Elementary Theory of Analytic Functions of One or Several Complex Variables

Henri Cartan

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486318677
• Topic: Mathematics
• Subtopic: Functional Analysis
• Index: Mathematics

Noted mathematician offers basic treatment of theory of analytic functions of a complex variable, touching on analytic functions of several real or complex variables as well as the existence theorem for solutions of differential systems where data is analytic. Also included is a systematic, though elementary, exposition of theory of abstract complex manifolds of one complex dimension. Topics include power series in one variable, holomorphic functions, Cauchy’s integral, more. Exercises. 1973 edition.

CHAPTER I

Power Series in One Variable

1. Formal Power Series

1. ALGEBRA OF POLYNOMIALS

Let K be a commutative field. We consider the formal polynomials in one symbol (or ‘indeterminate’) X with coefficients in K (for the moment we do not give a value to X). The laws of addition of two polynomials and of multiplication of a polynomial by a ‘scalar’ makes the set K[X] of polynomials into a vector space over K with the infinite base

Each polynomial is a finite linear combination of the Xⁿ with coefficients in K and we write it

, where it is understood that only a finite number of the coefficients a_n are non-zero in the infinite sequence of these coefficients. The multiplication table

defines a multiplication in K[X]; the product

is

, where

This multiplication is commutative and associative. It is bilinear in the sense that

for all polynomials P, P₁ P₂, Q, and all scalars λ. It admits as unit element (denoted by 1) the polynomial

such that a₀ = 1 and a_n = 0 for n > 0. We express all these properties by saying that K[X], provided with its vector space structure and its multiplication, is a commutative algebra with a unit element over the field K; it is, in particular, a commutative ring with a unit element.

2. THE ALGEBRA OF FORMAL SERIES

A formal power series in X is a formal expression

, where this time we no longer require that only a finite number of the coefficients a_n are non-zero. We define the sum of two formal series by

and the product of a formal series with a scalar by

The set K[[X]] of formal series then forms a vector space over K. The neutral element of the addition is denoted by 0; it is the formal series with all its coefficients zero.

The product of two formal series is defined by the formula (1.1), which still has a meaning because the sum on the right hand side is over a finite number of terms. The multiplication is still commutative, associative and bilinear with respect to the vector structure. Thus K[[X]] is an algebra over the field K with a unit element (denoted by 1), which is the series

such that a₀ = 1 and a_n = 0 for n > 0.

The algebra K[X] is identified with a subalgebra of K[[X]], the subalgebra of for...