Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $f$ be entire and non-constant. Assuming $f$ satisfies the functional equation $f(1-z)=1-f(z)$, can one show that the image of $f$ is $\mathbb{C}$?

The values $f$ takes on the unit disc seems to determine $f$...

Any ideas?

share|improve this question

2 Answers 2

up vote 11 down vote accepted

If $f({\mathbb C})$ misses $w$, then it also misses $1-w$. The only case where $w = 1-w$ is $1/2$, which is $f(1/2)$. Now use Picard's "little" theorem.

share|improve this answer

Since the function is entire, it misses at most one point by Picard's little theorem. If the point is say, $y$, then $1-y$ must be in the range, unless $y=1-y$. But, in this case, $y=\frac{1}{2}$ and we can deduce that $f(\frac{1}{2})=\frac{1}{2}$. So, $y\neq 1-y$. So, if there is a $z$ such that $f(z)=1-y$, then, $f(1-z)=1-f(z)=y$. So, the range of $f$ is $\mathbb C$.

share|improve this answer
1  
Didn't Robert just say that? –  Fly by Night Aug 28 '12 at 22:52
2  
@FlybyNight: I was trying to solve the problem by myself too, since I just took my first Complex Analysis class last semester, and really wanted to answer the question . If it is in breach of M.SE etiquette to answer a question a little late using strikingly similar methods, I would willingly delete my reply. I meant no offense. Please let me know. –  Galois Group Aug 28 '12 at 23:00
1  
@FortuonPaendrag: Don't delete your answer, it is on topic and correct. Quite often two people will write out similar answers, and post them at around the same time. I see no problem with this. –  Eric Naslund Aug 29 '12 at 3:49
    
It is possible that they were posted at the same time. But what usually happens, and what tends to annoy me, is that people reply without bothering to read the other answers. I'm not suggesting that this is the case here, and I hope I have not upset Fortuon. I've just become very cynical. –  Fly by Night Aug 29 '12 at 19:16

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.