1

Let's explain it more detail.

Suppose that $U\subseteq \mathbb{C}$ is an open set. Let $F\in C^0(U)$. Suppose that for every $\bar{D}(z,r) \subseteq U$ and $\gamma$ the curve surrounding this disc (with counterclockwise orientation) and all $w\in D(z,r)$ it holds that \begin{align} F(w) = \frac{1}{2\pi i} \oint_{\gamma} \frac{F(\zeta)}{\zeta-w} d\zeta \end{align} Textbook says that $F$ is holomorphic.

How do I prove this from the definition of holomorphic. $i.e$, $\frac{\partial F}{\partial \bar{w}}=0$.

BCLC
  • 13,459
phy_math
  • 6,448
  • Write down the definition of the integral (choose a smooth curve..) and check whether the result may differentiated under the integral. Then verify the condition you are after. – Thomas Sep 24 '16 at 09:47
  • @Thomas and phy_math related? https://math.stackexchange.com/questions/986776 – BCLC Apr 17 '21 at 11:08

1 Answers1

2

Your hypothesis allow you to take the derivative inside the integral sign: $$ \partial_{\bar{w}} F(w) = \frac{1}{2\pi i} \oint_\gamma F(\zeta) \left(\partial_{\bar{w}} \frac{1}{\zeta-w}\right) \; d\zeta = 0 $$

H. H. Rugh
  • 35,236
  • 1
    Seriously, $\partial_{\overline{w}}$ ? Don't you think it is more much pedagogic to look at $d/dw$ ?$ F(w+h)-F(w) = \frac{1}{2i\pi} \int_\gamma F(\zeta) (\frac{1}{\zeta-(w+h)}-\frac{1}{\zeta-w})d\zeta = \frac{1}{2i\pi} \int_\gamma F(\zeta) (h\frac{d}{dw}\frac{1}{\zeta-w} + o(h))d\zeta$ – reuns Sep 24 '16 at 09:57
  • 1
    @user1952009 Agree, it is a rather abstract approach but this was what was asked for in the question. – H. H. Rugh Sep 24 '16 at 10:42
  • 1
    Oh I didn't see it, sry. But I still hate $\partial / \partial \overline{w}$ (except in the context of antiholomorphic functions) – reuns Sep 24 '16 at 10:44
  • 1
    @user1952009 It is sometimes quite pratical. Any $C^1$ function of real $x,y$ may be written as a function of $w=x+iy$ and $\bar{w}=x-iy$. $f$ then admits partial derivatives wrt $w$ and $\bar{w}$ and the Cauchy Riemann equations are equivalent to $\partial_{\bar{w}} f=0$. – H. H. Rugh Sep 24 '16 at 11:16
  • 1
    Yes but this is tautological. – reuns Sep 24 '16 at 11:42
  • @reuns and H. H. Rugh related? https://math.stackexchange.com/questions/986776 – BCLC Apr 17 '21 at 11:07