Is there a non-constant real valued function in $D$ which is analytic in $D$?

Is there exists a non-constant real valued function in $D$ which is analytic in $D$?

• What is $D$? The disk? Some region? Other? Oct 29, 2014 at 7:20
• $D$ is a region in $\mathbb C$.@AdamHughes Oct 29, 2014 at 8:36
• Then the open mapping theorem doesn't allow this. Oct 29, 2014 at 8:50
• so???@AdamHughes Oct 29, 2014 at 8:59

No. For instance, from Cauchy-Riemann equations.

• So u mean that every analytic real valued function in a region is constant. right??????@orangeskid Oct 29, 2014 at 8:37
• @David: Yep $\ \ \ \ \ \ \ \$ Oct 29, 2014 at 9:18

Actually there's a theorem that says that if $$\mathbb{D}$$ is a region of $$\mathbb{C}$$ and $$f$$ is holomorphic in this region, then they're equivalent:

1. The real part of $$f$$ is constant
2. The imaginary part of $$f$$ is constant
3. $$f$$ is constant
4. $$\overline{f}$$ is holomorphic
5. $$|f|$$ is constant

So as you can see if $$f$$ is a non-constant real valued function then it can't be holomorphic.