Appendix:

The Elementary Transcendental Functions

A.1 On certain results assumed in Chapters i–iv.

It was convenient, in the first four chapters of this work, to assume some of the properties of the elementary transcendental functions, namely the exponential, logarithmic and circular functions; it was also convenient to make use of a number of results which the reader would be prepared to accept intuitively by reason of his familiarity with the geometrical representation of complex numbers by means of points in a plane.

To take two instances, (i) it was assumed (§2.7) that lim $(\exp z) = \exp (\lim z)$, and (ii) the geometrical concept of an angle in the Argand diagram made it appear plausible that the argument of a complex number was a many-valued function, possessing the property that any two of its values differed by an integer multiple of $2\pi$.

The assumption of results of the first type was clearly illogical; it was also illogical to base arithmetical results on geometrical reasoning. For, in order to put the foundations of geometry on a satisfactory basis, it is not only desirable to employ the axioms of arithmetic, but it is also necessary to utilise a further set of axioms of a more definitely geometrical character, concerning properties of points, straight lines and planes.[1] And, further, the arithmetical theory of the logarithm of a complex number appears to be a necessary preliminary to the development of a logical theory of angles.

Apart from this, it seems unsatisfactory to the aesthetic taste of the mathematician to employ one branch of mathematics as an essential constituent in the structure of another; particularly when the former has, to some extent, a material basis whereas the latter is of a purely abstract nature.[2]

The reasons for pursuing the somewhat illogical and unaesthetic procedure, adopted in the earlier part of this work, were, firstly, that the properties of the elementary transcendental functions were required gradually in the course of Chapter ii, and it seemed undesirable that the course of a general development of the various infinite processes should be frequently interrupted in order to prove theorems (with which the reader was, in all probability, already familiar) concerning a single particular function; and, secondly, that (in connexion with the assumption of results based on geometrical considerations) a purely arithmetical mode of development of Chapters i–iv, deriving no help or illustrations from geometrical processes, would have very greatly increased the difficulties of the reader unacquainted with the methods and the spirit of the analyst.

A.11 Summary of the Appendix.

The general course of the Appendix is as follows:

In §§A.2–A.22, the exponential function is defined by a power series. From this definition, combined with results contained in Chapter ii, are derived the elementary properties (apart from the periodic properties) of this function. It is then easy to deduce corresponding properties of logarithms of positive numbers (§§A.3–A.33).

Next, the sine and cosine are defined by power series from which follows the connexion of these functions with the exponential function. A brief sketch of the manner in which the formulae of elementary trigonometry may be derived is then given (§§A.4–A.42).

The results thus obtained render it possible to discuss the periodicity of the exponential and circular functions by purely arithmetical methods (§A.5, §A.51).

In §§A.52–A.522, we consider, substantially, the continuity of the inverse circular functions. When these functions have been investigated, the theory of logarithms of complex numbers (§A.6) presents no further difficulty.

Finally, in §A.7, it is shewn that an angle, defined in a purely analytical manner, possesses properties which are consistent with the ordinary concept of an angle, based on our experience of the material world.

It will be obvious to the reader that we do not profess to give a complete account of the elementary transcendental functions, but we have confined ourselves to a brief sketch of the logical foundations of the theory[3]. The developments have been given by writers of various treatises, such as Hobson, Plane Trigonometry; Hardy, A course of Pure Mathematics; and Bromwich, Theory of Infinite Series.

A.12 A logical order for development of the elements of Analysis.

The reader will find it instructive to read Chapters i–iv and the Appendix a second time in the following order:

• Chapter i (omitting all of §l.5 except the first two paragraphs).[4]
• Chapter ii to the end of §2.61 (omitting the examples in §§2.31–2.61).
• Chapter iii to the end of §3.34 and §§3.5–3.73.
• The Appendix, §§A.2–A.6 (omitting §A32, §A.33).
• Chapter ii, the examples of §§2.31–2.61.
• Chapter iii, §§3.341–3.4.
• Chapter iv, inserting §A.32, §A.33, §A.7 after §4.13.
• Chapter ii, §§2.7–2.82.

The reader could thus be convinced that (in that order) it is possible to elaborate a purely arithmetical development of the subject, in which the graphic and familiar language of geometry is to be regarded as merely conventional.[5]

A.2 The exponential function, $\exp z$.

The exponential function, of a complex variable $z$, is defined by the series[6] $$\exp z = 1+ \frac{z}{1!}+\frac{z^2}{2!}+\frac{z^3}{3!}+ \cdots = 1+\sum_{n=1}^\infty \frac{z^n}{n!}.$$ This series converges absolutely for all values of $z$ (real and complex) by D’Alembert’s ratio test (§2.36) since $\lim\limits_{n\rightarrow \infty} \left|\, (\left.z\middle/n\right.) \,\right| = 0 < 1$; so the definition is valid for all values of $z$.

Further, the series converges uniformly throughout any bounded domain of values of $z$; for, if the domain be such that $\left|\, z \,\right| \leq R$ when $z$ is in the domain, then $$\left|\,(\left.z^n\middle/n!\right.)\,\right| \leq \left.R^{\:\!n}\middle/n!\right.,$$ and the uniformity of the convergence is a consequence of the test of Weierstrass (§3.34), by reason of the convergence of the series $1+\sum\limits_{n=1}^\infty \dfrac{R^{\:\!n}}{n!}$, in which the terms are independent of $z$.

Moreover, since, for any fixed value of $n$, $\left.z^n\middle/n!\right.$ is a continuous function of $z$, it follows from §3.32 that the exponential function is continuous for all values of $z$; and hence (cf. §3.2), if $z$ be a variable which tends to the limit $\zeta$, we have $$\lim_{z \rightarrow \zeta} \exp z= \exp \zeta.$$

A.21 The addition-theorem for the exponential function, and its consequences.

From Cauchy’s theorem on multiplication of absolutely convergent series (§2.53), it follows that[7] $$\begin{align*} (\exp z_1)(\exp z_2) &= \left(1+ \frac{z_1}{1!}+\frac{z_1^2}{2!}+ \cdots \right)\left(1+ \frac{z_2}{1!}+\frac{z_2^2}{2!}+ \cdots \right) \\ &=1+ \frac{z_1+z_2}{1!}+\frac{z_1^2+2z_1z_2+z_2^2}{2!}+ \cdots \\ \\ &=\exp(z_1 + z_2), \end{align*}$$ so that $\exp (z_1 + z_2 )$ can be expressed in terms of exponential functions of $z_1$ and of $z_2$ by the formula $$\exp(z_1 + z_2) = (\exp z_1)(\exp z_2).$$ This result is known as the addition-theorem for the exponential function. From it, we see by induction that $$(\exp z_1) (\exp z_2 ) \cdots (\exp z_n ) = \exp (z_1 + z_2 + \cdots + z_n),$$ and, in particular, $$\{\exp z\} \{\exp (-z)\} = \exp 0 = 1.$$

From the last equation, it is apparent that there is no value of $z$ for which $\exp z= 0$; for, if there were such a value of $z$, since $\exp(-z)$ would exist for this value of $z$, we should have $0=1$.

It also follows that, when $x$ is real, $\exp x > 0$; for, from the series definition, $\exp x \geq 1$ when $x \geq 0$; and, when $x \leq 0$, $\exp x = \left.1\middle/\exp( - x)\right. > 0$.

Further, $\exp x$ is an increasing function of the real variable $x$; for, if $k > 0$, $$\exp (x + k) - \exp x = \{\exp x \} \{\exp k - 1\} > 0,$$ because $\exp x > 0$ and $\exp k > 1$.

Also, since $$\left.\{\exp h-1\}\middle/h\right. = 1+\left(\left.h\middle/2!\right.\right) + \left(\left.h^2\middle/3!\right.\right) + \cdots ,$$ and the series on the right is seen (by the methods of §A.2) to be continuous for all values of $h$, we have $$\lim_{h \rightarrow 0}\, \left.\{\exp h - 1\}\middle/h\right. = 1,$$ and so $$\frac{d \exp z}{dz} = \lim_{h \rightarrow 0}\frac{\exp(z+h) - \exp z}{h} = \exp z.$$

A.22 Various properties of the exponential function.

Returning to the formula $(\exp z_1) (\exp z_2 ) \cdots (\exp z_n ) = \exp (z_1 + z_2 + \cdots + z_n)$, we see that, when $n$ is a positive integer, $$(\exp z)^n = \exp(nz),$$ and $$(\exp z)^{-n} = \left.1\middle/(\exp z)^n \right. = \left.1\middle/\exp (nz) \right. = \exp ( - nz).$$ In particular, taking $z =1$ and writing $e$ in place of $\exp 1 = 2.71828 \dots$, we see that, when $m$ is an integer, positive or negative, $$e^m = \exp m = 1 + \left(\left.m^\vphantom{1}\middle/1!\right.\right) + \left(\left.m^2\middle/2!\right.\right) + \cdots .$$

Also, if $\mu$ be any rational number ($\mu=\left.p\middle/q\right.$, where $p$ and $q$ are integers, $q$ being positive) $$(\exp \mu)^q = \exp \mu q = \exp p = e^p ,$$ so that the $q$th. power of $\exp \mu$ is $e^p$; that is to say, $\exp \mu$ is a value of $e^{\:\!\left.p\middle/q\right.} = e^\mu$, and it is obviously (§A.21) the real positive value.

If $x$ be an irrational-real number, (defined by a section in which $\alpha_1$ and $\alpha_2$ are typical members of the $L$-class and the $R$-class respectively), the irrational power $e^x$ is most simply defined as $\exp x$; we thus have, for all real values of $x$, rational and irrational, $$e^x = 1+\frac{x}{1!}+\frac{x^2}{2!}+ \cdots ,$$ an equation first given by Newton.[8]

It is, therefore, legitimate to write $e^x$ for $\exp x$ when $x$ is real, and it is customary to write $e^z$ for $\exp z$ when $z$ is complex. The function $e^z$ (which, of course, must not be regarded as being a power of $e$), thus defined, is subject to the ordinary laws of indices, viz. $$e^{z} e^{\zeta} = e^{z+\zeta}, \quad e^{-z} = \left. 1\middle/ e^{z} \right. .$$

[Note. Tannery, Lecons d’Algebre et d’Analyse, i. (1906), p. 45, practically defines $e^x$, when $x$ is irrational, as the only number $X$ such that $e^{\alpha_1} \leq X \leq e^{a_2}$, for every $\alpha_1$ and $\alpha_2$. From the definition we have given it is easily seen that such a unique number exists. For $\exp x \,(= X)$ satisfies the inequality, and if $X' \,(\neq X)$ also did so, then $$\exp \alpha_2 - \exp \alpha_1 = e^{\alpha_2}-e^{\alpha_1} \geq \left|\, X'-X \,\right|,$$ so that, since the exponential function is. continuous, $\alpha_2 -\alpha_1$ cannot be chosen arbitrarily small, and so $(\alpha_1 , \alpha_2 )$ does not define a section.]

A.3 Logarithms of positive numbers.[9]

It has been seen (§A.2, §A.21) that, when $x$ is real, $\exp x$ is a positive continuous increasing function of $x$, and obviously $\exp x \rightarrow +\infty$ as $x \rightarrow +\infty$, while $$\exp x =\left.1\middle/\exp(-x)\right. \rightarrow 0\, \text{ as }\, x\rightarrow -\infty.$$

If, then, $a$ be any positive number, it follows from §3.63 that the equation in $x$, $$\exp x = a,$$ has one real root and only one. This root (which is, of course, a function of $a$) will be written $\mathrm{Log}_e\:\! a$ or simply $\mathrm{Log}\, a$;[10] it is called the Logarithm of the positive number $a$.

Since a one-one correspondence has been established between $x$ and $a$, and since $a$ is an increasing function of $x$, $x$ must be an increasing function of $a$; that is to say, the Logarithm is an increasing function.

Example. Deduce from §A.21 that $\mathrm{Log}\, a + \mathrm{Log}\, b = \mathrm{Log}\, ab$.

A.31 The continuity of the Logarithm.

It will now be shewn that, when $a$ is positive, $\mathrm{Log}\, a$ is a continuous function of $a$.

Let $$\mathrm{Log}\, a= x, \quad \mathrm{Log} (a + h)=x + k,$$ so that $$e^x = a,\quad e^{x + k} = a + h, \quad 1 + (\left.h\middle/ a\right.) = e^k .$$

First suppose that $h > 0$, so that $k > 0$, and then $$1 + (\left.h\middle/ a\right.) = l + k+\frac{1}{2}k^2 + \cdots > 1+ k,$$ and so $$0 < k <\left.h\middle/ a\right. ,$$ that is to say $$0 < \mathrm{Log} (a + h) - \mathrm{Log}\, a < \left.h\middle/ a\right. .$$

Hence, $h$ being positive, $\mathrm{Log} (a + h) - \mathrm{Log}\, a$ can be made arbitrarily small by taking $h$ sufficiently small.

Next, suppose that $h < 0$, so that $k < 0$, and then $\left.a\middle/(a + h)\right. = e^{-k}$. Hence (taking $0 < -h < \frac{1}{2}a$, as is obviously permissible) we get $$\left.a\middle/(a + h)\right. = 1 + (-k) + \frac{1}{2}k^2 + \cdots > 1 - k,$$ and so $$-k < -1 + \left.a\middle/(a + h)\right. = -\left.h\middle/(a + h)\right. <-2\left.h\middle/a\right. .$$

Therefore, whether $h$ be positive or negative, if $\epsilon$ be an arbitrary positive number and if $\left|\,h\,\right|$ be taken less than both $\frac{1}{2}a$ and $\frac{1}{2}a\epsilon$, we have $$\left|\, \mathrm{Log} (a + h) - \mathrm{Log}\, a \,\right| < \epsilon,$$ and so the condition for continuity (§3.2) is satisfied.

A.32 Differentiation of the Logarithm.

Retaining the notation of §A.31, we see, from results there proved, that, if $h \rightarrow 0$ ($a$ being fixed), then also $k \rightarrow 0$. Therefore, when $a > 0$, $$\frac{d\:\! \mathrm{Log}\, a}{da} = \lim_{k \rightarrow 0} \frac{k}{e^{x+k} -e^x} = \frac{1}{e^x} = \frac{1}{a}.$$

Since $\mathrm{Log}\, 1 = 0$, we have, by §4.13 example 3, $$\mathrm{Log}\, a = \int _1^a \! t^{-1} \,dt.$$

A.33 The expansion of $\:\!\mathrm{Log} (1 + a)$ in powers of $a$.

From §A.32 we have $$\begin{align*} \mathrm{Log}(1+a) &= \int _0^a \! (1+ t)^{-1} \,dt \\ &=\int _0^a \! \{1-t+t^2- \cdots +(-1)^{n-1}t^{n-1}+(-1)^n t^n (1+t)^{-1} \}\,dt \\ &=a-\frac{1}{2}a^2+\frac{1}{3}t^3- \cdots +(-1)^{n-1}\frac{1}{n}a^n +R_n, \end{align*}$$ where $$R_n = (-1)^n \int_0^a \! t^n (1+t)^{-1}\,dt.$$

Now, if $-1 < a < 1$, we have $$\begin{align*} \left|\,R_n\,\right|&\leq \int_0^{\left|\,a\,\right|} \! t^n (1-\left|\,a\,\right|)^{-1}dt \\ &= \left|\,a\,\right|^{n+1}\{(n+1)(1-\left|\,a\,\right|)\}^{-1} \\ \\ & \rightarrow 0\, \text{ as } \, n \rightarrow \infty . \end{align*}$$

Hence, when $-1 < a < 1$, $\mathrm{Log}(1 + a)$ can be expanded into the convergent series[11] $$\mathrm{Log}(1+a) = a-\frac{1}{2}a^2+\frac{1}{3}a^3- \cdots = \sum_{n=1}^\infty (-1)^{n-1} \left.a^n\middle/n\right. .$$ If $a = +1$, $$\left|\,R_n\,\right|=\int_0^1 \! t^n (1+t)^{-1} dt < \int_0^1 \! t^n \,dt =(n+1)^{-1} \rightarrow 0\, \text{ as } \, n \rightarrow \infty,$$ so the expansion is valid when $a= + 1$; it is not valid when $a = -1$.

Example. Shew that $\lim\limits_{n \rightarrow \infty} \left(1 + \dfrac{1}{n}\right)^n =e$.

[We have $\displaystyle \lim_{n \rightarrow \infty} n\, \mathrm{Log}\left(1 + \dfrac{1}{n}\right) = \lim_{n \rightarrow \infty} \left(1-\frac{1}{2n}+\frac{1}{3n^2}- \cdots \right) = 1,$ and the result required follows from the result of §A.2 that $\lim\limits_{z \rightarrow \zeta} e^z = e^\zeta$.]