Does there exists an entire function $f: \mathbb C \to \mathbb C$ which is bounded on real line and imaginary line?

Question

Does there exists a nonconstant entire function $f: \mathbb C \to \mathbb C$ which is bounded on real line and imaginary line?

Clearly,$ f(z)=sin(z)$ is an example of an entire function which is bounded on real line and $ f(z)= e^z$ is example of a function which is bounded on imaginary line.But I'm unable to find a function which is bounded on both the lines.Any ideas?

See this Math Overflow page for some really good answers towards this direction: http://mathoverflow.net/questions/190837/entire-function-bounded-at-every-line — Cameron Williams, Jun 28 '15 at 17:05
@CameronWilliams I think thats more general.There may be some easy example in OP's case. — Dontknowanything, Jun 28 '15 at 17:08

zhw. · Accepted Answer · 2019-06-05T17:59:09.287

36

$f(z) =e^{iz^2}$ will do that for you.

edited Jun 05 '19 at 17:59

answered Jun 28 '15 at 17:05

zhw.

105,693

2

You should remove some of the ellipses. – Cameron Williams Jun 28 '15 at 17:06
1

@CameronWilliams I have them to meet the character lower limit. – zhw. Jun 28 '15 at 17:11
2

Use , in $\rm\LaTeX$ to put in artificial spaces. – Cameron Williams Jun 28 '15 at 17:13
@zhw. Sorry..How did you thought of such example? – Dontknowanything Jun 28 '15 at 17:34
3

@Learner: I imagine that zhw considered first what functions send exactly the real and imaginary lines to the real line. One such function is $z\mapsto z^2.$ Hence $z\mapsto iz^2$ maps exactly the real and imaginary lines to the imaginary line, whence composition with $w\mapsto e^w$ yields the desired result. A similar approach is to consider $\sin(z^2)$. – Cameron Buie Jun 28 '15 at 18:06
@CameronBuie Thank you.Regards, – Dontknowanything Jun 28 '15 at 18:59
@Learner What CameronBuie said is pretty much right. – zhw. Jun 29 '15 at 16:58
@zhw. Thank you for your response.Regards, – Dontknowanything Jul 01 '15 at 11:58
Nice Example..... – neelkanth Dec 08 '17 at 10:25

score 0 · Answer 2 · answered Jul 31 '21 at 16:38

0

Another is $e^{ie^z}$. On real line it is $|e^{ie^x}|=1$ and on imaginary axix it is $|e^{i \cos y -\sin y}|\leq e$.

answered Jul 31 '21 at 16:38

mathemather

2,959

Does there exists an entire function $f: \mathbb C \to \mathbb C$ which is bounded on real line and imaginary line?

2 Answers2

Linked

Related