Several Variables
A complex analytic function of several complex variables is defined to be analytic and holomorphic at a point if it is locally expandable (within a polydisk, a Cartesian product of disks, centered at that point) as a convergent power series in the variables. This condition is stronger than the Cauchy–Riemann equations; in fact it can be stated as follows:
A function of several complex variables is holomorphic if and only if it satisfies the Cauchy–Riemann equations and is locally square-integrable.
Read more about this topic: Holomorphic Function
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