Introduction to Several Complex Variables: Lectures by by Lipman Bers, Marion S. Weiner, Joan Landman

By Lipman Bers, Marion S. Weiner, Joan Landman

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Definition 20. Let q; E C 2. The the Hessian of 1/) is defined to be the following matrix: Note that His Hermitian, if ,P is real valued. Proposition 1. Let DC «:n. IR, fjJ € c 2 . Then~ is pseudoconvex if and only if the Hessian of 4> is positive semidefinite. Proof. P(~) = ,P(z 1 + ta1, ... 1 £ «: ' =1, ... , n . Now 0(~) is subharmonic if and only if At~ 0 . J 4ar n a2! '\aj azka:Z. J Proposition 2. Let DC

S is locally arcwise connected. Corollary 3. If n > 1 and f· is holomorphic in D d: en, then the zeroes of f are not isolated. § 2. Rings of power series A. As we have remarked previously, the set of convergent power series in z1 , ••• ,zn at a point forms a ring, as does the set of formal power series. We shall now state some algebraic results which will prove useful in the sequel. g. van der Waerden, Moderne Algebra, for proofs and details. - Let R be a commutative unitary ring. R is said to be an integral domain if: a E R, b E R, ab = 0 implies a = 0 or b = o.

Zl Consider rxfz~; define tj = zj/z 1 , j = 2, ••. ,n. Then fxfz~ is a polynomial in the tj ; rK K = 1 +r , zl where r is a homogeneous polynomial in the t without a constant term. Therefore, for lt 3 1 < p1 , lrt < 1/2. Under these conditions: f r:K rK + (rK+l + ••• ) fK rK+l+ ••• /z~ = 1 + K fK/zl rK+l+ • • • 1 = 1 + K l+r zl = 1 = +q I where q is a power series in z1 and the ourselves to the above inequalities; hence < 1 tj. We restrict • Choosing some determination of log, we obtain: log ~ K = log (l+q) = q - q2/2 + q3/3 - q4/4 + ...

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