entire, sometimes also called "integral" (Knopp 1996, p. 112).'

I have questions about the terms 'finite points'.
Why are this terms here?
And If the terms are omitted, the function has another name or meaning?

Sometimes, in addition to the complex numbers, we add an extra point called infinity. The definition here is just emphasizing that the function is NOT required to be analytic at infinity. So, for example, the function $e^z$ is entire, even though it has an essential singularity at infinity.

 

[Count Iblis]

New Edit: I see that Edgar was a bit quicker than me.
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I think that is to exclude the point at infinity that is often added to the complex numbers. Any entire function will have a pole or essential singularity at infinity unless it is a constant. If it has a pole, then it is a polynomial. A function that is meromorphic including at infinity has to be a rational function.

So, unless you exclude the point at infinity, you are severely limiting the set of functions you can call analytic.

 

[emptyboat]

Thanks for both answers! I am more cleared about the definition.