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i have a problem regarding holomorphy of a complex valued functions.

there are a lot of applications where the real part or imaginary part of a function is given and you have to find the function which is holomorphic.

ok...now my problem:

  • we know that for a real function u(x,y) that is harmonic on a simply connected domain, D, exist a complex valued function, f, holomorphic on D and Re(f)=u(x,y);

  • what happens when my function is not defined on a simply connected domain, such as:

$$u(x,y)=\frac{x}{x^2+y^2}+x$$

i found guys that solved a lot of problems like that but they say nothing about the domain of holomorphy!!

thanks ;)

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If the region $D$ is not simply connected, the statement can fail. The canonical example is the radial function $$u(x, y) := \tfrac{1}{2}\log(x^2 + y^2)$$ on its maximal domain, $\Bbb C - \{ 0 \}$.

If $u$ had a harmonic conjugate $v$, by the Cauchy-Riemann equations it would satisfy \begin{align} \frac{\partial v}{\partial y} &= \phantom{-}\frac{\partial u}{\partial x} = \phantom{-}\frac{y}{x^2 + y^2}\\ \frac{\partial v}{\partial x} &= -\frac{\partial u}{\partial y} = -\frac{x}{x^2 + y^2} . \end{align} But we can conclude from this that no such conjugate exists: If $S^1$ is the unit circle (oriented anticlockwise), evaluating $\int_{S^1} dv$ by expanding $dv = \frac{\partial v}{\partial x} dx + \frac{\partial v}{\partial y} dy$, substituting the above formulas, and, e.g., parameterizing, gives $\int_{S^1} dv = 2 \pi$. But since $S^1$ is compact, the Fundamental Theorem of Calculus gives $\int_{S^1} dv = 0$, a contradiction.

On the other hand, if we consider some simply connected subset $D' \subset \Bbb C - \{ 0 \}$, then $u\vert_{D'}$ does have a harmonic conjugate, and any such harmonic conjugate has the form $v(x, y) = \arg(x + iy) + v_0$ for some constant $v_0$, where $\arg$ is a choice of branch of the argument on $D'$.

For example, the occurrence of the expression $x^2 + y^2$ in the equations for the partial derivatives of $v$ suggest changing to polar coordinates, for which the equations become $\frac{\partial v}{\partial r} = 0$, $\frac{\partial v}{\partial \theta} = 1$, giving $v = \theta + \theta_0$. Converting back to rectangular coordinates on a suitable domain (say, $D' = \{x > 0\})$ and choosing a convenient $\theta_0$ gives $v(x, y) = \arctan \frac{y}{x}$.

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  • $\begingroup$ i think that Cauchy-Riemann equations aren't quite right, but the reasoning after that is ok. so, in a general case i should consider a subset of the definition set for the function that is given and analyze it there? how do i choose that subset? i mean it can be almost anything suitable, isn't it? $\endgroup$ Apr 11 '18 at 12:04
  • $\begingroup$ @AndreiCiobanașu You're right about the C-R equations of course, I was just careless with copy-and paste. It's now fixed I'm not sure I understand your latter questions. The theorem in particular implies that given any point $(x_0, y_0)$ in $D$, there's an open set $E$ containing $(x_0, y_0)$ (a sufficiently small ball centered there will suffice) on which $u\vert_E$ has a harmonic conjugate. $\endgroup$ Apr 11 '18 at 19:02

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