The Exponential Function

Gerhard P. Hochschild
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Gerhard P. Hochschild: University of California, Department of Mathematics

Chapter Chapter VII in Perspectives of Elementary Mathematics, 1983, pp 79-92 from Springer

Abstract: Abstract Let z denote the identity map on C. For every non-negative integer n, we define a polynomial function E n by $$ {E_n} = \sum\limits_{{k = 0}}^n {\frac{1}{{k!}}{z^k}} $$ Given an arbitrary complex number c, let n be such that n + 1 ≧ 2|c|, and let q be an arbitrary positive integer. Then we have $$\begin{array}{*{20}{c}} {\left| {{E_{n + q}}\left( c \right) - {E_n}\left( c \right)} \right|\underline \leqslant \sum\limits_{k = 1}^q {\frac{1}{{\left( {n + k} \right)!}}{{\left| c \right|}^{n + k}}} } \\ {\underline \leqslant \frac{{{{\left| c \right|}^{n + 1}}}}{{\left( {n + 1} \right)!}}\sum\limits_{k = 1}^q {{{\left( {\frac{{\left| c \right|}}{{n + 1}}} \right)}^{k - 1}}} } \\ {\underline \leqslant 2\frac{{{{\left| c \right|}^{n + 1}}}}{{\left( {n + 1} \right)!}}} \end{array}$$ This shows that the sequence (E n (c)) n =0, 1,... is a Cauchy sequence of complex numbers. For every complex number c, we define the complex number exp(c) as the limit of this Cauchy sequence. The function exp from C to C so defined is the exponential function.

Date: 1983
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DOI: 10.1007/978-1-4612-5567-3_7

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