The Fundamental Properties of Analytic Functions;
Taylor’s, Laurent’s and Liouville’s Theorems

5.1 A Property of the elementary functions.

[1]The reader will observe that this is not the sense in which the term function is defined (§3.1) in this work. Thus e.g. \(x - iy\) and \(\left| \,z\, \right|\) are functions of \(z \,( = x + iy)\) in the sense of §3.1, but are not elementary functions of the type under consideration. ↩

The reader will be already familiar with the term elementary function, as used (in text-books on Algebra, Trigonometry, and the Differential Calculus) to denote certain analytical expressions[1] depending on a variable \(z\), the symbols involved therein being those of elementary algebra together with exponentials, logarithms and the trigonometrical functions;[2] examples of such expressions are \[z^2,\quad e^z, \quad \log z, \quad \arcsin z^{\frac{3}{2}}.\] Such combinations of the elementary functions of analysis have in common a remarkable property, which will now be investigated.

[2]Editor’s Note: Exponentials, logarithms and the trigonometrical functions are defined and their fundamental properties derived in the Appendix. ↩

Take as an example the function \(e^z\). Write \(e^z=f(x)\). Then, if \(z\) be a fixed point and if \(z'\) be any other point, we have \[\begin{align*} \frac{f(z')-f(z)}{z'-z} & = \frac{e^{z'}-e^z}{z'-z} = e^z \frac{e^{\{z'-z\}}-1}{z'-z}\\ & = e^z \left\{1+\frac{z'-z}{2!}+\frac{(z'-z)^2}{3!}+ \cdots \right\}; \end{align*}\] and since the last series in brackets is uniformly convergent for all values of \(z'\), it follows (§3.7) that, as \(z' \rightarrow z\), the quotient \[\frac{f(z')-f(z)}{z'-z}\] tends to the limit \(e^z\), uniformly for all values of \(\arg (z'- z)\). This shews that the limit of \[\frac{f(z')-f(z)}{z'-z}\] is in this case independent of the path by which the point \(z'\) tends towards coincidence with \(z\).

It will be found that this property is shared by many of the well-known elementary functions; namely, that if \(f(z)\) be one of these functions and \(h\) be any complex number, the limiting value of \[\frac{1}{h} \left\{f(z+h)-f(z) \right\}\] exists and is independent of the mode in which \(h\) tends to zero.

The reader will, however, easily prove that, if \(f(z)= x -iy\), where \(z = x + iy\), then \(\lim \dfrac{f(z+h)-f(z)}{h}\) is not independent of the mode in which \(h \rightarrow 0\).

5.11 Occasional failure of the property.

For each of the elementary functions, however, there will be certain points \(z\) at which this property will cease to hold good. Thus it does not hold for the function \(\left. 1 \middle/(z - a) \right.\) at the point \(z = a\), since \[\lim_{h \rightarrow 0} \frac{1}{h}\left\{\frac{1}{z-a+h} - \frac{1}{z-a} \right\}\] does not exist when \(z = a\). Similarly it does not hold for the functions \(\log z\) and \(z^{\frac{1}{2}}\) at the point \(z = 0\).

These exceptional points are called singular points or singularities of the function \(f(z)\) under consideration; at other points \(f(z)\) is said to be analytic.

The property does not hold good at any point for the function \( \left|\,z\,\right|\).

5.12 Cauchy’s definition[3] of an analytic function of a complex variable.

[3]See the memoir cited in §5.2 (1st margin note). ↩

The property considered in §5.11 will be taken as the basis of the definition of an analytic function, which may be stated as follows.

[4]The words ‘regular’ and ‘holomorphic’ are sometimes used. A distinction has been made by Borel between ‘monogenic’ and ‘analytic’ functions in the case of functions with an infinite number of singularities. See §5.51.

Editor’s Note: Modern usage is slightly different: a complex function is analytic, holomorphic (or regular) at a point \(z\) if it is differentiable in a open neighborhood of \(z\) and is holomorphic on set \(S\) if it is holomorphic at every point of an open set containing \(S\). ↩

Let a two-dimensional region in the \(z\)-plane be given; and let \(u\) be a function of \(z\) defined uniquely at all points of the region. Let \(z\), \(z + \delta z\) be values of the variable \(z\) at two points, and \(u\), \(u + \delta u\) the corresponding values of \(u\). Then, if, at any point \(z\) within the area, \(\dfrac{\delta u}{\delta z}\) tends to a limit when \(\delta x \rightarrow 0\), \(\delta y \rightarrow 0\), independently (where \(\delta z = \delta x + i\delta y\)), \(u\) is said to be a function of \(z\) which is monogenic or analytic[4] at the point. If the function is analytic and one-valued at all points of the region, we say that the function is analytic throughout the region.[5]

[5]See §5.2 cor. 2, margin note. ↩

We shall frequently use the word ‘function’ alone to denote an analytic function, as the functions studied in this work will be almost exclusively analytic functions.

In the foregoing definition, the function \(u\) has been defined only within a certain region in the \(z\)-plane. As will be seen subsequently, however, the function \(u\) can generally be defined for other values of \(z\) not included in this region; and (as in the case of the elementary functions already discussed) may have singularities, for which the fundamental property no longer holds, at certain points outside the limits of the region.

We shall now state the definition of analytic functionality in a more arithmetical form.

Let \(f(z)\) be analytic at \(z\), and let \(\epsilon\) be an arbitrary positive number; then we can find numbers \(l\) and \(\delta\), (\(\delta\) depending on \(\epsilon\)) such that \[\left|\,\frac{f(z')-f(z)}{z'-z} -l \, \right| < \epsilon\] whenever \(\left|\, z' - z \,\right| < \delta\).

If \(f(z)\) is analytic at all points \(z\) of a region, \(l\) obviously depends on \(z\); we consequently write \(l = f '(z)\).

Hence \[f(z')= f(z) + (z'-z)\;\! f'(z) + v(z'- z),\] where \(v\) is a function of \(z\) and \(z'\) such that \(\left|\,v\,\right| < \epsilon\) when \(\left|\,z' -z\,\right|< \delta\).

Example 1. Find the points at which the following functions are not analytic: \[\begin{align*} \text{(i)}& \quad z^2. \\ \text{(ii)}& \quad \mathrm{cosec}\, z \quad(z=n\pi, \, n\: \text{any integer}). \\ \text{(iii)}& \quad \frac{z-1}{z^2-5z+6} \quad (z=2,\, 3). \\ \text{(iv)}& \quad e^{\frac{1}{z}} \quad (z=0).\\ \text{(v)}& \quad \{(z-1)z\}^{\frac{1}{3}} \quad (z=0,\,1). \end{align*}\]

Example 2. If \(z = x+iy\), \(f(z) = u + iv\), where \(u\), \(v\), \(x\), \(y\) are real and \(f\) is an analytic function, shew that \[\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}.\](Riemann.)

5.13 An application of the modified Heine-Borel theorem.

Let \(f(z)\) be analytic at all points of a continuum; and on any point \(z\) of the boundary of the continuum let numbers \(f_1(z)\), \(\delta\) (\(\delta\) depending on \(z\)) exist such that \[\left|\,f(z')-f(z)-(z'-z)\;\! f_1(z)\,\right|\ < \epsilon\, \left|\, z'-z\,\right|\] whenever \(\left|\, z'-z\,\right| < \delta\) and \(z'\) is a point of the continuum or its boundary.[6]

[6]We write \(f_1(z)\) instead of \(f'(z)\) as the differential coefficient might not exist when \(z'\) approaches \(z\) from outside the boundary so that \(f_1(z)\) is not necessarily a unique derivate. ↩

The above inequality is obviously satisfied for all points \(z\) of the continuum as well as boundary points.

Applying the two-dimensional form of the theorem of §3.6, we see that the region formed by the continuum and its boundary can be divided into a finite number of parts (squares with sides parallel to the axes and their interiors, or portions of such squares) such that inside or on the boundary of any part (part \(n\))[7] there is one point \(z_n\) such that the inequality \[\left|\,f(z')-f(z_n)-(z'-z_n)\;\! f_1(z_n)\,\right|\ < \epsilon\, \left|\, z'-z_n\,\right|\] is satisfied by all points \(z'\) inside or on the boundary of that part.

[7]Editor’s Note: For clarity, I indexed the parts and the \(z_n\). ↩