# State why the function $f(z)=2z^2-3-ze^z+e^{-z}$ is entire

Consider the complex function

$f(z)=2z^2-3-ze^z+e^{-z}$

First of all, I want to get down with the definition of entire function. Based on Wikipedia, in complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane.

A holomorphic function to me is the same as analytic function.

I don't see the difference, so I am just wondering if we can use Cauchy–Riemann equations to do this exercise.

• It is Cauchy-Riemann equation, not Quotient-Riemann. – P Vanchinathan Aug 4 '16 at 0:47
• Correct, sorry XD – TheMathNoob Aug 4 '16 at 0:48
• Replace $z$ with $x+iy$ and split the function on the rhs collecting the coefficients of $i$ calling it $v(x,y)$ and the rest as $u(x,y)$. Now apply Cauchy-Riemann. – P Vanchinathan Aug 4 '16 at 0:50
• @Nobody, how about "firstiliciously"? :) – paul garrett Aug 4 '16 at 0:56
• I have seen it quite often in chat conversations and social networks. I know firstable=first of all. I thought they were equivalent because to be honest English is quite wrong. Some vowels don't correlate with some words. For example, infinite and finite sound different and I haven't found the reason why. – TheMathNoob Aug 4 '16 at 1:01