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Suppose function $F: \mathbb{C}^n \to \mathbb{C}$ is analytic everywhere and in every coodinate; i. e. for any $q \in \mathbb{C}^n$ and for any $j \in \{1,2,\ldots,n\}$ function $f_{q,j}: \mathbb{C} \to \mathbb{C}$ defined as $f_{q,j}(z) = F\left(q_1, \ldots, q_{j-1}, q_j+z, q_{j+1}, \ldots, q_n\right)$ is analytic at $0$.

Rudin finds it obvious that, under that assumption, the following holds: for any $a, b \in \mathbb{C}^n$ function $g_{a,b}: \mathbb{C} \to \mathbb{C}$ defined as $g(z) = F(a+bz)$ is an entire function. How does it follow?

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There are many equivalent definitions of analytic functions in several variables. For example, a function is analytic if it analytic of each variable separately (the definition given by the OP).

This is for example equivalent to having local power series expansions. (The proof is very similar to the one variable case.)

Or, if you prefer this is also equivalent to the fact that $F$ satisfies Cauchy-Riemann's equation, which for $n > 1$ is a system of linear partial differential equations: $$\frac{\partial F}{\partial \bar z_j} = 0, \qquad 1\le j \le n,$$ where $$ \frac{\partial}{\partial \bar z_j} = \frac12\left( \frac{\partial}{\partial x_j} - \frac{\partial}{\partial y_j}\right).$$

All of this should be worked out in detail in any textbook in several complex variables.

As a consequence, essentially by the chain rule and the third of the above equivalent definitions, if $\phi : \mathbb{C} \to \mathbb{C}^n$ and $F : \mathbb{C}^n \to \mathbb{C}$ are analytic, then the composition $F \circ \phi$ is also analytic. (This is also true in more generality: the composition of two analytic functions, regardless of dimension is analytic.)

In your particular case, $\phi(z) = a+bz$ is clearly an analytic function $\mathbb{C} \to \mathbb{C}^n$, so $g_{a,b}$ is entire.

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