Sign up ×
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.

I know $Aut(\mathbb{C})=\{f(z): f(z)=az+b\}$ for some constant $a,b$,$a\in\mathbb{R}$, my question is if an entire function which maps Horizontal line to Horizontal line then it is of the form $az+b$? or in general $f$ maps any lines to to lines then is it of the form $az+b$?

share|cite|improve this question

2 Answers 2

up vote 2 down vote accepted

An entire function $f$ that maps horizontal lines to horizontal lines must have a derivative $f'$ that is real and $\ne0$ at all points $z\in\Bbb C$. It follows that $f'$ has to be a real nonzero constant. Therefore the most general $f$ satisfying your description is of the form $$f(z)=\sigma z+ c,\qquad \sigma\in\dot{\mathbb R},\quad c\in \Bbb C\ .$$

share|cite|improve this answer

Any function that maps horizontal lines to horizontal lines must be of the form

$$ f(x + \mathbf{i} y) = u(x,y) + \mathbf{i} v(y)$$


share|cite|improve this answer
why ?add detail please. – La Belle Noiseuse Mar 22 '13 at 17:06
.... A horizontal line is completely determined by its imaginary part. – Hurkyl Mar 23 '13 at 3:53

Your Answer


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.