The question is from the following problem:
If $f(z)$ is an analytic function that maps the entire finite complex plane into the real axis, then the imaginary axis must be mapped onto
A. the entire real axis
B. a point
C. a ray
D. an open finite interval
E. the empty set
A search on Google didn't return any satisfying result of the concept "the entire finite complex plane". The word "entire" and "finite" seem to be a contradiction to me. Here are my questions:
- What is "the entire finite complex plane"?
- What properties does one need to use about analytic function to solve this problem?
Best Answer
The "entire finite complex plane" is just the complex plane. This phrase isn't used very often, but can be used if you want to differentiate between the complex plane $\mathbb{C}$ and the Riemann sphere $\mathbb{C}\cup \{\infty\}$.
The easiest way to answer this is to use Little Picard, but there are certainly easier ways using less powerful machinery. For example, you could consider $\exp(if(z))$ and use Liouville.
Edit: Since you've added the tag reference-request, I will mention that the term "finite complex plane" is used in Silverman's translation of Markushevich's monumental "Theory of Functions of a Complex Variable" which is one of the standard references in complex analysis.