0

Let $\operatorname{abs}$ denote the absolute value (of a complex number) and $\operatorname{sgn}$ the signum function. Also $\operatorname{Re}$ means the real part of a complex number.

Im looking for real-entire functions $f(z)$ such that for $\operatorname{abs}(\operatorname{Re}(z))>\frac{1}{3}$ and $z = a + bi$:

$$ \operatorname{sgn}( \operatorname{Re}(f(a+bi))) = \operatorname{sgn}(a) $$

or stated differently:

$$ f(a + bi) = g + h i $$ $$ 0 < ga $$

I was inspired by $\operatorname{Si}(z)$.

I also wondered how fast these functions can grow in the direction of positive or negative infinity.

Ps: for those unaware : A real-entire functions is a function that is entire(analytic on the entire complex plane) and maps all reals to reals.

mick
  • 15,946
  • @vitamind thanks – mick Mar 23 '21 at 23:44
  • At the end you are trying to find functions that depend on the signum function is the last step, so you're only checking if $1=1$ or $1≠1$ etc. It's seems clear to me that these functions f do not have anything more specific in common. – vitamin d Mar 24 '21 at 11:51
  • sign(-7) = -1 sign(-0.1) = -1 sign (2) = 1. Are you talking about the same sgn function ? If so , I am unclear what you meant. @vitamind – mick Mar 24 '21 at 11:57
  • the trivial solution $a z + b$ for $a,b$ positive reals is ofcourse known. Possibly related : https://math.stackexchange.com/questions/616393/entire-function-fz-bounded-for-mathrmrez2-1?rq=1 – mick Mar 24 '21 at 11:58
  • Yes: https://en.wikipedia.org/wiki/Sign_function. Also the question about growth "I also wondered how fast these functions can grow in the direction of positive or negative infinity" doesn't make sense. – vitamin d Mar 24 '21 at 12:03
  • that question does make sense , the lineair solution $az + b$ grows linear in the direction of positive or negative infinity. $Si$ grows towards constants etc – mick Mar 24 '21 at 12:05
  • Do you agree with me that there are infinitely many linear, quadratic, cubic... functions that can satisfy the conditions you stated? – vitamin d Mar 24 '21 at 12:06
  • I dont know. z^5 does not satisfy it. – mick Mar 24 '21 at 12:13
  • What do you mean by ”real-entire” functions? Are you perhaps interested in analytic functions $f:\mathbb{C}\to\mathbb{C}$ such that the real part is an even function? – AD - Stop Putin - Mar 24 '21 at 21:17
  • A real-entire functions is a function that is entire and maps all reals to reals. – mick Mar 24 '21 at 22:19
  • @mick What do you mean by "Im looking for real-entire functions"? (real-entire is called (real)-analytic) What answer do you expect? Examples or a class functions for example "linear functions" or...? – vitamin d Mar 30 '21 at 21:23
  • wait real-analytic implies analytic near the real line. It could have a pole at $i$. So that is not the same. @vitamind – mick Mar 31 '21 at 21:14
  • @mick That's why real analytic $f\colon\mathbb{R}\to\mathbb{R}$. But please answer my other question. – vitamin d Apr 04 '21 at 13:24
  • How do you mean " what do you expect " ? I ask for a function with properties so I want a function with those properties. Or many of them. – mick Apr 08 '21 at 20:09

0 Answers0