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Let $\phi$ be a continuous (complex valued) function on the real interval [−1, 1] inside C, and define

$$f(z)=\int_{-1}^1\frac{\phi(t)}{t-z}dt$$

Show that f is analytic on C less the interval [−1, 1].

I thought about CR equation to prove analytic but the function is not of the form $f(z) = u(x, y) + iv(x, y)$

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  • $\begingroup$ What are the bounds of integration? $\endgroup$ – copper.hat Feb 5 '16 at 15:02
  • $\begingroup$ I added the bound. $\endgroup$ – user102239 Feb 5 '16 at 15:06
  • $\begingroup$ Since $f$ is defined only on the real interval $[-1,1]$, I speculate that the integral is along the real line, from $-1$ to $1$. In that case, the statement, if true, is a bit surprising. $\endgroup$ – Mark Fischler Feb 5 '16 at 15:06
  • $\begingroup$ Hi, can you elaborate on that? $\endgroup$ – user102239 Feb 5 '16 at 15:09
  • $\begingroup$ @MarkFischler Why would it be surprising? It's definitely meant to be the real integral. $\endgroup$ – Thomas Andrews Feb 5 '16 at 15:14
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If you wanted to use CR:

(This answer only works for $\phi$ real, but if $\phi=\phi_1+i\phi_2$ with $\phi_1,\phi_2$ real, you can see that it follows from the real case.)

Write $z=a+bi$ then $$\frac{\phi(t)}{(t-a)-bi} = \frac{\phi(t)(t-a +bi)}{(t-a)^2+b^2}$$

So $$\int_{-1}^{1} \frac{\phi(t)}{t-z}\,dt = \int_{-1}^1 \frac{\phi(t)(t-a)}{(t-a)^2+b^2}\,dt + i\int_{-1}^1 \frac{b\phi(t)}{(t-a)^2+b^2}\,dt$$

So there are your $u(a,b)$ and $v(a,b)$.

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Let $\gamma$ be a closed curve in $\mathbb{C} \setminus [-1,1]$, not enclosing $[-1,1]$. Then \begin{align} \int_\gamma f(z)\,dz &= \int_\gamma \left( \int_{-1}^1 \frac{\phi(t)}{z-t}\,dt \right) \, dz \\ &= \int_{-1}^1 \left( \int_{\gamma} \frac{\phi(t)}{z-t}\,dz \right) \, dt \\ &= \int_{-1}^1 \phi(t) \left( \int_{\gamma} \frac{1}{z-t}\,dz \right) \, dt \\ &= \int_{-1}^1 \phi(t) \cdot 0 \, dt = 0 \end{align} since the inner integral vanishes for every $t$ (by Cauchy's integral theorem). Morera's theorem now shows that $f$ is holomorphic on $\mathbb{C} \setminus [-1,1]$. (I'll leave it to you to verify that $f$ is continuous.)

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The functions $\eta(z,t) = {\phi(t) \over t-z}$ and ${\partial \eta \over \partial z}$are continuous on $[-1,1]^c \times [-1,1]$, hence by the Leibnitz rule we see that $f$ is differentiable and $f'(z) = \int_{-1,1]} {\partial \eta(z,t) \over \partial z} dt = \int_{-1,1]} {\phi(t) \over (t-z)^2} dt$.

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    $\begingroup$ This is the best answer :) $\endgroup$ – Vincenzo Zaccaro Feb 5 '16 at 15:35
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Hint:

$$u(x,y)=\Re\int_{-1}^1\frac{\phi(t)\,dt}{t-z},\\v(x,y)=\Im\int_{-1}^1\frac{\phi(t)\,dt}{t-z}.$$

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  • $\begingroup$ Crossed with @mrf. $\endgroup$ – Yves Daoust Feb 5 '16 at 15:15

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