Entire function of several complex variables

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 $\mathbb C^d$ .

For the Taylor claim, Cauchy's coefficient estimates on an arbitrary finite polydisc bound each coefficient by the supremum divided by the corresponding radii. Since the radii are arbitrary, the resulting multiple series converges at every point. The converse is Hartogs' line theorem: local boundedness and line holomorphy supply a joint power series on each coordinate polydisc.

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.