Do there exist functions which are differentiable everywhere but analytic nowhere?

Asked Oct 07 '18 at 16:24

Active Oct 07 '18 at 16:34

Viewed 209 times

Once upon a time, when the earth was still young and innocent I studied one complex variable. In this course I learned that a function can be complex analytic, but that there are far fewer functions than the real differentiable ones which are complex analytic.

Can we find an example of a function which is real differentiable everywhere but complex analytic nowhere?

If I recall right, the requirements for analyticity of function $x+yi\to u(x,y)+ iv(x,y)$ are known as the Cauchy-Riemann equations:

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0\\ \frac{\partial u}{\partial y} - \frac{\partial v}{\partial x} = 0$$

Let us disqualify $z\to \bar z$, it is too uninteresting example.

edited Oct 07 '18 at 16:34

asked Oct 07 '18 at 16:24

mathreadler

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$f(z)=\bar{z}$, for example. – user10354138 Oct 07 '18 at 16:30
@user10354138 ah yes, i came to think of it right after posting, i will try to edit question. its far too simple now. – mathreadler Oct 07 '18 at 16:31
use it or lose it :) – AlvinL Oct 07 '18 at 16:32
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If you rewrite CR equations in the form $$\frac{\partial f}{\partial \overline z}=0, $$ it will become more apparent why $f(z)=\overline z$ fails to be a solution. – Giuseppe Negro Oct 07 '18 at 16:34
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I think $e^{-x^2}$ is another example. – MJD Oct 07 '18 at 16:34
@MJD : Mhm, maybe, it will be chirps in quadrature in imaginary direction, no? Quite interesting function. – mathreadler Oct 07 '18 at 16:36
@AlvinLepik Probably the course I remember most from all-in-all from those days. :) – mathreadler Oct 07 '18 at 16:38
@GiuseppeNegro that is an interesting form, I was not aware of it. – mathreadler Oct 07 '18 at 16:38
Then consider $f(z) = |z|^2$, which is complex differentiable only at the origin. – Giuseppe Negro Oct 07 '18 at 16:43
@GuiseppeNegro But it involves conjugate since $|z|^2=z \cdot \bar z$ – mathreadler Oct 07 '18 at 16:46
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"real differentiable everywhere but complex analytic nowhere" --- Since this is rather trivial (conjugation function, as pointed out by @user10354138), I wonder if you were thinking of real-valued $C^{\infty}$ functions that are nowhere real-analytic? – Dave L. Renfro Oct 07 '18 at 16:52
@DaveL.Renfro I was just trying to reformulate it as I realized it was too easy. Hmm, maybe something like that. Wow cool, did you investigate it? – mathreadler Oct 07 '18 at 16:56
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This is something I collected some literature on during the mid to late 1990s, and I wrote a summary of the topic in May 2002, here and here. – Dave L. Renfro Oct 07 '18 at 18:58

Do there exist functions which are differentiable everywhere but analytic nowhere?

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