Holomorphic Parameter Integral

Let $U\subseteq\mathbb{C}$ be open, $\gamma$ a way in $\mathbb{C}$ which is picewise differentiable continiously and $f\colon rg(\gamma)\times U\to\mathbb{C}$ a continious function. Consider the parameter integral $$ F(z):=\int_{\gamma}f(\omega,z), d\omega, z\in U. $$ Show: Let the function $z\mapsto f(\omega,z)$ be holomorphic in $U$ for every $\omega\in rg(\gamma)$ with continious derivation $\frac{\partial}{\partial z}f(\omega,z)$ on $rg(\gamma)\times U$. Then $F(z)$ is holomorphic in $U$, and one can differentiate under the integral: $$ F'(z)=\int_{\gamma}\frac{\partial}{\partial z}f(\omega,z)\, d\omega, z\in U $$

Unfortunately I do not know how to show that. I know the proof when $f$ is a function with values in $\mathbb{R}$ but cannot proof it here when $f$ has values in $\mathbb{C}$.

Solution 1:

Firstly, you can prove it with real methods, splitting the function and integrals into real and imaginary parts, and using the characterisation of holomorphic functions by the Cauchy-Riemann equations.

We have, however, nicer methods at our disposal in complex analysis. One very nice tool is

Theorem (Morera):

Let $U \subset \mathbb{C}$ open, and $f \colon U \to \mathbb{C}$ a continuous function. If, for all closed triangles $\Delta \subset U$, you have

$$\int_{\partial \Delta} f(z)\, dz = 0,$$

then $f$ is holomorphic on $U$.

(Cauchy's integral theorem provides the converse.)

Armed with that, we let $\Delta$ an arbitrary closed triangle in $U$ and compute

$$\begin{align} \int_{\partial \Delta} F(z)\,dz &= \int_{\partial \Delta} \int_\gamma f(\omega,\,z) \, d\omega\,dz\\ &= \int_\gamma \int_{\partial \Delta} f(\omega,\, z)\,dz\, d\omega & \text{(Fubini)}\\ &= \int_\gamma 0\, d\omega & \text{(Cauchy)}\\ &= 0, \end{align}$$

hence conclude by Morera's theorem that $F$ is holomorphic in $U$.

Since the partial derivative $\frac{\partial f}{\partial z}$ is continuous by the premise, the function

$$G(z) = \int_\gamma \frac{\partial f}{\partial z}(\omega,\, z)\,d\omega,$$

is continuous on $U$ (even holomorphic, by Morera), and we can compute

$$\begin{align} \int_\alpha G(z)\,dz &= \int_\alpha \int_\gamma \frac{\partial f}{\partial z}(\omega,\, z)\,d\omega\,dz\\ &= \int_\gamma \int_\alpha \frac{\partial f}{\partial z}(\omega,\, z)\,dz\,d\omega & \text{(Fubini)}\\ &= \int_\gamma f(\omega,\, \alpha(1)) - f(\omega,\,\alpha(0))\,d\omega\\ &= F(\alpha(1)) - F(\alpha(0)) \end{align}$$

for each path of integration $\alpha \colon [0,\,1] \to U$. That implies that $F'(z) = G(z)$.

(To make it explicit, fix $z_0 \in U$ and choose $\alpha$ a straight line segment connecting $z_0$ and $z \in U$ - for $r := \lvert z-z_0\rvert$ small enough, the segment is contained in $U$. Then

$$\begin{align} \left\lvert\frac{F(z) - F(z_0)}{z-z_0} - G(z_0)\right\rvert &= \left\lvert\frac{1}{z-z_0}\int_\alpha G(\zeta) - G(z_0)\, d\zeta\right\rvert\\ &\leqslant \frac{1}{\lvert z-z_0\rvert}\int_0^1 \lvert G(z_0 + t(z-z_0)) - G(z_0)\rvert\cdot \lvert z-z_0\rvert\, dt\\ &\leqslant \max_{\lvert w-z_0\rvert \leqslant r} \lvert G(w) - G(z_0)\rvert, \end{align}$$

and the continuity of $G$ in $z_0$ shows the desired convergence.)

Solution 2:

Using Cauchy's integral formula , we have$$f(w,z)=\frac{1}{2\pi i} \oint_C \frac{f(w,\zeta)}{\zeta-z}\, \mathrm{d} \zeta,$$ where $C$ is the boundary of a disc contained in $U$, which encloses $z$.

Therefore

$$F(z)=\frac{1}{2\pi i}\int_\gamma \oint_C \frac{f(w,\zeta)}{\zeta-z}\, \mathrm{d} \zeta \mathrm{d} w. $$ Using Fubini's theorem we can change the order of integration, so that

$$F(z)=\oint_C \left[ \frac{1}{2\pi i} \int_\gamma f(w,\zeta)\, \mathrm{d}w \right] \frac{\mathrm{d} \zeta}{\zeta-z}. $$

Integrals of the form $\oint_C \frac{\varphi(\zeta)}{\zeta-z} \mathrm{d} \zeta$, with continuous $\varphi$ can be differentiated under the integral symbol (cf. Lemma 3 in Chapter 4 of Ahlfors' text).

Thus $$F'(z)=\oint_C \left[ \frac{1}{2\pi i} \int_\gamma f(w,\zeta)\, \mathrm{d}w \right] \frac{\mathrm{d} \zeta}{(\zeta-z)^2}, $$ and applying Fubini's theorem once more we find $$F'(z)= \int_\gamma \left[ \frac{1}{2 \pi i} \oint_C \frac{f(w,z)}{(\zeta-z)^2} \mathrm{d} \zeta \right] \mathrm{d} w $$

We recognize the inner integral as $\frac{\partial f}{\partial z}$ (Cauchy's integral formula). Overall we have shown $$F'(z)=\int_\gamma \frac{\partial}{\partial z} f(w,z) \mathrm{d} w $$