ABSTRACT
It was shown that if a function F (z) is holomorphic in a region or satisfies the Cauchy conditions (stated in Section 15.1 and proven in Sections 19.1-19.6), then it has derivates of all orders at all interior points (Sections 15.1-15-4). Further it can be shown that the holomorphic function also has a convergent Taylor series, that is, a complex differentiable function is also an analytic function (Section 23.7). This is not true of real functions, since the existence of first-order derivate does not imply the existence of higher-order derivates; nor the convergence of the Taylor series for infinitely differentiable (or smooth) functions. The Taylor (Maclaurin) series allows the calculation, from the value of the function and all its derivates at a point z = a (the origin z = 0) its value at all other points z in its circle of convergence; the region of convergence of a Taylor (Maclaurin) series is a circle, because the expansion proceeds in integral ascending powers of z − a(z), that is, holds for |z − a| > R(|z| < R) where R is the radius of convergence. If instead of f(z) = z − a or f(z) = z another holomorphic function f(z) is used, then the region of convergence |f(z)| < R has a noncircular shape, and expanding the analytic function F (z) in powers of f(z) leads to a Lagrange-Burmann series (Section 23.6) valid in this region. Thus the Taylor series is the particular case of the Lagrange-Burmann series, in which the expansion proceeds in powers of the auxiliary function, f(z) = z − a. If the series is summed with N terms, the accuracy of the approximation to the function is specified by its remainder, for which more than one form may be found (Sections 23.8 and 23.9). A holomorphic function is not only analytic, but also harmonic (Section 23.1), since its value at one point is the average value on a circle with center at that point. This defining property of harmonic functions can also be extended to noncircular regions: in this case the function has maximum modulus on the boundary (Section 23.2). The proof of all the results stated (Sections 23.1-23.2 and 23.6-23.9) ultimately rests on the harmonic series (Sections 21.8 and 21.9) applied to an auxiliary holomorphic function (Section 23.3) that is used for the ascending power series expansion (Section 23.4) in the Lagrange-Burmann series; the latter is more general than Taylor or Maclaurin series since the region of convergence need not be a circle, and can have any shape specified by (Section 23.5) the auxiliary function. The ascending power series or Taylor’s series shows that any potential flow without singularity at a point can be represented by a superposition of corner flows of submultiples of π, and likewise for all potential fields; and the Lagrange-Burmann series allows the replacement of the corners by regions with other shapes.
