9 The Exponential and Logarithmic Functions. Undetermined Coeﬃcients. Involution and Evolution. The Binomial Theorem

Chapter 9
The Exponential and Logarithmic Functions. Undetermined Coeﬃcients. Involution and Evolution. The Binomial Theorem

70. Function. A variable $w$ is said to be a function of a second variable $z$ for the area $A$ of the $z$ -plane (§42), when to the $z$ belonging to every point of $A$ there corresponds a determinate value or set of values of $w$ .

Thus if $w = 2 z$ , $w$ is a function of $z$ . For when $z = 1$ , $w = 2$ ; when $z = 2$ , $w = 4$ ; and there is in like manner a determinate value of $w$ for every value of $z$ . In this case $A$ is coextensive with the entire $z$ -plane.

Similarly $w$ is a function of $z$ , if

w = a_{0} + a_{1} z + a_{2} z^{2} + \dots + a_{n} z^{n} + \dots,

so long as this inﬁnite series is convergent, i. e. for the portion of the

z

-plane bounded by a circle having the null-point for centre, and for radius the modulus of the smallest value of

z

for which the series diverges.

It is customary to use for $w$ when a function of $z$ the symbol $f (z)$ , read “function $z$ .”

71. Functional Equation of the Exponential Function. For positive integral values of $z$ and $t$ , $a^{z} \cdot a^{t} = a^{z + t}$ . The question naturally suggests itself, is there a function of $z$ which will satisfy the condition expressed by this equation, or the “functional equation” $f (z) f (t) = f (z + t)$ , for all values of $z$ and $t$ ?

We proceed to the investigation of this question and another which it suggests, not only because they lead to deﬁnitions of the important functions $a^{z}$ and ${log}_{a} z$ for complex values of $a$ and $z$ , and so give the operations of involution, evolution, and the taking of logarithms the perfectly general character already secured to the four fundamental operations,—but because they aﬀord simple examples of a large class of mathematical investigations.²⁶

72. Undetermined Coeﬃcients. In investigations of this sort, the method commonly used in one form or another is that of undetermined coeﬃcients. This method consists in assuming for the function sought an expression involving a series of unknown but constant quantities—coeﬃcients,—in substituting this expression in the equation or equations which embody the conditions which the function must satisfy, and in so determining these unknown constants that these equations shall be identically satisﬁed, that is to say, satisﬁed for all values of the variable or variables.

The method is based on the following theorem, called “the theorem of undetermined coeﬃcients,” viz.:

If the series $A + B z + C z^{2} + \dots$ be equal to the series $A^{'} + B^{'} z + C^{'} z^{2} + \dots$ for all values of $z$ which make both convergent, and the coeﬃcients be independent of $z$ , the coeﬃcients of like powers of $z$ in the two are equal.

For, since

A + B z + C z^{2} + \dots = A^{'} + B^{'} z + C^{'} z^{2} + \dots,

A - A^{'} + (B - B^{'}) z + (C - C^{'}) z^{2} + \dots = 0

throughout the circle of convergence common to the two given series (§§ 67, 69, 1).

And being convergent within this circle, the series

A - A^{'} + (B - B^{'}) z + (C - C^{'}) z^{2} + \dots

can be made to diﬀer as little as we please from its ﬁrst term,

A - A^{'}

(§ 68).

∴ A - A^{'} = 0 (§ 30, Cor.), or A = A^{'} .

Therefore

(B - B^{'}) z + (C - C^{'}) z^{2} + \dots = 0

throughout the common circle of convergence, and hence (at least, for values of

z

diﬀerent from 0)

B - B^{'} + (C - C^{'}) z + \dots = 0

Therefore by the reasoning which proved that

A - A^{'} = 0, B - B^{'} = 0, or B = B^{'} .

In like manner it may be proved that $C = C^{'}$ , $D = D^{'}$ , etc.

\begin{array}{l} COR.If = A^{'} + B^{'} z + C^{'} t + D^{'} z^{2} + E^{'} z t + F^{'} t^{2} + \dots for all values of z and t which make both series convergent, and z be independent of t,
and the coeﬃcients independent of both z and t, the coeﬃcients of like powers of z
and t in the two series are equal. For, arrange both series with reference to the powers of either variable. The
coeﬃcients of like powers of this variable are then equal, by the preceding theorem.
These coeﬃcients are series in the other variable, and by applying the theorem to
each equation between them the corollary is demonstrated. 73. The Exponential Function. To apply this method to the case in hand, assume f (z) = A_{0} + A_{1} z + A_{2} z^{2} + \dots + A_{n} z^{n} + \dots, and determine whether
values of the coeﬃcients A_{i} can be found capable of satisfying the “functional equation,” f (z) f (t) = f (z + t), (1) for all values of z and t . On substituting in this equation, we have, for all values of z and t for which the

series converge, (A_{0} + A_{1} z + A_{2} z^{2} + \dots A_{n} z^{n} + \dots) (A_{0} + A_{1} t + A_{2} t^{2} + \dots A_{n} t^{n} + \dots) = A_{0} + A_{1} (z + t) + A_{2} {(z + t)}^{2} + \dots A_{n} {(z + t)}^{n} + \dots; or, expanding and arranging the terms with reference to the powers of z and t, \begin{array}{l} A_{0} A_{0} & + A_{1} A_{0} z + A_{0} A_{1} t + A_{2} A_{0} z^{2} + A_{1} A_{1} z t + A_{0} A_{2} t^{2} + \dots \\ + A_{n} A_{0} z^{n} + A_{n - 1} A_{1} z^{n - 1} t + \dots + A_{n - k} A_{k} z^{n - k} t^{k} + \dots + A_{0} A_{n} t^{n} \\ + \dots \\ = A_{0} + A_{1} z + A_{1} t + A_{2} z^{2} + 2 A_{2} z t + A_{2} t^{2} + \dots \\ + A_{n} z^{n} + A_{n} n z^{n - 1} t + \dots + A_{n} n_{k} z^{n - k} t^{k} + \dots + A_{n} t^{n} + \dots, \\ where & n_{k} = \frac{(n (n - 1) \dots (n - k + 1)}{k!} \end{array} Equating the coeﬃcients of like powers of z and t in the
two members of this equation, we get \begin{array}{l} A_{n - 1} A_{k} equal always to A_{n} n_{k} . \\ In particular & A_{0} A_{0} = A_{0}, therefore A_{0} = 1 . Also \\ A_{1} A_{1} = 2 A_{2}, A_{2} A_{1} = 3 A_{3}, \\ A_{3} A_{1} = 4 A_{4}, \dots, A_{n - 1} A_{1} = n A_{n}; \end{array} or, multiplying these equations together member by member, A_{1}^{n} = A_{n} n!, or A_{n} = \frac{A_{1}^{n}}{n!} . A part of the equations among the coeﬃcients are, therefore, suﬃcient
to determine the values of all of them in terms of the one coeﬃcient A_{1} . But
these values will satisfy the remaining equations; for substituting them in the general
equation \begin{array}{l} A_{n - k} A_{k} = A_{n} n_{k}, \\ we get & \frac{A_{1}^{n - k}}{(n - k)!} \times \frac{A_{1}^{k}}{k!} = \frac{A_{1}^{n}}{n!} \times \frac{n (n - 1) \dots (n - k + 1)}{k!}, \end{array} which is obviously an identical equation. The coeﬃcient A_{1} or,
more simply written, A,
remains undetermined. It has been demonstrated, therefore, that to satisfy equation (1), it is only necessary
that, f (z) be the sum of an inﬁnite series of the form 1 + A z + \frac{A^{2}}{2!} z^{2} + \frac{A^{3}}{3!} z^{3} + \dots, (2) where A is undetermined; a series which has a sum, i. e. is convergent, for all ﬁnite values of z and A .
(§ 63, 2, § 66.) By properly determining A, f (z) may be identiﬁed
with a^{z}, for any
particular value of a . If a^{z} is to be identically
equal to the series (2), A must have such a value that \begin{array}{l} a = 1 + A + \frac{A^{2}}{2!} + \frac{A^{3}}{3!} + \dots . \\ Let & e^{z} = 1 + z + \frac{z^{2}}{2!} + \frac{z^{3}}{3!} + \dots, (3) \\ where & e = 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \dots;27 \\ Then & e^{A} = 1 + A + \frac{A^{2}}{2!} + \frac{A^{3}}{3!} + \dots . \\ Therefore & a = e^{A}; \end{array} or, calling any number which satisﬁes the equation e^{z} = a the logarithm of a to the base e and writing it {log}_{e} a, A = {log}_{e} a . Whence ﬁnally, a^{z} = 1 + ({log}_{e} a) z + \frac{{({log}_{e} a)}^{2} z^{2}}{2!} + \frac{{({log}_{e} a)}^{3} z^{3}}{3!} + \dots, (4) a deﬁnition of a^{z}, valid for
all ﬁnite complex values of a and z, if it may be assumed
that {log}_{e} a is a number,
whatever the value of a . The series (3) is commonly called the exponential series, and its sum e^{z} the exponential function . It is much more useful than the more general
series (2), or (4), because of its greater simplicity; its coeﬃcients do not
involve the logarithm, a function not yet fully justiﬁed and, as will
be shown, to a certain extent indeterminate. Inasmuch, however, as e^{z} is a particular
function of the class a^{z}, a^{z} is
sometimes called the general exponential function, and series (4) the general
exponential series. 74. The Functions Sine and Cosine. It was shown in § 51 that when 𝜃 is a
real number, \begin{array}{l} e^{i 𝜃} & = cos 𝜃 + i sin 𝜃 . \\ But e^{i 𝜃} & = 1 + i 𝜃 + \frac{{(i 𝜃)}^{2}}{2!} + \frac{{(i 𝜃)}^{3}}{3!} + \frac{{(i 𝜃)}^{4}}{4!} + \dots \\ = 1 - \frac{𝜃^{2}}{2!} + \frac{𝜃^{4}}{4!} - \dots \\ + i (𝜃 - \frac{𝜃^{3}}{3!} + \dots) . \end{array} Therefore (by § 36, 2, Cor.), for real values of 𝜃 cos 𝜃 = 1 - \frac{𝜃^{2}}{2!} + \frac{𝜃^{4}}{4!} - \dots, (9.1) and sin 𝜃 = 𝜃 - \frac{𝜃^{3}}{3!} + \frac{𝜃^{5}}{5!} - \dots, (9.2) series which both converge for all ﬁnite values of 𝜃 . Though cos 𝜃 and sin 𝜃 only admit of geometrical
interpretation when 𝜃 is real, it is convenient to continue to use these names for the sums of the series (5) and
(6) when 𝜃 is complex. 75. Periodicity. When 𝜃 is real, evidently neither its sine nor its cosine will be changed if it
be increased or diminished by any multiple of four right angles, or 2 π; or, if n be any positive
integer, cos (𝜃 \pm 2 n π) = cos 𝜃, sin (𝜃 \pm 2 n π) = sin 𝜃, and hence e^{i (𝜃 \pm 2 n π)} = e^{i 𝜃} . The functions e^{i 𝜃}, cos 𝜃, sin 𝜃, are
on this account called periodic functions, with the modulus of periodicity 2 π . 76. The Logarithmic Function. If z = e^{z} and t = e^{T}, z t = e^{z} e^{T} = e^{Z + T}, § 73 or {log}_{e} z t = {log}_{e} z + {log}_{e} t . (7) The question again is whether a function exists capable of satisfying this
equation, or, more generally, the “functional equation,” f (z t) = f (z) + f (t), (8) for complex values of z and t . When z = 0, (7) becomes {log}_{e} 0 = {log}_{e} 0 + {log}_{e} t, an equation which cannot
hold for any value of t for which {log}_{e} t is
not zero unless {log}_{e} 0 is numerically greater than any ﬁnite number whatever. Therefore {log}_{e} 0 is
inﬁnite. On the other hand, when z = 1,
(7) becomes {log}_{e} t = {log}_{e} 1 + {log}_{e} t, so that {log}_{e} 1 is zero. Instead, therefore, of assuming a series with undetermined coeﬃcients for f (z) itself, we assume
one for f (1 + z), setting f (1 + z) = A_{1} z + A_{2} z^{2} + \dots + A_{n} z^{n} + \dots, and inquire whether
the coeﬃcients A_{i} admit of values which satisfy the functional equation (8) for complex values of z and t . Now 1 + z + t = (1 + z) (1 + \frac{t}{1 + z}), identically . ∴ f [1 + (z + t)] = f (1 + z) + f (1 + \frac{t}{1 + z}), or \begin{array}{l} A_{1} (z + t) + A_{2} {(z + t)}^{2} + \dots + A_{n} {(z + t)}^{n} + \dots \\ = & A_{1} z + A_{2} z^{2} + \dots + A_{n} z^{n} + \dots \\ + & A_{1} {(1 + z)}^{- 1} t + A_{2} {(1 + z)}^{- 2} t^{2} + \dots + A_{n} {(1 + z)}^{- n} t^{n} + \dots \end{array} Equating the coeﬃcients of the ﬁrst power of t (§ 72) in the two members of this equation, \begin{array}{l} A_{1} + 2 A_{2} z + 3 A_{3} z^{2} + \dots + (n + 1) A_{n + 1} z^{n} + \dots \\ = & A_{1} (1 - z + z^{2} - z^{3} + \dots + {(- 1)}^{n} z^{n} + \dots); \end{array} whence, equating the coeﬃcients of like powers of z, \begin{array}{l} A_{1} = A_{1}, 2 A_{2} = - A_{1}, \dots, n A_{n} = {(- 1)}^{n - 1} A_{1}, \dots, \\ or & A_{2} = - \frac{A_{1}}{2}, \dots, A_{n} = {(- 1)}^{n - 1} \frac{A_{1}}{n}, \dots . \end{array} As in the case of the exponential function, a part of the equations among the
coeﬃcients are suﬃcient to determine them all in terms of the one coeﬃcient A_{1} . But
as in that case (by assuming the truth of the binomial theorem for negative integral
values of the exponent) it can be readily shown that these values will satisfy the
remaining equations also. The series z - \frac{z^{2}}{2} + \frac{z^{3}}{3} - \dots + {(- 1)}^{n - 1} \frac{z^{n}}{n} + \dots converges for all values of z whose moduli are less than 1 (§ 62, 3) For such values, therefore, the function A (z - \frac{z^{2}}{2} + \dots + {(- 1)}^{n - 1} \frac{z^{n}}{n} + \dots) (9) satisﬁes the functional equation f [(1 + z) (1 + t)] = f (1 + z) + f (1 + t) . \begin{array}{l} And since & z \equiv 1 - (1 - z) and t \equiv 1 - (1 - t), \\ the function & - A (1 - z + \frac{{(1 - z)}^{2}}{2} + \dots + \frac{{(1 - z)}^{n}}{n} + \dots) \end{array} satisﬁes this equation when written in the simpler form f (z t) = f (z) + f (t), for values of 1 - z and 1 - t whose moduli are both less than 1. 1. L o g_{e} b . To identify the
general function f (1 + z) with
the particular function {log}_{e} (1 + z) it is only necessary to give the undetermined coeﬃcient A the
value 1. For since {log}_{e} (1 + z) belongs to the class of functions which satisfy the equation (8), {log}_{e} (1 + z) = A (z - \frac{z^{2}}{2} + \dots) . Therefore \begin{array}{l} e^{{log}_{e} (1 + z)} & = e^{A (z - \frac{z^{2}}{2} + \dots)} \\ = 1 + A (z - \frac{z^{2}}{2} + \dots) + \frac{1}{2!} A^{2} {(z - \frac{z^{2}}{2} + \dots)}^{2} + \dots . \\ But e^{{log}_{e} (1 + z)} & = 1 + z . \end{array} Hence 1 + z = 1 + A (z - \frac{z^{2}}{2} + \dots) + \frac{1}{2!} A^{2} {(z - \frac{z^{2}}{2} + \dots)}^{2} + \dots; or, equating the coeﬃcients of the ﬁrst power of z, A = 1 . The coeﬃcients of the higher powers of z in the
right number are then identically 0. It has thus been demonstrated that {log}_{e} b is a number (real or
complex), if when b is
written in the form 1 + z,
the absolute value of z is less than 1. To prove that it is a number for other than such values of b, let b = ρ e^{i 𝜃}, (§ 51), where ρ, as being the
modulus of b,
is positive. Then {log}_{e} b = {log}_{e} ρ + i 𝜃, and it only
remains to prove that {log}_{e} ρ is a number. Let ρ be written
in the form e^{n} - (e^{n} - ρ),
where e^{n} is the ﬁrst
integral power of e greater than ρ . \begin{array}{l} Then since & e^{n} - (e^{n} - ρ) \equiv e^{n} (1 - \frac{e^{n} - ρ}{e^{n}}), \\ {log}_{e} ρ = {log}_{e} e^{n} + {log}_{e} (1 - \frac{e^{n} - ρ}{e^{n}}) \\ = n + {log}_{e} (1 - \frac{e^{n} - ρ}{e^{n}}), \end{array} and {log}_{e} (1 - \frac{e^{n} - ρ}{e^{n}}) is a
number since \frac{e^{n} - ρ}{e^{n}} is less than 1. 2. L o g_{a} b .
It having now been fully demonstrated that a^{z} is a number satisfying
the equation a^{Z} a^{T} = a^{Z + T} for
all ﬁnite values of a, Z, T; let a^{Z} = z, a^{T} = t, and call Z the logarithm
of z to the
base a, or {log}_{a} z, and in like
manner T, {log}_{a} t . \begin{array}{l} Then, since & z t = a^{Z} a^{T} = a^{z + T}, \\ {log}_{a} (z t) = {log}_{a} z + {log}_{a} t, \end{array} or {log}_{a} z belongs,
like {log}_{e} z,
to the class of functions which satisfy the functional equation (8). Pursuing the method followed in the case of {log}_{e} b, it will be found
that {log}_{a} (1 + z) is equal
to the series A (z - \frac{z^{2}}{2} + \dots) when A = \frac{1}{l o g_{e} a} .
This number is called the modulus of the system of logarithms of which a is
base. 77. Indeterminateness of log a . Since any complex number a may be thrown into the form ρ e^{i 𝜃}, {log}_{e} a = {log}_{e} ρ + i 𝜃 . (10) This, however, is only one of an inﬁnite series of possible values of {log}_{e} a . For, since e^{i 𝜃} = e^{i (𝜃 \pm 2 n π)} (§ 75), {log}_{e} a = {log}_{e} ρ e^{i (𝜃 \pm 2 n π)} = {log}_{e} ρ + i (𝜃 \pm 2 n π), where n may be any
positive integer. Log_{e} a is, therefore, to a certain extent indeterminate; a fact which must be carefully
regarded in using and studying this function. 28 The value given it in (10), for which n = 0, is
called its principal value. When a is a positive
real number, 𝜃 = 0, so that the
principal value of {log}_{e} a is real;
on the other hand, when a is a negative real number, 𝜃 = π,
or the principal value of {log}_{e} a is the logarithm of the positive number corresponding to a, plus i π . 78. Permanence of the Remaining Laws of Exponents. Besides the law a^{z} a^{t} = a^{z + t} which led to its
deﬁnition, the function a^{z} is subject to the laws: \begin{array}{l} 1 . {(a^{z})}^{t} & = a^{z t} . \\ 2 . {(a b)}^{z} & = a^{z} b^{z} .29 \\ 1 . {(a^{z})}^{t} & = a^{z t} . \\ For a^{z} = {(e^{{log}_{e} a})}^{z} & = 1 + ({log}_{e} a) z + \frac{{({log}_{e} a)}^{2} z^{2}}{2!} + \dots & § 73, (4) \\ = 1 + z {log}_{e} a + \frac{{(z {log}_{e} a)}^{2}}{2!} + \dots \\ = e^{z {log}_{e} a} . & § 73, (3) \\ ∴ {(e^{{log}_{e} a})}^{z} & = e^{z {log}_{e} a}, and {log}_{e} a^{z} = z {log}_{e} a . \end{array} From these results it follows that \begin{array}{l} {(a^{z})}^{t} & = e^{{log}_{e} {(a^{z})}^{t}} \\ = e^{t {log}_{e} a^{z}} \\ = e^{t z {log}_{e} a} \\ = a^{z t} . \\ 2 . {(a b)}^{z} & = a^{z} b^{z} . \\ For {(a b)}^{z} & = e^{{log}_{e} {(a b)}^{z}} \\ = e^{z {log}_{e} a b} \\ = e^{z {log}_{e} a + z {log}_{e} b} & § 76, (7) \\ = e^{z {log}_{e} a} \cdot e^{z {log}_{e} b} & § 73, (1) \\ = a^{z} \cdot b^{z} . \end{array} 79. Permanence of the Remaining Law of Logarithms. In like manner, the
function {log}_{a} z is subject not only to the law \begin{array}{l} {log}_{a} (z t) & = {log}_{a} z + {log}_{a} t, \\ but also to the law \\ {log}_{a} z^{t} & = t {log}_{a} z . \\ For & z & = a^{{log}_{a} z}, \\ and hence & z^{t} & = {(a^{{log}_{a} z})}^{t} \\ = a^{t {log}_{a} z} . & § 78, 1 \end{array} 80. Evolution. Consider three complex numbers ζ, z, Z, connected by
the equation ζ^{Z} = z . This equation gives rise to three problems, each of which is the inverse of the other two.
For Z and ζ may be given
and z sought;
or ζ and z may be given
and Z sought;
or, ﬁnally, z and Z may be
given and ζ sought. The exponential function is the general solution of the ﬁrst problem (involution),
and the logarithmic function of the second. For the third (evolution) the symbol \sqrt[Z]{z} has been
devised. This symbol does not represent a new function; for it is deﬁned by the equation {(\sqrt[Z]{z})}^{Z} = z,
an equation which is satisﬁed by the exponential function z^{\frac{1}{Z}} . Like the logarithmic function, \sqrt[Z]{z} is indeterminate, though not always to the same extent. When Z is a positive integer, ζ^{Z} = z is an algebraic equation,
and by § 56 has Z roots
for any one of which \sqrt[Z]{z} is, by deﬁnition, a symbol. From the mere fact that z = t, therefore, it cannot be
inferred that \sqrt[Z]{z} = \sqrt[Z]{t}, but only that
one of the values of \sqrt[Z]{z} is equal
to one of the values of \sqrt[Z]{t} .
The same remark, of course, applies to the equivalent symbols z^{\frac{1}{Z}}, t^{\frac{1}{Z}} . 81. Permanence of the Binomial Theorem. By aid of the results just
obtained, it may readily be demonstrated that the binomial theorem is valid for
general complex as well as for rational values of the exponent. For b being any complex number whatsoever, and the absolute value of z being
supposed less than 1, \begin{array}{l} {(1 + z)}^{b} & = e^{b {log}_{e} (1 + z)} \\ = e^{b (z - \frac{z^{2}}{2} + \dots)} \\ = 1 + b z + terms involving higher powers of z . \end{array} Therefore let {(1 + z)}^{b} = 1 + b z + A_{2} z^{2} + \dots + A_{n} z^{n} + \dots . (11) Since, then, {(a + Z)}^{b} = a^{b} {(1 + \frac{z}{a})}^{b}, § 78, 2 if \frac{z}{a} be
substituted for z in (11), and the equation be multiplied throughout by a^{b}, {(a + z)}^{b} = a^{b} + b a^{b - 1} z + A_{2} a^{b - 2} z^{2} + \dots + A_{n} a^{b - n} z^{n} + \dots . (12) Starting with the identity {(1 + \underline{z + t})}^{b} = {(\underline{1 + z} + t)}^{b}, developing {(1 + \underline{z + t})}^{b} by (11) and {(\underline{1 + z} + t)}^{b} by (12), equating the coeﬃcients of the ﬁrst power of t in these developments, multiplying the resultant equation by 1 + z, and equating the
coeﬃcients of like powers of z in this product, equations are obtained from which values may be derived for the
coeﬃcients A_{i} identical in form with those occurring in the development for {(1 + z)}^{b} when b is a
positive integer. It may also be shown that these values of the coeﬃcients satisfy the
equations which result from equating the coeﬃcients of higher powers of t . up prev ptail top \end{array}

Chapter 9The Exponential and Logarithmic Functions. Undetermined Coeﬃcients. Involution and Evolution. The Binomial Theorem

Chapter 9
The Exponential and Logarithmic Functions. Undetermined Coeﬃcients. Involution and Evolution. The Binomial Theorem