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

Question

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.

Answer 1

An entire function is defined as a function holomorphic on all of the complex plane. This is correct. Further, yes this is equivalent to the function being analytic. In particular an entire function is given by a power series (and conversely a powerseries converging on all of the complex plane gives an entire function). Thus an entire function is (the same as) a function given by a power series with infinite radius of convergence.

I am not sure what you mean by "Quotient Riemann equations." You could use the Cauchy-Riemmann equation to solve this problem. But this seems a tedious way.

Instead I assume the intended solution is to exploit facts about entire functions. Such as:

• The sum and product of two entire functions is entire.
• A polynomial function is entire.
• The exponential function is entire.
• The inverse of a non-zero entire function is entire.

With these at hand the result follows quite readily. If you do not know those, try to show them first.