By Hirzebruch F., Scharlau W.
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OLONIORPHIC FUNCTIONS from the maximum principle for harmonic functions that for any y > 0. log If(z)I 5 u(z) - u(--)< -i log If(z)I, z E A. Hence u(z) = u(z) on w. Similarly, we have i(z) = v(z) in w for u(z) = 3'f(z). We set f (z) := 11(z) + i%(z) in A. Since f (z) = f (z) in w. it follows that f (z) is holomorphic in A. On the other hand, both f(z) and f(z) are continuous on A and the zero set of f (z) is isolated in ,: hence. e is isolated and f (z) f (z) in J. 5. Holomorphic Mappings. We let z = (z,.....
Therefore w(z,w) is a holomorphic function in A and w(z, w) 0 0 in 0° x r. 1 that w(z, w) 34 0 on A. Summarizing these results, we have the following theorem.
Contrary to the case of one complex variable, there exist domains D in C". n > 1, with C" \D having non-empty interior but such that every bounded holomorphic function in D is constant. 2) that the complement of a ball has this property. 1. HOLO\MORPHIC FUNCTIONS AND DOMAINS OF HOLOMORPHY 16 2. Identity theorem. Let f (z) and gg(z) be holomorphic functions in a domain D in C". If f (z) = g(z) for all z in a non-empty open set 6 in D. then f (z) __ g(z) in D. Hence, analytic continuation of holomorphic functions in several complex variables can be performed as in the case of one complex variable.