Finding conditions that assure that a holomorphic function i

[lolittaFarhat]

Let H be a real-valued function of two real variables with continuous first partial derivatives. If h(z)=h(x+i y)= u(x,y)+iv(x,y)is holomorphic in a region V, and H(u,v)=0. Find a condition on H under which one can conclude that h is a constant . Of, course H is the zero function can do the job, but i do not want a trivial condition, any clue?

 

[Samy_A]

Let H be a real-valued function of two real variables with continuous first partial derivatives. If h(z)=h(x+i y)= u(x,y)+iv(x,y)is holomorphic in a region V, and H(u,v)=0. Find a condition on H under which one can conclude that h is a constant . Of, course H is the zero function can do the job, but i do not want a trivial condition, any clue?

If H is the zero function, H(u,v)=0 for any choice of functions u,v, and hence for every holomorphic function in V. So H=0 can't be an answer.

Hint:
1) What can you tell about $\frac{dH}{dx}$ and $\frac{dH}{dy}$ in V?
2) What do you know about the first partial derivatives of the real an imaginary part of an holomorphic function?

 

[lolittaFarhat]

$\frac{dH}{dx}$ and $\frac{dH}{dy}$ in V are continuous, how can this help?

 

[Samy_A]

$\frac{dH}{dx}$ and $\frac{dH}{dy}$ in V are continuous, how can this help?

You know that H(u,v)=0.
More precisely, for $x+iy \in V$, $H(u(x,y),v(x,y))=0$.
That should tell you more about the derivatives (when $x+iy \in V$) than that they are continuous.
Also, the first partial derivatives of the real an imaginary part of an holomorphic function satisfy a very specific set of equations.