# Power series expansion without Cauchy theorem

How is the power series expansion of an analytic function at a point constructed, without using Cauchy's theorem(or formula)?

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Below are a couple of possible approaches.

MR0123687 (23 #A1010) 30.20
Porcelli, P.; Connell, E. H.
A proof of the power series expansion without Cauchy's formula.
Bull. Amer. Math. Soc. 67 1961 177--181.
http://projecteuclid.org/euclid.bams/1183524076

Starting with the basic result of topological analysis that a differentiable function of a complex variable generates an open mapping, the authors succeed in establishing in remarkably simple fashion, independent of any integration theory, the validity of the power-series expansion of such a function. A series of lemmas leads up to the theorem that differentiability of a function $f(z)$ on $0<|z-z_0| < r$ plus continuity at $z_0$ implies differentiability also at $z_0$. Using this, along with existence of higher-ordered derivatives, it is then shown that if $f(z)$ is continuous and $|f(z)|\leq 1$ for $|z|\leq 1$ and differentiable for $|z|<1$, then $|f^{(n)}(0)|\leq n!2^n$, for all $n$, and the Taylor series for $f$ converges to $f$ for $|z|<1/2$.
Reviewed by G. T. Whyburn

MR0993637 (90m:30002a) 30A99 (26A39 30B10 42A20)
Shisha, Oved(1-RI)
Proof of power series and Laurent expansions of complex differentiable functions without use of Cauchy's integral formula or Cauchy's integral theorem.
J. Approx. Theory 57 (1989), no. 2, 117--135.

MR1006337 (90m:30002b) 30A99 (26A39 30B10 42A20)
Shisha, Oved
Erratum: Proof of power series and Laurent expansions of complex differentiable functions without use of Cauchy's integral formula or Cauchy's integral theorem''.
J. Approx. Theory 58 (1989), no. 2, 246.

The author shows how to establish the basic results of complex function theory in the context of real variable Fourier analysis. His basic tool is a generalization of the classical Riemann integral, which he proposes should become the standard integral of the working analyst in place of currently used integrals (including the Lebesgue integral), all of which it contains as a special case. A topological development, eliminating all use of integrals, is due to \n E. Connell\en, \n R. L. Plunket\en, \n P. Porcelli\en, \n A. H. Read\en and \n G. T. Whyburn\en [Whyburn, Topological analysis, revised edition, Princeton Univ. Press, Princeton, NJ, 1964; MR0165476 (29 #2758)]. The present development generalizes the work of \n P. R. Beesack\en [Canad. Math. Bull. 15 (1972), 473--480; MR0310199 (46 #9301)], who placed various conditions on the derivative to achieve his results.

The relevant property of the new integral is the fact that if $f$ is an arbitrary differentiable function on an interval $[a, b]$, then $\int^b_a f'(x)\,dx=f(b)-f(a)$. For any function $f$ on $[a,b]$, the new integral is defined as the number $I$ such that for every $\varepsilon >0$, there exists a positive function $\delta_\varepsilon(x)$ on $[a,b]$, such that for every partition $a=x_0 < x_1 < \cdots < x_n=b$ and sequence $s_1,\cdots, s_n$, $x_{k-1} \le s_k\le x_k$, for $k=1,\cdots,n$, such that $x_k-x_{k-1}<\delta_{\epsilon}(k)$ for $k=1,\cdots,n$, one has $|I-\sum ^n_{k=1}f(s_k)(x_k-x_{k-1})|<\varepsilon$. Let $f$ be a complex differentiable function on an annulus $0\le R' < |z| < R''<\infty$. Then by Dini's test, since $f$ is everywhere differentiable, for each $r$, $R' < r < R''$, the function $f(re^{i\phi})$, $0\le \phi\le 2\pi$, can be expanded in a Fourier series $f(re^{i\phi}) =\sum^{+\infty}_{n=-\infty}c_k(r)e^{i\phi}$. The objective of the paper is achieved if it can be shown that the coefficients $a_k(r)$ are independent of $r$. The basic lemma used states that $$\iint_{\begin{array}{} R_1\le r\le R \\ 0\le \varphi\le 2\pi \end{array}} f'(re^{i\varphi})drd\varphi =\int_0^{2\pi}e^{-i\varphi}[f(Re^{i\varphi})-F(R_1e^{-i\varphi})]d\varphi,$$ $$\widehat{\iint}_{\begin{array}{} R_1\le r\le R \\ 0\le \varphi\le 2\pi \end{array}} f'(re^{i\varphi})drd\varphi =\int_{R_1}^{R}r^{-1}\int_0^{2\pi}e^{-i\varphi}f(re^{i\varphi})d\varphi dr,$$ where the integrals on the left are two-dimensional generalized Riemann integrals. The exposition is quite detailed and fully self-contained, including the development of all relevant characteristics of the generalized Riemann integral.
\edref {Two minor errors are corrected in the erratum.}
Reviewed by Kenneth O. Leland

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