Holomorphic Functions

  • Kunihiko Kodaira
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 283)


We begin by defining holomorphic functions of n complex variables. The n-dimensional complex number space is the set of all n-tuples (z1,…, z n ) of complex numbers z i , i = 1,…, n, denoted by ℂ n . ℂ n is the Cartesian product of n copies of the complex plane: ℂ n = ℂ × … × ℂ. Denoting (z1,…, z n ) by z, we call z = (z1,…, z n ) a point of ℂ n , and zl,…, z n the complex coordinates of z. Letting z j = x2j−1 + ix2j by decomposing z j into its real and imaginary parts (where \(i = \sqrt { - 1} \)), we can express z as
$$ z = ({x_1},{x_2}, \ldots ,{x_{2n - 1}},{x_{2n}}). $$


Power Series Holomorphic Function Analytic Continuation Power Series Expansion Irreducible Factorization 
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Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • Kunihiko Kodaira
    • 1
  1. 1.Shinjuku-Ku, TokyoJapan

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