From Titchmarsh, Theory of Functions, 2nd ed 1939:
[His definition of an analytic function appears to be that $f(z)$ is analytic at $z_0$ if $\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}$ exists (sections 2.13 and 2.14). He then goes on in section 2.2 to say:]
$``$The reader might expect that we should now proceed after the manner of the real differential calculus. There, having distinguished the special class of differentiable functions, we next consider the still more special class of functions which have a second differential coefficient. Some of these functions have differential coefficients of the third order, and so on. Finally, from among functions with have differential coefficients of all orders, we pick out those which can be expanded in a power series by Taylor's Theorem. There is no such process of successive specialisation for analytic functions of a complex variable. A function which is analytic in a region has differential coefficients of all orders at all points of the region, and the function can be expanded in a power series ... about any point of the region. All these facts follow from the definition of an analytic function by means of its first differential coefficient.
The reader would perhaps expect us to begin by proving that an analytic function has a second differential coefficient. We are unable to do this. The results have of course been proved... but they have never been proved directly. They all depend on the complex integral calculus.$''$ [My italics]
Is that still true? If not, could someone sketch out a direct proof or provide a reference?
I notice Complex differentiable implies analytic but the question is not quite the same and the answer does not really help me. Similarly, the Looman-Menchoff seems to address a different point.
[Edward Charles Titchmarsh (1899-1963) was a leading British mathematician. His most famous work was his book/monograph on the Riemann Zeta Function (1st ed 1930). He was Savilian Prof of Geometry at Oxford 1932-1963.]
[Silly TeX question: why haven't my opening and closing quotes worked properly? Is it a MathJax thing?]