9.1 Consequences of Holomorphy
Obviously, we will first formally define the concept of holomorphy in higher dimensions.
Definition 9.1.1.
A function is holomorphic if it is holomorphic in each variable when the others are held constant.
If we consider to be a function of , then is holomorphic iff for all and has all continuous partial derivatives.
Theorem 9.1.1 (Cauchy's Integral Formula on Polydisks).
Fix arbitrarily and suppose are the radii of the polydisk defined by (where the product here is the Cartesian product). Suppose is holomorphic. For fixed , we have that
for any .
Proof.
By Cauchy–Goursat (Theorem 3.2.1), we have
for , which is holomorphic. Thus, by the same application on , we have
By reiterating times and reversing the order of differentiation and integration, the conclusion follows.
By the boundedness assumption for , we have:
Corollary 9.1.1 (Cauchy's Estimate on Polydisks).
Let be fixed and suppose are the radii of the polydisk defined by (where the product here is the Cartesian product). Suppose is holomorphic. For fixed , we have that
for any .
Proof.
For each , let satisfy . By Cauchy’s Integral Formula (Theorem 9.1.1), we have
For each , let , and it follows that . Because , we have, after substitution,
since for all .
Similar to the univariate case, there are Taylor expansions of holomorphic functions in several complex variables.
Theorem 9.1.2.
Let be holomorphic on (a neighborhood of) the closure of a polydisk centered at . Then, for any , we have the expansion
where ,
The series converges absolutely and uniformly on .
Proof.
By Theorem 9.1.1 we have
For each , since on , the geometric series expansion holds:
which converges uniformly in on . Hence, we have
where uniform convergence has allowed the interchange of summation and integration. Reiteration of this process gives
By the Cauchy Integral Formula (Theorem 9.1.1), we have
and hence if we let
then (9.1.1) follows. Cauchy’s Estimate (Corollary 9.1.1) gives that
where for some for all . Hence,
By the Weierstrass –Test (Theorem 2.3.2), the series converges absolutely and uniformly on .
Theorem 9.1.3 (Identity).
Let be a holomorphic function on . If the set has an accumulation point in , then on .
Theorem 9.1.4 (Maximum Modulus Principle).
Let be a open bounded region, and suppose that is holomorphic. If
then for all , unless is constant.
Theorem 9.1.5 (Weierstrass).
Suppose that is a region and that is a sequence of holomorphic functions . If converges locally uniformly to on , then is holomorphic on . Moreover, ,
on compact subsets of .
Theorem 9.1.6 (Montel).
A family of holomorphic functions on some region is normal iff it is locally uniformly bounded on .