Entire function of several complex variables

Definition

An entire function of $d$ complex variables is a function $F:\mathbb C^d\to\mathbb C$ that is holomorphic at every point. Equivalently, near every point it has a convergent power series in $d$ variables; by analytic continuation the Taylor series about any point has a domain of convergence covering all of $\mathbb C^d$ .

Relation to complex lines

An entire function restricts to an entire function of one variable on every affine complex line. Conversely, a locally bounded function whose restriction to every complex line is holomorphic is holomorphic on $\mathbb C^d$ .

Exponential type

Growth conditions of the form $|F(x+iy)|\le A e^{\sigma|y|}$ encode bounded Fourier support of the restriction $F|_{\mathbb R^d}$ through the Paley–Wiener theorem.

References

Steven G. Krantz, Function Theory of Several Complex Variables, 2nd ed., AMS Chelsea, 2001. Publisher record.