# Exercises

## Functions of Several Complex Variables

### Question

Let $$n \geq 2$$, let $$\Omega$$ be an open subset of $$\mathbb{C}^n$$ and let $$f: \Omega \mapsto \mathbb{C}$$ a continuous function. Show that $$f$$ is complex-differentiable in $$\Omega$$ if and only if for any $$(z_1, \dots, z_n) \in \Omega$$, the partial function $f_{k,z}: w \mapsto f(z_1, \dots, z_{k-1}, w, z_{k+1}, \dots, z_n)$ is holomorphic.

We may define the embedding functions $$e_{k,z}: \mathbb{C} \to \mathbb{C}^n$$ by $e_{k,z}(w) = (z_1, \dots, z_{k-1}, w, z_{k+1}, \dots, z_n).$ It is plain that the $$e_{k,z}$$ are continuous. A function $$f_{k,z}$$ is defined on the preimage of the open set $$\Omega$$ by $$e_{k,z}$$ which is therefore an open set.
Assume that $$f$$ is complex-differentiable; it is continuous. Additionally, $$f_{k,z} = f \circ e_{k,z}$$; as the function $$e_{k,z}$$ is complex-linear, it is complex-differentiable and $$f_{k,z}$$ is complex-differentiable (or holomorphic) as the composition of complex-differentiable functions.
Conversely, if $$f$$ if continuous and every partial function $$f_{k,z}$$ is complex-differentiable, the function $$f$$ itself is complex-differentiable as every partial derivative $$z \in \Omega \mapsto (\partial f/\partial z_k) (z)$$ is continuous – not merely as a function of its $$k$$-th variable which is plain, but as a function of all its variables.
Let $$z=(z_1,\dots, z_n) \in \Omega$$, let $$c \in \mathbb{C}$$ and $$r>0$$ such that $\forall \, w \in \mathbb{C}, \; |w - c| \leq r \; \to \; (z_1,\dots,z_{k-1},w,z_k,\dots, z_n) \in \Omega.$ Cauchy’s formula, applied to the partial function $$f_{k,z}$$ for the path $$\gamma =c + r[\circlearrowleft]$$, provides $f(z_1,\dots, z_n) = \frac{1}{i2\pi} \int_{\gamma} \frac{f(z_1, \dots, z_{k-1}, w, z_{k+1}, \dots, z_n)}{w-z_k} dw$ The integrand is continuous with respect to the pair $$(z_1,w_1)$$ and complex-differentiable with respect to $$z_1$$, thus we may compute the partial derivative of $$f$$ with respect to $$z_k$$ satisfies by differentiation under the integral sign: $\frac{\partial f}{\partial z_k} (z_1,\dots, z_n) = \frac{1}{i2\pi} \int_{\gamma} \frac{f(z_1, \dots, z_{k-1}, w, z_{k+1}, \dots, z_n)}{(w-z_k)^2} dw.$ As the function $$f$$ is continuous, the partial derivative is also continous.