Editing Liouville's theorem (complex analysis) (section)

===Entire functions have dense images===
If <math>f</math> is a non-constant entire function, then its image is [[Dense set|dense]] in <math>\Complex</math>. This might seem to be a much stronger result than Liouville's theorem, but it is actually an easy corollary. If the image of <math>f</math> is not dense, then there is a complex number <math>w</math> and a real number 
<math>r > 0 </math> such that the open disk centered at <math>w</math> with radius <math>r</math> has no element of the image of <math>f</math>. Define 

:<math>g(z) = \frac{1}{f(z) - w}.</math>

Then <math>g</math> is a bounded entire function, since for all <math>z</math>,

:<math>|g(z)|=\frac{1}{|f(z)-w|} < \frac{1}{r}.</math>

So, <math>g</math> is constant, and therefore <math>f</math> is constant.