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How is $f(z)=e^{5z}$ ENTIRE?

This is a part of the solution in between a bigger question. But i don't seem to understand on how to check the above $f(z)$ holomorphicity?

I know that when $f(z)$ is entire means that it is holomorphic in the entire complex plane.

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    $f'(z) = 5 e^{5z}$ et voila.2012-10-22
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    Or you can just say it is the composition of two entire functions.2012-10-22
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    i think i am thinking too much since the question involves complex planes.2012-10-22
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    I'm tempted to say that the answer is entirely obvious.2012-10-22

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