Chapter XThe General Theory of the Logarithmic, Exponential, and Circular Functions

217Functions of a complex variable

In Ch. III we defined the complex variable¹

z = x + i y,

and we considered a few simple properties of some classes of expressions involving $z$ , such as the polynomial $P (z)$ . It is natural to describe such expressions as functions of $z$ , and in fact we did describe the quotient $P (z) / Q (z)$ , where $P (z)$ and $Q (z)$ are polynomials, as a ‘rational function’. We have however given no general definition of what is meant by a function of $z$ .

It might seem natural to define a function of $z$ in the same way as that in which we defined a function of the real variable $x$ , i.e. to say that $Z$ is a function of $z$ if any relation subsists between $z$ and $Z$ in virtue of which a value or values of $Z$ corresponds to some or all values of $z$ . But it will be found, on closer examination, that this definition is not one from which any profit can be derived. For if $z$ is given, so are $x$ and $y$ , and conversely: to assign a value of $z$ is precisely the same thing as to assign a pair of values of $x$ and $y$ . Thus a ‘function of $z$ ’, according to the definition suggested, is precisely the same thing as a complex function

f (x, y) + i g (x, y),

of the two real variables $x$ and $y$ . For example

x - i y, x y, ∣ z ∣ = x^{2} + y^{2}, am z = arc tan (y / x)

are ‘functions of $z$ ’. The definition, although perfectly legitimate, is futile because it does not really define a new idea at all. It is therefore more convenient to use the expression ‘function of the complex variable $z$ ’ in a more restricted sense, or in other words to pick out, from the general class of complex functions of the two real variables $x$ and $y$ , a special class to which the expression shall be restricted. But if we were to attempt to explain how this selection is made, and what are the characteristic properties of the special class of functions selected, we should be led far beyond the limits of this book. We shall therefore not attempt to give any general definitions, but shall confine ourselves entirely to special functions defined directly.

Footnotes

In this chapter we shall generally find it convenient to write $x + i y$ rather than $x + y i$ . ↩

218

We have already defined polynomials in $z$ (§ 39), rational functions of $z$ (§ 46), and roots of $z$ (§ 47). There is no difficulty in extending to the complex variable the definitions of algebraical functions, explicit and implicit, which we gave (§§ 26–27) in the case of the real variable $x$ . In all these cases we shall call the complex number $z$ , the argument (§ 44) of the point $z$ , the argument of the function $f (z)$ under consideration. The question which will occupy us in this chapter is that of defining and determining the principal properties of the logarithmic, exponential, and trigonometrical or circular functions of $z$ . These functions are of course so far defined for real values of $z$ only, the logarithm indeed for positive values only.

We shall begin with the logarithmic function. It is natural to attempt to define it by means of some extension of the definition

lo g x = \int_{1}^{x} \frac{d t}{t} (x > 0);

and in order to do this we shall find it necessary to consider briefly some extensions of the notion of an integral.

219Real and complex curvilinear integrals

Let $A B$ be an arc $C$ of a curve defined by the equations

x = ϕ (t), y = ψ (t),

where $ϕ$ and $ψ$ are functions of $t$ with continuous differential coefficients $ϕ^{'}$ and $ψ^{'}$ ; and suppose that, as $t$ varies from $t_{0}$ to $t_{1}$ , the point $(x, y)$ moves along the curve, in the same direction, from $A$ to $B$ .

Then we define the curvilinear integral

\int_{C} {g (x, y) d x + h (x, y) d y}, (1)

where $g$ and $h$ are continuous functions of $x$ and $y$ , as being equivalent to the ordinary integral obtained by effecting the formal substitutions $x = ϕ (t)$ , $y = ψ (t)$ , i.e. to

\int_{t_{0}}^{t_{1}} {g (ϕ, ψ) ϕ^{'} + h (ϕ, ψ) ψ^{'}} d t .

We call $C$ the path of integration.

Let us suppose now that

z = x + i y = ϕ (t) + i ψ (t),

so that $z$ describes the curve $C$ in Argand’s diagram as $t$ varies. Further let us suppose that

f (z) = u + i v

is a polynomial in $z$ or rational function of $z$ .

Then we define

\int_{C} f (z) d z (2)

as meaning

\int_{C} (u + i v) (d x + i d y),

which is itself defined as meaning

\int_{C} (u d x - v d y) + i \int_{C} (v d x + u d y) .

220The definition of $\operatorname{Log} \zeta$

Now let $ζ = ξ + i η$ be any complex number. We define $Log ζ$ , the general logarithm of $ζ$ , by the equation

Log ζ = \int_{C} \frac{d z}{z},

where $C$ is a curve which starts from $1$ and ends at $ζ$ and does not pass through the origin. Thus (Fig. 54) the paths (a), (b), (c) are paths such as are contemplated in the definition. The value of $Log z$ is thus defined when the particular path of integration has been chosen. But at present it is not clear how far the value of $Log z$ resulting from the definition depends upon what path is chosen. Suppose for example that $ζ$ is real and positive, say equal to $ξ$ . Then one possible path of integration is the straight line from $1$ to $ξ$ , a path which we may suppose to be defined by

the equations $x = t$ , $y = 0$ . In this case, and with this particular choice of the path of integration, we have

Log ξ = \int_{1}^{ξ} \frac{d t}{t},

so that $Log ξ$ is equal to $lo g ξ$ , the logarithm of $ξ$ according to the definition given in the last chapter. Thus one value at any rate of $Log ξ$ , when $ξ$ is real and positive, is $lo g ξ$ . But in this case, as in the general case, the path of integration can be chosen in an infinite variety of different ways. There is nothing to show that every value of $Log ξ$ is equal to $lo g ξ$ ; and in point of fact we shall see that this is not the case. This is why we have adopted the notation $Log ζ$ , $Log ξ$ instead of $lo g ζ$ , $lo g ξ$ . $Log ξ$ is (possibly at any rate) a many valued function, and $lo g ξ$ is only one of its values. And in the general case, so far as we can see at present, three alternatives are equally possible, viz. that

(1) we may always get the same value of $Log ζ$ , by whatever path we go from $1$ to $ζ$ ;

(2) we may get a different value corresponding to every different path;

(3) we may get a number of different values each of which corresponds to a whole class of paths:

and the truth or falsehood of any one of these alternatives is in no way implied by our definition.

221The values of $\operatorname{Log} \zeta$

Let us suppose that the polar coordinates of the point $z = ζ$ are $ρ$ and $ϕ$ , so that

ζ = ρ (cos ϕ + i sin ϕ) .

We suppose for the present that $- π < ϕ < π$ , while $ρ$ may have any positive value. Thus $ζ$ may have any value other than zero or a real negative value.

The coordinates $(x, y)$ of any point on the path $C$ are functions of $t$ , and so also are its polar coordinates $(r, θ)$ . Also

Log ζ = \int_{C} \frac{d z}{z} = \int_{C} \frac{d x + i d y}{x + i y} = \int_{t_{0}}^{t_{1}} \frac{1}{x + i y} (\frac{d x}{d t} + i \frac{d y}{d t}) d t,

in virtue of the definitions of § 219. But $x = r cos θ$ , $y = r sin θ$ , and

\frac{d x}{d t} + i \frac{d y}{d t} = (cos θ \frac{d r}{d t} - r sin θ \frac{d θ}{d t}) + i (sin θ \frac{d r}{d t} + r cos θ \frac{d θ}{d t}) = (cos θ + i sin θ) (\frac{d r}{d t} + i r \frac{d θ}{d t});

so that

Log ζ = \int_{t_{0}}^{t_{1}} \frac{1}{r} \frac{d r}{d t} d t + i \int_{t_{0}}^{t_{1}} \frac{d θ}{d t} d t = [lo g r] + i [θ],

where $[lo g r]$ denotes the difference between the values of $lo g r$ at the points corresponding to $t = t_{1}$ and $t = t_{0}$ , and $[θ]$ has a similar meaning.

It is clear that

[lo g r] = lo g ρ - lo g 1 = lo g ρ;

but the value of $[θ]$ requires a little more consideration. Let us suppose first that the path of integration is the straight line from $1$ to $ζ$ . The initial value of $θ$ is the amplitude of $1$ , or rather

one of the amplitudes of $1$ , viz. $2 k π$ , where $k$ is any integer. Let us suppose that initially $θ = 2 k π$ . It is evident from the figure that $θ$ increases from $2 k π$ to $2 k π + ϕ$ as $t$ moves along the line. Thus

[θ] = (2 k π + ϕ) - 2 k π = ϕ,

and, when the path of integration is a straight line, $Log ζ = lo g ρ + i ϕ$ .

We shall call this particular value of $Log ζ$ the principal value. When $ζ$ is real and positive, $ζ = ρ$ and $ϕ = 0$ , so that the principal value of $Log ζ$ is the ordinary logarithm $lo g ζ$ . Hence it will be convenient in general to denote the principal value of $Log ζ$ by $lo g ζ$ . Thus

lo g ζ = lo g ρ + i ϕ,

and the principal value is characterised by the fact that its imaginary part lies between $- π$ and $π$ .

Next let us consider any path (such as those shown in Fig. 56) such that the area or areas included

between the path and the straight line from $1$ to $ζ$ does not include the origin. It is easy to see that $[θ]$ is still equal to $ϕ$ . Along the curve shown in the figure by a continuous line, for example, $θ$ , initially equal to $2 k π$ , first decreases to the value

2 k π - X O P

and then increases again, being equal to $2 k π$ at $Q$ , and finally to $2 k π + ϕ$ . The dotted curve shows a similar but slightly more complicated case in which the straight line and the curve bound two areas, neither of which includes the origin. Thus

if the path of integration is such that the closed curve formed by it and the line from $1$ to $ζ$ does not include the origin, then

Log ζ = lo g ζ = lo g ρ + i ϕ .

On the other hand it is easy to construct paths of integration such that $[θ]$ is not equal to $ϕ$ . Consider, for example, the curve indicated by a continuous line in Fig. 57. If $θ$ is initially equal to $2 k π$ , it will have increased

by $2 π$ when we get to $P$ and by $4 π$ when we get to $Q$ ; and its final value will be $2 k π + 4 π + ϕ$ , so that $[θ] = 4 π + ϕ$ and

Log ζ = lo g ρ + i (4 π + ϕ) .

In this case the path of integration winds twice round the origin in the positive sense. If we had taken a path winding $k$ times round the origin we should have found, in a precisely similar way, that $[θ] = 2 k π + ϕ$ and

Log ζ = lo g ρ + i (2 k π + ϕ) .

Here $k$ is positive. By making the path wind round the origin in the opposite direction (as shown in the dotted path in Fig. 57), we obtain a similar series of values in which $k$ is negative. Since $∣ ζ ∣ = ρ$ , and the different angles $2 k π + ϕ$ are the different values of $am ζ$ , we conclude that every value of $lo g ∣ ζ ∣ + i am ζ$ is a value of $Log ζ$ ; and it is clear from the preceding discussion that every value of $Log ζ$ must be of this form.

We may summarise our conclusions as follows:

the general value of $Log ζ$ is

lo g ∣ ζ ∣ + i am ζ = lo g ρ + i (2 k π + ϕ),

where $k$ is any positive or negative integer. The value of $k$ is determined by the path of integration chosen. If this path is a straight line then $k = 0$ and

Log ζ = lo g ζ = lo g ρ + i ϕ .

In what precedes we have used $ζ$ to denote the argument of the function $Log ζ$ , and $(ξ, η)$ or $(ρ, ϕ)$ to denote the coordinates of $ζ$ ; and $z$ , $(x, y)$ , $(r, θ)$ to denote an arbitrary point on the path of integration and its coordinates. There is however no reason now why we should not revert to the natural notation in which $z$ is used as the argument of the function $Log z$ , and we shall do this in the following examples.

Examples XCIII.

1. We supposed above that $- π < θ < π$ , and so excluded the case in which $z$ is real and negative. In this case the straight line from $1$ to $z$ passes through $0$ , and is therefore not admissible as a path of integration. Both $π$ and $- π$ are values of $am z$ , and $θ$ is equal to one or other of them: also $r = - z$ . The values of $Log z$ are still the values of $lo g ∣ z ∣ + i am z$ , viz.

lo g (- z) + (2 k + 1) π i,

where $k$ is an integer. The values $lo g (- z) + π i$ and $lo g (- z) - π i$ correspond to paths from $1$ to $z$ lying respectively entirely above and entirely below the real axis. Either of them may be taken as the principal value of $Log z$ , as convenience dictates. We shall choose the value $lo g (- z) + π$ i corresponding to the first path.

2. The real and imaginary parts of any value of $Log z$ are both continuous functions of $x$ and $y$ , except for $x = 0$ , $y = 0$ .

3. The functional equation satisfied by $Log z$ . The function $Log z$ satisfies the equation

Log z_{1} z_{2} = Log z_{1} + Log z_{2}, (1)

in the sense that every value of either side of this equation is one of the values of the other side. This follows at once by putting

z_{1} = r_{1} (cos θ_{1} + i sin θ_{1}), z_{2} = r_{2} (cos θ_{2} + i sin θ_{2}),

and applying the formula of p. 401. It is however not true that

lo g z_{1} z_{2} = lo g z_{1} + lo g z_{2} (2)

in all circumstances. If, e.g.,

z_{1} = z_{2} = \frac{1}{2} (- 1 + i 3) = cos \frac{2}{3} π + i sin \frac{2}{3} π,

then $lo g z_{1} = lo g z_{2} = \frac{2}{3} π i$ , and $lo g z_{1} + lo g z_{2} = \frac{4}{3} π i$ , which is one of the values of $Log z_{1} z_{2}$ , but not the principal value. In fact $lo g z_{1} z_{2} = - \frac{2}{3} π i$ .

An equation such as (1), in which every value of either side is a value of the other, we shall call a complete equation, or an equation which is completely true.

4. The equation $Log z^{m} = m Log z$ , where $m$ is an integer, is not completely true: every value of the right-hand side is a value of the left-hand side, but the converse is not true.

5. The equation $Log (1/ z) = - Log z$ is completely true. It is also true that $lo g (1/ z) = - lo g z$ , except when $z$ is real and negative.

6. The equation

lo g (\frac{z - a}{z - b}) = lo g (z - a) - lo g (z - b)

is true if $z$ lies outside the region bounded by the line joining the points $z = a$ , $z = b$ , and lines through these points parallel to $O X$ and extending to infinity in the negative direction.

7. The equation

lo g (\frac{a - z}{b - z}) = lo g (1 - \frac{a}{z}) - lo g (1 - \frac{b}{z})

is true if $z$ lies outside the triangle formed by the three points $O$ , $a$ , $b$ .

8. Draw the graph of the function $I (Log x)$ of the real variable $x$ . [The graph consists of the positive halves of the lines $y = 2 k π$ and the negative halves of the lines $y = (2 k + 1) π$ .]

9. The function $f (x)$ of the real variable $x$ , defined by

π f (x) = p π + (q - p) I (lo g x),

is equal to $p$ when $x$ is positive and to $q$ when $x$ is negative.

10. The function $f (x)$ defined by

π f (x) = p π + (q - p) I {lo g (x - 1)} + (r - q) I (lo g x)

is equal to $p$ when $x > 1$ , to $q$ when $0 < x < 1$ , and to $r$ when $x < 0$ .

11. For what values of $z$ is (i) $lo g z$ (ii) any value of $Log z$ (a) real or (b) purely imaginary?

12. If $z = x + i y$ then $Log Log z = lo g R + i (Θ + 2 k^{'} π)$ , where

R^{2} = (lo g r)^{2} + (θ + 2 k π)^{2}

and $Θ$ is the least positive angle determined by the equations

cos Θ : sin Θ : 1 :: lo g r : θ + 2 k π : (lo g r)^{2} + (θ + 2 k π)^{2} .

Plot roughly the doubly infinite set of values of $Log Log (1 + i 3)$ , indicating which of them are values of $lo g Log (1 + i 3)$ and which of $Log lo g (1 + i 3)$ .

222The exponential function

In Ch. IX we defined a function $e^{y}$ of the real variable $y$ as the inverse of the function $y = lo g x$ . It is naturally suggested that we should define a function of the complex variable $z$ which is the inverse of the function $Log z$ .

Definition. If any value of $Log z$ is equal to $ζ$ , we call $z$ the exponential of $ζ$ and write

z = exp ζ .

Thus $z = exp ζ$ if $ζ = Log z$ . It is certain that to any given value of $z$ correspond infinitely many different values of $ζ$ . It would not be unnatural to suppose that, conversely, to any given value of $ζ$ correspond infinitely many values of $z$ , or in other words that $exp ζ$ is an infinitely many-valued function of $ζ$ . This is however not the case, as is proved by the following theorem.

Theorem. The exponential function $exp ζ$ is a one-valued function of $ζ$ .

For suppose that

z_{1} = r_{1} (cos θ_{1} + i sin θ_{1}), z_{2} = r_{2} (cos θ_{2} + i sin θ_{2})

are both values of $exp ζ$ . Then

ζ = Log z_{1} = Log z_{2},

and so

lo g r_{1} + i (θ_{1} + 2 mπ) = lo g r_{2} + i (θ_{2} + 2 nπ),

where $m$ and $n$ are integers. This involves

lo g r_{1} = lo g r_{2}, θ_{1} + 2 mπ = θ_{2} + 2 nπ .

Thus $r_{1} = r_{2}$ , and $θ_{1}$ and $θ_{2}$ differ by a multiple of $2 π$ . Hence $z_{1} = z_{2}$ .

Corollary. If $ζ$ is real then $exp ζ = e^{ζ}$ , the real exponential function of $ζ$ defined in Ch. IX.

For if $z = e^{ζ}$ then $lo g z = ζ$ , i.e. one of the values of $Log z$ is $ζ$ . Hence $z = exp ζ$ .

223The value of $\exp \zeta$

Let $ζ = ξ + i η$ and

z = exp ζ = r (cos θ + i sin θ) .

Then

ξ + i η = Log z = lo g r + i (θ + 2 mπ),

where $m$ is an integer. Hence $ξ = lo g r$ , $η = θ + 2 mπ$ , or

r = e^{ξ}, θ = η - 2 mπ;

and accordingly

exp (ξ + i η) = e^{ξ} (cos η + i sin η) .

If $η = 0$ then $exp ξ = e^{ξ}$ , as we have already inferred in § 222. It is clear that both the real and the imaginary parts of $exp (ξ + i η)$ are continuous functions of $ξ$ and $η$ for all values of $ξ$ and $η$ .

224The functional equation satisfied by $\exp \zeta$

Let $ζ_{1} = ξ_{1} + i η_{1}$ , $ζ_{2} = ξ_{2} + i η_{2}$ . Then

exp ζ_{1} \times exp ζ_{2} = e^{ξ_{1}} (cos η_{1} + i sin η_{1}) \times e^{ξ_{2}} (cos η_{2} + i sin η_{2}) = e^{ξ_{1} + ξ_{2}} {cos (η_{1} + η_{2}) + i sin (η_{1} + η_{2})} = exp (ζ_{1} + ζ_{2}) .

The exponential function therefore satisfies the functional relation $f (ζ_{1} + ζ_{2}) = f (ζ_{1}) f (ζ_{2})$ , an equation which we have proved already (§ 205) to be true for real values of $ζ_{1}$ and $ζ_{2}$ .

225The general power $a^{\zeta}$

It might seem natural, as $exp ζ = e^{ζ}$ when $ζ$ is real, to adopt the same notation when $ζ$ is complex and to drop the notation $exp ζ$ altogether. We shall not follow this course because we shall have to give a more general definition of the meaning of the symbol $e^{ζ}$ : we shall find then that $e^{ζ}$ represents a function with infinitely many values of which $exp ζ$ is only one.

We have already defined the meaning of the symbol $a^{ζ}$ in a considerable variety of cases. It is defined in elementary Algebra in the case in which $a$ is real and positive and $ζ$ rational, or $a$ real and negative and $ζ$ a rational fraction whose denominator is odd. According to the definitions there given $a^{ζ}$ has at most two values. In Ch. III we extended our definitions to cover the case in which $a$ is any real or complex number and $ζ$ any rational number $p / q$ ; and in Ch. IX we gave a new definition, expressed by the equation

a^{ζ} = e^{ζ l o g a},

which applies whenever $ζ$ is real and $a$ real and positive.

Thus we have, in one way or another, attached a meaning to such expressions as

3^{1/2}, (- 1)^{1/3}, (3 + \frac{1}{2} i)^{- 1/2}, (3.5)^{1 + 2};

but we have as yet given no definitions which enable us to attach any meaning to such expressions as

(1 + i)^{2}, 2^{i}, (3 + 2 i)^{2 + 3 i} .

We shall now give a general definition of $a^{ζ}$ which applies to all values of $a$ and $ζ$ , real or complex, with the one limitation that $a$ must not be equal to zero.

Definition. The function $a^{ζ}$ is defined by the equation

a^{ζ} = exp (ζ Log a)

where $Log a$ is any value of the logarithm of $a$ .

We must first satisfy ourselves that this definition is consistent with the previous definitions and includes them all as particular cases.

(1) If $a$ is positive and $ζ$ real, then one value of $ζ Log a$ , viz. $ζ lo g a$ , is real: and $exp (ζ lo g a) = e^{ζ l o g a}$ , which agrees with the definition adopted in Ch. IX. The definition of Ch. IX is, as we saw then, consistent with the definition given in elementary Algebra; and so our new definition is so too.

(2) If $a = e^{τ} (cos ψ + i sin ψ)$ , then

Log a = τ + i (ψ + 2 mπ), exp {(p / q) Log a} = e^{p τ / q} Cis {(p / q) (ψ + 2 mπ)},

where $m$ may have any integral value. It is easy to see that if $m$ assumes all possible integral values then this expression assumes $q$ and only $q$ different values, which are precisely the values of $a^{p / q}$ found in § 48. Hence our new definition is also consistent with that of Ch. III.

226The general value of $a^{\zeta}$

Let

ζ = ξ + i η, a = σ (cos ψ + i sin ψ)

where $- π < ψ ≦ π$ , so that, in the notation of § 225, $σ = e^{τ}$ or $τ = lo g σ$ .

Then

ζ Log a = (ξ + i η) {lo g σ + i (ψ + 2 mπ)} = L + i M,

where

L = ξ lo g σ - η (ψ + 2 mπ), M = η lo g σ + ξ (ψ + 2 mπ);

and

a^{ζ} = exp (ζ Log a) = e^{L} (cos M + i sin M) .

Thus the general value of $a^{ζ}$ is

e^{ξ l o g σ - η (ψ + 2 mπ)} [cos {η lo g σ + ξ (ψ + 2 mπ)} + i sin {η lo g σ + ξ (ψ + 2 mπ)}] .

In general $a^{ζ}$ is an infinitely many-valued function. For

∣ a^{ζ} ∣ = e^{ξ l o g σ - η (ψ + 2 mπ)}

has a different value for every value of $m$ , unless $η = 0$ . If on the other hand $η = 0$ , then the moduli of all the different values of $a^{ζ}$ are the same. But any two values differ unless their amplitudes are the same or differ by a multiple of $2 π$ . This requires that $ξ (ψ + 2 mπ)$ and $ξ (ψ + 2 nπ)$ , where $m$ and $n$ are different integers, shall differ, if at all, by a multiple of $2 π$ . But if

ξ (ψ + 2 mπ) - ξ (ψ + 2 nπ) = 2 k π,

then $ξ = k / (m - n)$ is rational. We conclude that $a^{ζ}$ is infinitely many-valued unless $ζ$ is real and rational. On the other hand we have already seen that, when $ζ$ is real and rational, $a^{ζ}$ has but a finite number of values.

The principal value of $a^{ζ} = exp (ζ Log a)$ is obtained by giving $Log a$ its principal value, i.e. by supposing $m = 0$ in the general formula. Thus the principal value of $a^{ζ}$ is

e^{ξ l o g σ - η ψ} {cos (η lo g σ + ξ ψ) + i sin (η lo g σ + ξ ψ)} .

Two particular cases are of especial interest. If $a$ is real and positive and $ζ$ real, then $σ = a$ , $ψ = 0$ , $ξ = ζ$ , $η = 0$ , and the principal value of $a^{ζ}$ is $e^{ζ l o g a}$ , which is the value defined in the last chapter. If $∣ a ∣ = 1$ and $ζ$ is real, then $σ = 1$ , $ξ = ζ$ , $η = 0$ , and the principal value of $(cos ψ + i sin ψ)^{ζ}$ is $cos ζ ψ + i sin ζ ψ$ . This is a further generalisation of De Moivre’s Theorem (§§ 45, 49).

Examples XCIV.

1. Find all the values of $i^{i}$ . [By definition

i^{i} = exp (i Log i) .

But

i = cos \frac{1}{2} π + i sin \frac{1}{2} π, Log i = (2 k + \frac{1}{2}) π i,

where $k$ is any integer. Hence

i^{i} = exp {- (2 k + \frac{1}{2}) π} = e^{- (2 k + \frac{1}{2}) π} .

All the values of $i^{i}$ are therefore real and positive.]

2. Find all the values of $(1 + i)^{i}$ , $i^{1 + i}$ , $(1 + i)^{1 + i}$ .

3. The values of $a^{ζ}$ , when plotted in the Argand diagram, are the vertices of an equiangular polygon inscribed in an equiangular spiral whose angle is independent of $a$ . (Math. Trip. 1899.)

[If $a^{ζ} = r (cos θ + i sin θ)$ we have

r = e^{ξ l o g σ - η (ψ + 2 mπ)}, θ = η lo g σ + ξ (ψ + 2 mπ);

and all the points lie on the spiral $r = σ^{(ξ^{2} + η^{2}) / ξ} e^{- η θ / ξ}$ .]

4. The function $e^{ζ}$ . If we write $e$ for $a$ in the general formula, so that $lo g σ = 1$ , $ψ = 0$ , we obtain

e^{ζ} = e^{ξ - 2 mπ η} {cos (η + 2 mπ ξ) + i sin (η + 2 mπ ξ)} .

The principal value of $e^{ζ}$ is $e^{ξ} (cos η + i sin η)$ , which is equal to $exp ζ$ (§ 223). In particular, if $ζ$ is real, so that $η = 0$ , we obtain

e^{ζ} (cos 2 mπ ζ + i sin 2 mπ ζ)

as the general and $e^{ζ}$ as the principal value, $e^{ζ}$ denoting here the positive value of the exponential defined in Ch. IX.

5. Show that $Log e^{ζ} = (1 + 2 mπ i) ζ + 2 nπ i$ , where $m$ and $n$ are any integers, and that in general $Log a^{ζ}$ has a double infinity of values.

6. The equation $1/ a^{ζ} = a^{- ζ}$ is completely true (Ex. xciii. 3): it is also true of the principal values.

7. The equation $a^{ζ} \times b^{ζ} = (ab)^{ζ}$ is completely true but not always true of the principal values.

8. The equation $a^{ζ} \times a^{ζ^{'}} = a^{ζ + ζ^{'}}$ is not completely true, but is true of the principal values. [Every value of the right-hand side is a value of the left-hand side, but the general value of $a^{ζ} \times a^{ζ^{'}}$ , viz.

exp {ζ (lo g a + 2 mπ i) + ζ^{'} (lo g a + 2 nπ i)},

is not as a rule a value of $a^{ζ + ζ^{'}}$ unless $m = n$ .]

9. What are the corresponding results as regards the equations

Log a^{ζ} = ζ Log a, (a^{ζ})^{ζ^{'}} = (a^{ζ^{'}})^{ζ} = a^{ζ ζ^{'}} ?

10. For what values of $ζ$ is (a) any value (b) the principal value of $e^{ζ}$ (i) real (ii) purely imaginary (iii) of unit modulus?

11. The necessary and sufficient conditions that all the values of $a^{ζ}$ should be real are that $2 ξ$ and ${η lo g ∣ a ∣ + ξ am a} / π$ , where $am a$ denotes any value of the amplitude, should both be integral. What are the corresponding conditions that all the values should be of unit modulus?

12. The general value of $∣ x^{i} + x^{- i} ∣$ , where $x > 0$ , is

e^{- (m - n) π} 2 {cosh 2 (m + n) π + cos (2 lo g x)} .

13. Explain the fallacy in the following argument: since $e^{2 mπ i} = e^{2 nπ i} = 1$ , where $m$ and $n$ are any integers, therefore, raising each side to the power $i$ we obtain $e^{- 2 mπ} = e^{- 2 nπ}$ .

14. In what circumstances are any of the values of $x^{x}$ , where $x$ is real, themselves real? [If $x > 0$ then

x^{x} = exp (x Log x) = exp (x lo g x) Cis 2 mπ x,

the first factor being real. The principal value, for which $m = 0$ , is always real.

If $x$ is a rational fraction $p / (2 q + 1)$ , or is irrational, then there is no other real value. But if $x$ is of the form $p /2 q$ , then there is one other real value, viz. $- exp (x lo g x)$ , given by $m = q$ .

If $x = - ξ < 0$ then

x^{x} = exp {- ξ Log (- ξ)} = exp (- ξ lo g ξ) Cis {- (2 m + 1) π ξ} .

The only case in which any value is real is that in which $ξ = p / (2 q + 1)$ , when $m = q$ gives the real value

exp (- ξ lo g ξ) Cis (- p π) = (- 1)^{p} ξ^{- ξ} .

The cases of reality are illustrated by the examples

(\frac{1}{3})^{1/3} = 3 \frac{1}{3}, (\frac{1}{2})^{\frac{1}{2}} = \pm \frac{1}{2}, (- \frac{2}{3})^{- \frac{2}{3}} = 3 \frac{9}{4}, (- \frac{1}{3})^{- \frac{1}{3}} = - 33 .]

15. Logarithms to any base. We may define $ζ = Log_{a} z$ in two different ways. We may say (i) that $ζ = Log_{a} z$ if the principal value of $a^{ζ}$ is equal to $z$ ; or we may say (ii) that $ζ = Log_{a} z$ if any value of $a^{ζ}$ is equal to $z$ .

Thus if $a = e$ then $ζ = Log_{e} z$ , according to the first definition, if the principal value of $e^{ζ}$ is equal to $z$ , or if $exp ζ = z$ ; and so $Log_{e} z$ is identical with $Log z$ . But, according to the second definition, $ζ = Log_{e} z$ if

e^{ζ} = exp (ζ Log e) = z, ζ Log e = Log z,

or $ζ = (Log z) / (Log e)$ , any values of the logarithms being taken. Thus

ζ = Log_{e} z = \frac{lo g ∣ z ∣ + ( am z + 2 mπ ) i}{1 + 2 nπ i},

so that $ζ$ is a doubly infinitely many-valued function of $z$ . And generally, according to this definition, $Log_{a} z = (Log z) / (Log a)$ .

16. $Log_{e} 1 = 2 mπ i / (1 + 2 nπ i)$ , $Log_{e} (- 1) = (2 m + 1) π i / (1 + 2 nπ i)$ , where $m$ and $n$ are any integers.

227The exponential values of the sine and cosine

From the formula

exp (ξ + i η) = exp ξ (cos η + i sin η),

we can deduce a number of extremely important subsidiary formulae. Taking $ξ = 0$ , we obtain $exp (i η) = cos η + i sin η$ ; and, changing the sign of $η$ , $exp (- i η) = cos η - i sin η$ . Hence

cos η sin η = = - \frac{1}{2} \frac{1}{2} i {exp (i η) + exp (- i η)}, {exp (i η) - exp (- i η)} .

We can of course deduce expressions for any of the trigonometrical ratios of $η$ in terms of $exp (i η)$ .

228Definition of $\sin\zeta$ and $\cos\zeta$ for all values of $\zeta$

We saw in the last section that, when $ζ$ is real,

cos ζ sin ζ = = - \frac{1}{2} \frac{1}{2} i {exp (i ζ) + exp (- i ζ)}, {exp (i ζ) - exp (- i ζ)} . (1a) (1b)

The left-hand sides of these equations are defined, by the ordinary geometrical definitions adopted in elementary Trigonometry, only for real values of $ζ$ . The right-hand sides have, on the other hand, been defined for all values of $ζ$ , real or complex. We are therefore naturally led to adopt the formulae (1) as the definitions of $cos ζ$ and $sin ζ$ for all values of $ζ$ . These definitions agree, in virtue of the results of § 227, with the elementary definitions for real values of $ζ$ .

Having defined $cos ζ$ and $sin ζ$ , we define the other trigonometrical ratios by the equations

tan ζ = \frac{sin ζ}{cos ζ}, cot ζ = \frac{cos ζ}{sin ζ}, sec ζ = \frac{1}{cos ζ}, cosec ζ = \frac{1}{sin ζ} . (2)

It is evident that $cos ζ$ and $sec ζ$ are even functions of $ζ$ , and $sin ζ$ , $tan ζ$ , $cot ζ$ , and $cosec ζ$ odd functions. Also, if $exp (i ζ) = t$ , we have

cos ζ = \frac{1}{2} {t + (1/ t)}, sin ζ = - \frac{1}{2} i {t - (1/ t)}, cos^{2} ζ + sin^{2} ζ = \frac{1}{4} [{t + (1/ t)}^{2} - {t - (1/ t)}^{2}] = 1. (3)

We can moreover express the trigonometrical functions of $ζ + ζ^{'}$ in terms of those of $ζ$ and $ζ^{'}$ by precisely the same formulae as those which hold in elementary trigonometry. For if $exp (i ζ) = t$ , $exp (i ζ^{'}) = t^{'}$ , we have

cos (ζ + ζ^{'}) = \frac{1}{2} (t t^{'} + \frac{1}{t t ^{'}}) = \frac{1}{4} {(t + \frac{1}{t}) (t^{'} + \frac{1}{t ^{'}}) + (t - \frac{1}{t}) (t^{'} - \frac{1}{t ^{'}})} = cos ζ cos ζ^{'} - sin ζ sin ζ^{'}; (4)

and similarly we can prove that

sin (ζ + ζ^{'}) = sin ζ cos ζ^{'} + cos ζ sin ζ^{'} . (5)

In particular

cos (ζ + \frac{1}{2} π) = - sin ζ, sin (ζ + \frac{1}{2} π) = cos ζ . (6)

All the ordinary formulae of elementary Trigonometry are algebraical corollaries of the equations (2)–(6); and so all such relations hold also for the generalised trigonometrical functions defined in this section.

229The generalised hyperbolic functions

In Ex. lxxxvii. 19, we defined $cosh ζ$ and $sinh ζ$ , for real values of $ζ$ , by the equations

cosh ζ = \frac{1}{2} {exp ζ + exp (- ζ)}, sinh ζ = \frac{1}{2} {exp ζ - exp (- ζ)} . (1)

We can now extend this definition to complex values of the variable; i.e. we can agree that the equations (1) are to define $cosh ζ$ and $sinh ζ$ for all values of $ζ$ real or complex. The reader will easily verify the following relations:

cos i ζ = cosh ζ, sin i ζ = i sinh ζ, cosh i ζ = cos ζ, sinh i ζ = i sin ζ .

We have seen that any elementary trigonometrical formula, such as the formula $cos 2 ζ = cos^{2} ζ - sin^{2} ζ$ , remains true when $ζ$ is allowed to assume complex values. It remains true therefore if we write $cos i ζ$ for $cos ζ$ , $sin i ζ$ for $sin ζ$ and $cos 2 i ζ$ for $cos 2 ζ$ ; or, in other words, if we write $cosh ζ$ for $cos ζ$ , $i sinh ζ$ for $sin ζ$ , and $cosh 2 ζ$ for $cos 2 ζ$ . Hence

cosh 2 ζ = cosh^{2} ζ + sinh^{2} ζ .

The same process of transformation may be applied to any trigonometrical identity. It is of course this fact which explains the correspondence noted in Ex. lxxxvii. 21 between the formulae for the hyperbolic and those for the ordinary trigonometrical functions.

230Formulae for $\cos(\xi + i\eta)$ , $\sin(\xi + i\eta)$ , etc

It follows from the addition formulae that

cos (ξ + i η) sin (ξ + i η) = cos ξ cos i η = sin ξ cos i η - sin ξ sin i η + cos ξ sin i η = cos ξ cosh η = sin ξ cosh η - i sin ξ sinh η, + i cos ξ sinh η .

These formulae are true for all values of $ξ$ and $η$ . The interesting case is that in which $ξ$ and $η$ are real. They then give expressions for the real and imaginary parts of the cosine and sine of a complex number.

Examples XCV.

1. Determine the values of $ζ$ for which $cos ζ$ and $sin ζ$ are (i) real (ii) purely imaginary. [For example $cos ζ$ is real when $η = 0$ or when $ξ$ is any multiple of $π$ .]

∣ cos (ξ + i η) ∣ ∣ sin (ξ + i η) ∣ = cos^{2} ξ + sinh^{2} η = sin^{2} ξ + sinh^{2} η = \frac{1}{2} (cosh 2 η + cos 2 ξ), = \frac{1}{2} (cosh 2 η - cos 2 ξ) .

[Use (e.g.) the equation $∣ cos (ξ + i η) ∣ = cos (ξ + i η) cos (ξ - i η)$ .]

3. $tan (ξ + i η) = \frac{sin 2 ξ + i sinh 2 η}{cosh 2 η + cos 2 ξ}$ , $cot (ξ + i η) = \frac{sin 2 ξ - i sinh 2 η}{cosh 2 η - cos 2 ξ}$ .

[For example

tan (ξ + i η) = \frac{sin ( ξ + i η ) cos ( ξ - i η )}{cos ( ξ + i η ) cos ( ξ - i η )} = \frac{sin 2 ξ + sin 2 i η}{cos 2 ξ + cos 2 i η},

which leads at once to the result given.]

sec (ξ + i η) cosec (ξ + i η) = \frac{cos ξ cosh η + i sin ξ sinh η}{\frac{1}{2} ( cosh 2 η + cos 2 ξ )}, = \frac{sin ξ cosh η - i cos ξ sinh η}{\frac{1}{2} ( cosh 2 η - cos 2 ξ )} .

5. If $∣ cos (ξ + i η) ∣ = 1$ then $sin^{2} ξ = sinh^{2} η$ , and if $∣ sin (ξ + i η) ∣ = 1$ then $cos^{2} ξ = sinh^{2} η$ .

6. If $∣ cos (ξ + i η) ∣ = 1$ , then

sin {am cos (ξ + i η)} = \pm sin^{2} ξ = \pm sinh^{2} η .

7. Prove that $Log cos (ξ + i η) = A + i B$ , where

A = \frac{1}{2} lo g {\frac{1}{2} (cosh 2 η + cos 2 ξ)}

and $B$ is any angle such that

\frac{cos B}{cos ξ cosh η} = - \frac{sin B}{sin ξ sinh η} = \frac{1}{\frac{1}{2} ( cosh 2 η + cos 2 ξ )} .

Find a similar formula for $Log sin (ξ + i η)$ .

8. Solution of the equation $cos ζ = a$ , where $a$ is real. Putting $ζ = ξ + i η$ , and equating real and imaginary parts, we obtain

cos ξ cosh η = a, sin ξ sinh η = 0.

Hence either $η = 0$ or $ξ$ is a multiple of $π$ . If (i) $η = 0$ then $cos ξ = a$ , which is impossible unless $- 1 ≦ a ≦ 1$ . This hypothesis leads to the solution

ζ = 2 k π \pm arc cos a,

where $arc cos a$ lies between $0$ and $\frac{1}{2} π$ . If (ii) $ξ = mπ$ then $cosh η = (- 1)^{m} a$ , so that either $a ≧ 1$ and $m$ is even, or $a ≦ - 1$ and $m$ is odd. If $a = \pm 1$ then $η = 0$ , and we are led back to our first case. If $∣ a ∣ > 1$ then $cosh η = ∣ a ∣$ , and we are led to the solutions

ζ ζ = = 2 k (2 k + 1) π \pm i lo g {π \pm i lo g {- a + a^{2} - 1} a + a^{2} - 1} (a > 1), (a < - 1) .

For example, the general solution of $cos ζ = - \frac{5}{3}$ is $ζ = (2 k + 1) π \pm i lo g 3$ .

9. Solve $sin ζ = α$ , where $α$ is real.

10. Solution of $cos ζ = α + i β$ , where $β \neq = 0$ . We may suppose $β > 0$ , since the results when $β < 0$ may be deduced by merely changing the sign of $i$ . In this case

cos ξ cosh η = α, sin ξ sinh η = - β, (1)

and

(α / cosh η)^{2} + (β / sinh η)^{2} = 1.

If we put $cosh^{2} η = x$ we find that

x^{2} - (1 + α^{2} + β^{2}) x + α^{2} = 0

or $x = (A_{1} \pm A_{2})^{2}$ , where

A_{1} = \frac{1}{2} (α + 1)^{2} + β^{2}, A_{2} = \frac{1}{2} (α - 1)^{2} + β^{2} .

Suppose $α > 0$ . Then $A_{1} > A_{2} > 0$ and $cosh η = A_{1} \pm A_{2}$ . Also

cos ξ = α / (cosh η) = A_{1} \mp A_{2},

and since $cosh η > cos ξ$ we must take

cosh η = A_{1} + A_{2}, cos ξ = A_{1} - A_{2} .

The general solutions of these equations are

ξ = 2 k π \pm arc cos M, η = \pm lo g {L + L^{2} - 1}, (2)

where $L = A_{1} + A_{2}$ , $M = A_{1} - A_{2}$ , and $arc cos M$ lies between $0$ and $\frac{1}{2} π$ .

The values of $η$ and $ξ$ thus found above include, however, the solutions of the equations

cos ξ cosh η = α, sin ξ sinh η = β, (3)

as well as those of the equations (1), since we have only used the second of the latter equations after squaring it. To distinguish the two sets of solutions we observe that the sign of $sin ξ$ is the same as the ambiguous sign in the first of the equations (2), and the sign of $sinh η$ is the same as the ambiguous sign in the second. Since $β > 0$ , these two signs must be different. Hence the general solution required is

ζ = 2 k π \pm [arc cos M - i lo g {L + L^{2} - 1}] .

11. Work out the cases in which $α < 0$ and $α = 0$ in the same way.

12. If $β = 0$ then $L = \frac{1}{2} ∣ α + 1∣ + \frac{1}{2} ∣ α - 1∣$ and $M = \frac{1}{2} ∣ α + 1∣ - \frac{1}{2} ∣ α - 1∣$ . Verify that the results thus obtained agree with those of Ex. 8.

13. Show that if $α$ and $β$ are positive then the general solution of $sin ζ = α + i β$ is

ζ = k π + (- 1)^{k} [arc sin M + i lo g {L + L^{2} - 1}],

where $arc sin M$ lies between $0$ and $\frac{1}{2} π$ . Obtain the solution in the other possible cases.

14. Solve $tan ζ = α$ , where $α$ is real. [All the roots are real.]

15. Show that the general solution of $tan ζ = α + i β$ , where $β \neq = 0$ , is

ζ = k π + \frac{1}{2} θ + \frac{1}{4} i lo g {\frac{α ^{2} + ( 1 + β ) ^{2}}{α ^{2} + ( 1 - β ) ^{2}}},

where $θ$ is the numerically least angle such that

cos θ : sin θ : 1 :: 1 - α^{2} - β^{2} : 2 α : (1 - α^{2} - β^{2})^{2} + 4 α^{2} .

16. If $z = ξ exp (\frac{1}{4} π i)$ , where $ξ$ is real, and $c$ is also real, then the modulus of $cos 2 π z - cos 2 π c$ is

[b] \sqrt [\frac{1}{2} {1 + cos 4 π c + cos (2 π ξ 2) + cosh (2 π ξ 2) - 4 cos 2 π c cos (π ξ 2) cosh (π ξ 2)}] .

17. Prove that

∣ exp exp (ξ + i η) ∣ = exp (exp ξ cos η), R {cos cos (ξ + i η)} I {sin sin (ξ + i η)} = cos (cos ξ cosh η) cosh (sin ξ sinh η), = cos (sin ξ cosh η) sinh (cos ξ sinh η) .

18. Prove that $∣ exp ζ ∣$ tends to $\infty$ if $ζ$ moves away towards infinity along any straight line through the origin making an angle less than $\frac{1}{2} π$ with $O X$ , and to $0$ if $ζ$ moves away along a similar line making an angle greater than $\frac{1}{2} π$ with $O X$ .

19. Prove that $∣ cos ζ ∣$ and $∣ sin ζ ∣$ tend to $\infty$ if $ζ$ moves away towards infinity along any straight line through the origin other than either half of the real axis.

20. Prove that $tan ζ$ tends to $- i$ or to $i$ if $ζ$ moves away to infinity along the straight line of Ex. 19, to $- i$ if the line lies above the real axis and to $i$ if it lies below.

231The connection between the logarithmic and the inverse trigonometrical functions

We found in Ch. VI that the integral of a rational or algebraical function $ϕ (x, α, β, \dots)$ , where $α$ , $β$ , … are constants, often assumes different forms according to the values of $α$ , $β$ , …; sometimes it can be expressed by means of logarithms, and sometimes by means of inverse trigonometrical functions. Thus, for example,

\int \frac{d x}{x ^{2} + α} = \frac{1}{α} arc tan \frac{x}{α} (1)

if $α > 0$ , but

\int \frac{d x}{x ^{2} + α} = \frac{1}{2 - α} lo g \frac{x - - α}{x + - α} (2)

if $α < 0$ . These facts suggest the existence of some functional connection between the logarithmic and the inverse circular functions. That there is such a connection may also be inferred from the facts that we have expressed the circular functions of $ζ$ in terms of $exp i ζ$ , and that the logarithm is the inverse of the exponential function.

Let us consider more particularly the equation

\int \frac{d x}{x ^{2} - α ^{2}} = \frac{1}{2 α} lo g (\frac{x - α}{x + α}),

which holds when $α$ is real and $(x - α) / (x + α)$ is positive. If we could write $i α$ instead of $α$ in this equation, we should be led to the formula

arc tan (\frac{x}{α}) = \frac{1}{2 i} lo g (\frac{x - i α}{x + i α}) + C, (3)

where $C$ is a constant, and the question is suggested whether, now that we have defined the logarithm of a complex number, this equation will not be found to be actually true.

Now (§ 221)

Log (x \pm i α) = \frac{1}{2} lo g (x^{2} + α^{2}) \pm i (ϕ + 2 k π),

where $k$ is an integer and $ϕ$ is the numerically least angle such that $cos ϕ = x / x^{2} + α^{2}$ and $sin ϕ = α / x^{2} + α^{2}$ . Thus

\frac{1}{2 i} Log (\frac{x - i α}{x + i α}) = - ϕ - l π,

where $l$ is an integer, and this does in fact differ by a constant from any value of $arc tan (x / α)$ .

The standard formula connecting the logarithmic and inverse circular functions is

arc tan x = \frac{1}{2 i} Log (\frac{1 + i x}{1 - i x}), (4)

where $x$ is real. It is most easily verified by putting $x = tan y$ , when the right-hand side reduces to

\frac{1}{2 i} Log (\frac{cos y + i sin y}{cos y - i sin y}) = \frac{1}{2 i} Log (exp 2 i y) = y + k π,

where $k$ is any integer, so that the equation (4) is ‘completely’ true (Ex. xciii. 3). The reader should also verify the formulae

arc cos x = - i Log {x \pm i 1 - x^{2}}, arc sin x = - i Log {i x \pm 1 - x^{2}}, (5)

where $- 1 ≦ x ≦ 1$ : each of these formulae also is ‘completely’ true.

Example. Solving the equation

cos u = x = \frac{1}{2} {y + (1/ y)},

where $y = exp (i u)$ , with respect to $y$ , we obtain $y = x \pm i 1 - x^{2}$ . Thus:

u = - i Log y = - i Log {x \pm i 1 - x^{2}},

which is equivalent to the first of the equations (5). Obtain the remaining equations (4) and (5) by similar reasoning.

232The power series for $\exp z$

¹ We saw in § 212 that when $z$ is real

exp z = 1 + z + \frac{z ^{2}}{2 !} + \dots . (1)

Moreover we saw in § 191 that the series on the right-hand side remains convergent (indeed absolutely convergent) when $z$ is complex. It is naturally suggested that the equation (1) also remains true, and we shall now prove that this is the case.

Let the sum of the series (1) be denoted by $F (z)$ . The series being absolutely convergent, it follows by direct multiplication (as in Ex. lxxxi. 7) that $F (z)$ satisfies the functional equation

F (z) F (h) = F (z + h) . (2)

Now let $z = i y$ , where $y$ is real, and $F (z) = f (y)$ . Then

f (y) f (k) = f (y + k);

and so

\frac{f ( y + k ) - f ( y )}{k} = f (y) {\frac{f ( k ) - 1}{k}} .

But

\frac{f ( k ) - 1}{k} = i {1 + \frac{ik}{2 !} + \frac{( ik ) ^{2}}{3 !} + \dots};

and so, if $∣ k ∣ < 1$ ,

\frac{f ( k ) - 1}{k} - i < (\frac{1}{2 !} + \frac{1}{3 !} + \dots) ∣ k ∣ < (e - 2) ∣ k ∣.

Hence ${f (k) - 1} / k \to i$ as $k \to 0$ , and so

f^{'} (y) = k \to 0 lim \frac{f ( y + k ) - f ( y )}{k} = i f (y) . (3)

Now

f (y) = F (i y) = 1 + (i y) + \frac{( i y ) ^{2}}{2} + \dots = ϕ (y) + i ψ (y),

where $ϕ (y)$ is an even and $ψ (y)$ an odd function of $y$ , and so

∣ f (y) ∣ = {ϕ (y)}^{2} + {ψ (y)}^{2} = {ϕ (y) + i ψ (y)} {ϕ (y) - i ψ (y)} = F (i y) F (- i y) = F (0) = 1;

and therefore

f (y) = cos Y + i sin Y,

where $Y$ is a function of $y$ such that $- π < Y ≦ π$ . Since $f (y)$ has a differential coefficient, its real and imaginary parts $cos Y$ and $sin Y$ have differential coefficients, and are a fortiori continuous functions of $y$ . Hence $Y$ is a continuous function of $y$ . Suppose that $Y$ changes to $Y + K$ when $y$ changes to $y + k$ . Then $K$ tends to zero with $k$ , and

\frac{K}{k} = {\frac{cos ( Y + K ) - cos Y}{k}} / {\frac{cos ( Y + K ) - cos Y}{K}} .

Of the two quotients on the right-hand side the first tends to a limit when $k \to 0$ , since $cos Y$ has a differential coefficient with respect to $y$ , and the second tends to the limit $- sin Y$ . Hence $K / k$ tends to a limit, so that $Y$ has a differential coefficient with respect to $y$ .

Further

f^{'} (y) = (- sin Y + i cos Y) \frac{d Y}{d y} .

But we have seen already that

f^{'} (y) = i f (y) = - sin Y + i cos Y .

Hence

\frac{d Y}{d y} = 1, Y = y + C,

where $C$ is a constant, and

f (y) = cos (y + C) + i sin (y + C) .

But $f (0) = 1$ when $y = 0$ , so that $C$ is a multiple of $2 π$ , and $f (y) = cos y + i sin y$ . Thus $F (i y) = cos y + i sin y$ for all real values of $y$ . And, if $x$ also is real, we have

F (x + i y) = F (x) F (i y) = exp x (cos y + i sin y) = exp (x + i y),

exp z = 1 + z + \frac{z ^{2}}{2 !} + \dots,

for all values of $z$ .

Footnotes

It will be convenient now to use $z$ instead of $ζ$ as the argument of the exponential function. ↩

233The power series for $\cos z$ and $\sin z$

From the result of the last section and the equations (1) of § 228 it follows at once that

cos z = 1 - \frac{z ^{2}}{2 !} + \frac{z ^{4}}{4 !} - \dots, sin z = z - \frac{z ^{3}}{3 !} + \frac{z ^{5}}{5 !} - \dots

for all values of $z$ . These results were proved for real values of $z$ in Ex. lvi. 1.

Examples XCVI.

1. Calculate $cos i$ and $sin i$ to two places of decimals by means of the power series for $cos z$ and $sin z$ .

2. Prove that $∣ cos z ∣ ≦ cosh ∣ z ∣$ and $∣ sin z ∣ ≦ sinh ∣ z ∣$ .

3. Prove that if $∣ z ∣ < 1$ then $∣ cos z ∣ < 2$ and $∣ sin z ∣ < \frac{6}{5} ∣ z ∣$ .

4. Since $sin 2 z = 2 sin z cos z$ we have

(2 z) - \frac{( 2 z ) ^{3}}{3 !} + \frac{( 2 z ) ^{5}}{5 !} - \dots = 2 (z - \frac{z ^{3}}{3 !} + \dots) (1 - \frac{z ^{2}}{2 !} + \dots) .

Prove by multiplying the two series on the right-hand side (§ 195) and equating coefficients (§ 194) that

(1 2 n + 1) + (3 2 n + 1) + \dots + (2 n + 1 2 n + 1) = 2^{2 n} .

Verify the result by means of the binomial theorem. Derive similar identities from the equations

cos^{2} z + sin^{2} z = 1, cos 2 z = 2 cos^{2} z - 1 = 1 - 2 sin^{2} z .

5. Show that

exp {(1 + i) z} = 0 \sum \infty 2^{\frac{1}{2} n} exp (\frac{1}{4} nπ i) \frac{z ^{n}}{n !} .

6. Expand $cos z cosh z$ in powers of $z$ . [We have

cos z cosh z + i sin z sinh z = cos {(1 - i) z} = \frac{1}{2} [exp {(1 + i) z} + exp {- (1 + i) z}] = \frac{1}{2} 0 \sum \infty 2^{\frac{1}{2} n} {1 + (- 1)^{n}} exp (\frac{1}{4} nπ i) \frac{z ^{n}}{n !},

and similarly

cos z cosh z - i sin z sinh z = cos (1 + i) z = \frac{1}{2} 0 \sum \infty 2^{\frac{1}{2} n} {1 + (- 1)^{n}} exp (- \frac{1}{4} nπ i) \frac{z ^{n}}{n !} .

Hence

cos z cosh z = \frac{1}{2} 0 \sum \infty 2^{\frac{1}{2} n} {1 + (- 1)^{n}} cos \frac{1}{4} nπ \frac{z ^{n}}{n !} = 1 - \frac{2 ^{2} z ^{4}}{4 !} + \frac{2 ^{4} z ^{8}}{8 !} - \dots .]

7. Expand $sin z sinh z$ , $cos z sinh z$ , and $sin z cosh z$ in powers of $z$ .

8. Expand $sin^{2} z$ and $sin^{3} z$ in powers of $z$ . [Use the formulae

sin^{2} z = \frac{1}{2} (1 - cos 2 z), sin^{3} z = \frac{1}{4} (3 sin z - sin 3 z), \dots .

It is clear that the same method may be used to expand $cos^{n} z$ and $sin^{n} z$ , where $n$ is any integer.]

9. Sum the series

C = 1 + \frac{cos z}{1 !} + \frac{cos 2 z}{2 !} + \frac{cos 3 z}{3 !} + \dots, S = \frac{sin z}{1 !} + \frac{sin 2 z}{2 !} + \frac{sin 3 z}{3 !} + \dots .

[Here

C + i S = 1 + \frac{exp ( i z )}{1 !} + \frac{exp ( 2 i z )}{2 !} + \dots = exp {exp (i z)} = exp (cos z) {cos (sin z) + i sin (sin z)},

and similarly

C - i S = exp {exp (- i z)} = exp (cos z) {cos (sin z) - i sin (sin z)} .

Hence

C = exp (cos z) cos (sin z), S = exp (cos z) sin (sin z) .]

10. Sum

1 + \frac{a cos z}{1 !} + \frac{a ^{2} cos 2 z}{2 !} + \dots, \frac{a sin z}{1 !} + \frac{a ^{2} sin 2 z}{2 !} + \dots .

11. Sum

1 - \frac{cos 2 z}{2 !} + \frac{cos 4 z}{4 !} - \dots, \frac{cos z}{1 !} - \frac{cos 3 z}{3 !} + \dots

and the corresponding series involving sines.

12. Show that

1 + \frac{cos 4 z}{4 !} + \frac{cos 8 z}{8 !} + \dots = \frac{1}{2} {cos (cos z) cosh (sin z) + cos (sin z) cosh (cos z)} .

13. Show that the expansions of $cos (x + h)$ and $sin (x + h)$ in powers of $h$ (Ex. lvi. 1) are valid for all values of $x$ and $h$ , real or complex.

234The logarithmic series

We found in § 213 that

lo g (1 + z) = z - \frac{1}{2} z^{2} + \frac{1}{3} z^{3} - \dots (1)

when $z$ is real and numerically less than unity. The series on the right-hand side is convergent, indeed absolutely convergent, when $z$ has any complex value whose modulus is less than unity. It is naturally suggested that the equation (1) remains true for such complex values of $z$ . That this is true may be proved by a modification of the argument of § 213. We shall in fact prove rather more than this, viz. that (1) is true for all values of $z$ such that $∣ z ∣ ≦ 1$ , with the exception of the value $- 1$ .

It will be remembered that $lo g (1 + z)$ is the principal value of $Log (1 + z)$ , and that

lo g (1 + z) = \int_{C} \frac{d u}{u},

where $C$ is the straight line joining the points $1$ and $1 + z$ in the plane of the complex variable $u$ . We may suppose that $z$ is not real, as the formula (1) has been proved already for real values of $z$ .

If we put

z = r (cos θ + i sin θ) = ζ r,

so that $∣ r ∣ ≦ 1$ , and

u = 1 + ζ t,

then $u$ will describe $C$ as $t$ increases from $0$ to $r$ . And

\int_{C} \frac{d u}{u} = \int_{0}^{r} \frac{ζ d t}{1 + ζ t} = \int_{0}^{r} {ζ - ζ^{2} t + ζ^{3} t^{2} - \dots + (- 1)^{m - 1} ζ^{m} t^{m - 1} + \frac{( - 1 ) ^{m} ζ ^{m + 1} t ^{m}}{1 + ζ t}} d t = ζ r - \frac{( ζ r ) ^{2}}{2} + \frac{( ζ r ) ^{3}}{3} - \dots + (- 1)^{m - 1} \frac{( ζ r ) ^{m}}{m} + R_{m} = z - \frac{z ^{2}}{2} + \frac{z ^{3}}{3} - \dots + (- 1)^{m - 1} \frac{z ^{m}}{m} + R_{m}, (2)

where

R_{m} = (- 1)^{m} ζ^{m + 1} \int_{0}^{r} \frac{t ^{m} d t}{1 + ζ t} . (3)

It follows from (1) of § 164 that

∣ R_{m} ∣ ≦ \int_{0}^{r} \frac{t ^{m} d t}{∣1 + ζ t ∣} . (4)

Now $∣1 + ζ t ∣$ or $∣ u ∣$ is never less than $ϖ$ , the perpendicular from $O$ on to the line $C$ .¹ Hence

∣ R_{m} ∣ ≦ \frac{1}{ϖ} \int_{0}^{r} t^{m} d t = \frac{r ^{m + 1}}{( m + 1 ) ϖ} ≦ \frac{1}{( m + 1 ) ϖ},

and so $R_{m} \to 0$ as $m \to \infty$ . It follows from (2) that

lo g (1 + z) = z - \frac{1}{2} z^{2} + \frac{1}{3} z^{3} - \dots . (5)

We have of course shown in the course of our proof that the series is convergent: this however has been proved already (Ex. lxxx. 4). The series is in fact absolutely convergent when $∣ z ∣ < 1$ and conditionally convergent when $∣ z ∣ = 1$ .

Changing $z$ into $- z$ we obtain

lo g (\frac{1}{1 - z}) = - lo g (1 - z) = z + \frac{1}{2} z^{2} + \frac{1}{3} z^{3} + \dots . (6)

Footnotes

Since $z$ is not real, $C$ cannot pass through $O$ when produced. The reader is recommended to draw a figure to illustrate the argument. ↩

235

Now

lo g (1 + z) = lo g {(1 + r cos θ) + i r sin θ} = \frac{1}{2} lo g (1 + 2 r cos θ + r^{2}) + i arc tan (\frac{r sin θ}{1 + r cos θ}) .

That value of the inverse tangent must be taken which lies between $- \frac{1}{2} π$ and $\frac{1}{2} π$ . For, since $1 + z$ is the vector represented by the line from $- 1$ to $z$ , the principal value of $am (1 + z)$ always lies between these limits when $z$ lies within the circle $∣ z ∣ = 1$ .¹

Since $z^{m} = r^{m} (cos m θ + i sin m θ)$ , we obtain, on equating the real and imaginary parts in equation (5) of § 234,

\frac{1}{2} lo g (1 + 2 r cos θ + r^{2}) arc tan (\frac{r sin θ}{1 + r cos θ}) = r cos θ - \frac{1}{2} r^{2} cos 2 θ + \frac{1}{3} r^{3} cos 3 θ - \dots, = r sin θ - \frac{1}{2} r^{2} sin 2 θ + \frac{1}{3} r^{3} sin 3 θ - \dots .

These equations hold when $0 ≦ r ≦ 1$ , and for all values of $θ$ , except that, when $r = 1$ , $θ$ must not be equal to an odd multiple of $π$ . It is easy to see that they also hold when $- 1 ≦ r ≦ 0$ , except that, when $r = - 1$ , $θ$ must not be equal to an even multiple of $π$ .

A particularly interesting case is that in which $r = 1$ . In this case we have

lo g (1 + z) = lo g (1 + Cis θ) = \frac{1}{2} lo g (2 + 2 cos θ) + i arc tan (\frac{sin θ}{1 + cos θ}) = \frac{1}{2} lo g (4 cos^{2} \frac{1}{2} θ) + \frac{1}{2} i θ,

if $- π < θ < π$ , and so

cos θ sin θ - \frac{1}{2} cos 2 θ - \frac{1}{2} sin 2 θ + \frac{1}{3} cos 3 θ + \frac{1}{3} sin 3 θ - \dots - \dots = \frac{1}{2} lo g (4 cos^{2} \frac{1}{2} θ), = \frac{1}{2} θ .

The sums of the series, for other values of $θ$ , are easily found from the consideration that they are periodic functions of $θ$ with the period $2 π$ . Thus the sum of the cosine series is $\frac{1}{2} lo g (4 cos^{2} \frac{1}{2} θ)$ for all values of $θ$ save odd multiples of $π$ (for which values the series is divergent), while the sum of the sine series is $\frac{1}{2} (θ - 2 k π)$ if $(2 k - 1) π < θ < (2 k + 1) π$ , and zero if $θ$ is an odd multiple of $π$ . The graph of the function represented by the sine series is shown in Fig. 58. The function is discontinuous for $θ = (2 k + 1) π$ .

If we write $i z$ and $- i z$ for $z$ in (5), and subtract, we obtain

\frac{1}{2 i} lo g (\frac{1 + i z}{1 - i z}) = z - \frac{1}{3} z^{3} + \frac{1}{5} z^{5} - \dots .

If $z$ is real and numerically less than unity, we are led, by the results of § 231, to the formula

arc tan z = z - \frac{1}{3} z^{3} + \frac{1}{5} z^{5} - \dots,

already proved in a different manner in § 214.

Examples XCVII.

1. Prove that, in any triangle in which $a > b$ ,

lo g c = lo g a - \frac{b}{a} cos C - \frac{b ^{2}}{2 a ^{2}} cos 2 C - \dots .

[Use the formula $lo g c = \frac{1}{2} lo g (a^{2} + b^{2} - 2 ab cos C)$ .]

2. Prove that if $- 1 < r < 1$ and $- \frac{1}{2} π < θ < \frac{1}{2} π$ then

r sin 2 θ - \frac{1}{2} r^{2} sin 4 θ + \frac{1}{3} r^{3} sin 6 θ - \dots = θ - arc tan {(\frac{1 - r}{1 + r}) tan θ},

the inverse tangent lying between $- \frac{1}{2} π$ and $\frac{1}{2} π$ . Determine the sum of the series for all other values of $θ$ .

3. Prove, by considering the expansions of $lo g (1 + i z)$ and $lo g (1 - i z)$ in powers of $z$ , that if $- 1 < r < 1$ then

r sin θ r cos θ + \frac{1}{2} r^{2} cos 2 θ + \frac{1}{2} r^{2} sin 2 θ - \frac{1}{3} r^{3} sin 3 θ - \frac{1}{3} r^{3} cos 3 θ - \frac{1}{4} r^{4} cos 4 θ + \dots - \frac{1}{4} r^{4} sin 4 θ + \dots = \frac{1}{2} lo g (1 + 2 r sin θ + r^{2}), = arc tan (\frac{r cos θ}{1 - r sin θ}), [1] r sin θ r cos θ - \frac{1}{3} r^{3} sin 3 θ + \dots - \frac{1}{3} r^{3} cos 3 θ + \dots = \frac{1}{4} lo g (\frac{1 + 2 r sin θ + r ^{2}}{1 - 2 r sin θ + r ^{2}}), = \frac{1}{2} arc tan (\frac{2 r cos θ}{1 - r ^{2}}),

the inverse tangents lying between $- \frac{1}{2} π$ and $\frac{1}{2} π$ .

4. Prove that

cos θ cos θ sin θ sin θ - \frac{1}{2} cos 2 θ cos^{2} θ - \frac{1}{2} sin 2 θ sin^{2} θ + \frac{1}{3} cos 3 θ cos^{3} θ - \dots + \frac{1}{3} sin 3 θ sin^{3} θ - \dots = \frac{1}{2} lo g (1 + 3 cos^{2} θ), = arc cot (1 + cot θ + cot^{2} θ),

the inverse cotangent lying between $- \frac{1}{2} π$ and $\frac{1}{2} π$ ; and find similar expressions for the sums of the series

cos θ sin θ - \frac{1}{2} cos 2 θ sin^{2} θ + \dots, sin θ cos θ - \frac{1}{2} sin 2 θ cos^{2} θ + \dots .

Footnotes

See the preceding footnote. ↩

236Some applications of the logarithmic series. The exponential limit

Let $z$ be any complex number, and $h$ a real number small enough to ensure that $∣ h z ∣ < 1$ . Then

lo g (1 + h z) = h z - \frac{1}{2} (h z)^{2} + \frac{1}{3} (h z)^{3} - \dots,

and so

\frac{lo g ( 1 + h z )}{h} = z + ϕ (h, z),

where

ϕ (h, z) = - \frac{1}{2} h z^{2} + \frac{1}{3} h^{2} z^{3} - \frac{1}{4} h^{3} z^{4} + \dots, ∣ ϕ (h, z) ∣ < ∣ h z^{2} ∣ (1 + ∣ h z ∣ + ∣ h^{2} z^{2} ∣ + \dots) = \frac{∣ h z ^{2} ∣}{1 - ∣ h z ∣},

so that $ϕ (h, z) \to 0$ as $h \to 0$ . It follows that

h \to 0 lim \frac{lo g ( 1 + h z )}{h} = z . (1)

If in particular we suppose $h = 1/ n$ , where $n$ is a positive integer, we obtain

n \to \infty lim n lo g (1 + \frac{z}{n}) = z,

and so

n \to \infty lim (1 + \frac{z}{n})^{n} = n \to \infty lim exp {n lo g (1 + \frac{z}{n})} = exp z . (2)

This is a generalisation of the result proved in § 208 for real values of $z$ .

From (1) we can deduce some other results which we shall require in the next section. If $t$ and $h$ are real, and $h$ is sufficiently small, we have

\frac{lo g ( 1 + t z + h z ) - lo g ( 1 + t z )}{h} = \frac{1}{h} lo g (1 + \frac{h z}{1 + t z})

which tends to the limit $z / (1 + t z)$ as $h \to 0$ . Hence

\frac{d}{d t} {lo g (1 + t z)} = \frac{z}{1 + t z} . (3)

We shall also require a formula for the differentiation of $(1 + t z)^{m}$ , where $m$ is any number real or complex, with respect to $t$ . We observe first that, if $ϕ (t) = ψ (t) + i χ (t)$ is a complex function of $t$ , whose real and imaginary parts $ϕ (t)$ and $χ (t)$ possess derivatives, then

\frac{d}{d t} (exp ϕ) = \frac{d}{d t} {(cos χ + i sin χ) exp ψ} = {(cos χ + i sin χ) ψ^{'} + (- sin χ + i cos χ) χ^{'}} exp ψ = (ψ^{'} + i χ^{'}) (cos χ + i sin χ) exp ψ = (ψ^{'} + i χ^{'}) exp (ψ + i χ) = ϕ^{'} exp ϕ,

so that the rule for differentiating $exp ϕ$ is the same as when $ϕ$ is real. This being so we have

\frac{d}{d t} (1 + t z)^{m} = \frac{d}{d t} exp {m lo g (1 + t z)} = \frac{m z}{1 + t z} exp {m lo g (1 + t z)} = m z (1 + t z)^{m - 1} . (4)

Here both $(1 + t z)^{m}$ and $(1 + t z)^{m - 1}$ have their principal values

237The general form of the Binomial Theorem

We have proved already (§ 215) that the sum of the series

1 + (1 m) z + (2 m) z^{2} + \dots

is $(1 + z)^{m} = exp {m lo g (1 + z)}$ , for all real values of $m$ and all real values of $z$ between $- 1$ and $1$ . If $a_{n}$ is the coefficient of $z^{n}$ then

\frac{a _{n + 1}}{a _{n}} = \frac{m - n}{n + 1} \to 1,

whether $m$ is real or complex. Hence (Ex. lxxx. 3) the series is always convergent if the modulus of $z$ is less than unity, and we shall now prove that its sum is still $exp {m lo g (1 + z)}$ , i.e. the principal value of $(1 + z)^{m}$ .

It follows from § 236 that if $t$ is real then

\frac{d}{d t} (1 + t z)^{m} = m z (1 + t z)^{m - 1},

$z$ and $m$ having any real or complex values and each side having its principal value. Hence, if $ϕ (t) = (1 + t z)^{m}$ , we have

ϕ^{(n)} (t) = m (m - 1) \dots (m - n + 1) z^{n} (1 + t z)^{m - n} .

This formula still holds if $t = 0$ , so that

\frac{ϕ ^{n} ( 0 )}{n !} = (n m) z^{n} .

Now, in virtue of the remark made at the end of § 164, we have

ϕ (1) = ϕ (0) + ϕ^{'} (0) + \frac{ϕ ^{''} ( 0 )}{2 !} + \dots + \frac{ϕ ^{(n - 1)} ( 0 )}{( n - 1 )!} + R_{n},

where

R_{n} = \frac{1}{( n - 1 )!} \int_{0}^{1} (1 - t)^{n - 1} ϕ^{(n)} (t) d t .

But if $z = r (cos θ + i sin θ)$ then

∣1 + t z ∣ = 1 + 2 t r cos θ + t^{2} r^{2} ≧ 1 - t r,

and therefore

∣ R_{n} ∣ < \frac{∣ m ( m - 1 ) \dots ( m - n + 1 ) ∣}{( n - 1 )!} r^{n} \int_{0}^{1} \frac{( 1 - t ) ^{n - 1}}{( 1 - t r ) ^{n - m}} d t < \frac{∣ m ( m - 1 ) \dots ( m - n + 1 ) ∣}{( n - 1 )!} \frac{( 1 - θ ) ^{n - 1} r ^{n}}{( 1 - θ r ) ^{n - m}},

where $0 < θ < 1$ ; so that (cf. § 163)

∣ R_{n} ∣ < K \frac{∣ m ( m - 1 ) \dots ( m - n + 1 ) ∣}{( n - 1 )!} r^{n} = ρ_{n},

say. But

\frac{ρ _{n + 1}}{ρ _{n}} = \frac{∣ m - n ∣}{n} r \to r,

and so (Ex. xxvii. 6) $ρ_{n} \to 0$ , and therefore $R_{n} \to 0$ , as $n \to \infty$ . Hence we arrive at the following theorem.

Theorem. The sum of the binomial series

1 + (1 m) z + (2 m) z^{2} + \dots

is $exp {m lo g (1 + z)}$ , where the logarithm has its principal value, for all values of $m$ , real or complex, and all values of $z$ such that $∣ z ∣ < 1$ .

A more complete discussion of the binomial series, taking account of the more difficult case in which $∣ z ∣ = 1$ , will be found on pp. 225 et seq. of Bromwich’s Infinite Series.

Examples XCVIII.

1. Suppose $m$ real. Then since

lo g (1 + z) = \frac{1}{2} lo g (1 + 2 r cos θ + r^{2}) + i arc tan (\frac{r sin θ}{1 + r cos θ}),

we obtain

0 \sum \infty (n m) z^{n} = exp {\frac{1}{2} m lo g (1 + 2 r cos θ + r^{2})} Cis {m arc tan (\frac{r sin θ}{1 + r cos θ})} = (1 + 2 r cos θ + r^{2})^{\frac{1}{2} m} Cis {m arc tan (\frac{r sin θ}{1 + r cos θ})},

all the inverse tangents lying between $- \frac{1}{2} π$ and $\frac{1}{2} π$ . In particular, if we suppose $θ = \frac{1}{2} π$ , $z = i r$ , and equate the real and imaginary parts, we obtain

1 - (2 m) r^{2} + (4 m) r^{4} - \dots (1 m) r - (3 m) r^{3} + (5 m) r^{5} - \dots = (1 + r^{2})^{\frac{1}{2} m} cos (m arc tan r), = (1 + r^{2})^{\frac{1}{2} m} sin (m arc tan r) .

2. Verify the formulae of Ex. 1 when $m = 1$ , $2$ , $3$ . [Of course when $m$ is a positive integer the series is finite.]

3. Prove that if $0 ≦ r < 1$ then

1 - \frac{1 \cdot 3}{2 \cdot 4} r^{2} + \frac{1 \cdot 3 \cdot 5 \cdot 7}{2 \cdot 4 \cdot 6 \cdot 8} r^{4} - \dots \frac{1}{2} r - \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} r^{3} + \frac{1 \cdot 3 \cdot 5 \cdot 7 \cdot 9}{2 \cdot 4 \cdot 6 \cdot 8 \cdot 10} r^{5} - \dots = \frac{1 + r ^{2} + 1}{2 ( 1 + r ^{2} )}, = \frac{1 + r ^{2} - 1}{2 ( 1 + r ^{2} )} .

[Take $m = - \frac{1}{2}$ in the last two formulae of Ex. 1.]

4. Prove that if $- \frac{1}{4} π < θ < \frac{1}{4} π$ then

cos m θ sin m θ = cos^{m} θ {1 - (2 m) tan^{2} θ + (4 m) tan^{4} θ - \dots}, = cos^{m} θ {(1 m) tan θ - (3 m) tan^{3} θ + \dots},

for all real values of $m$ . [These results follow at once from the equations

cos m θ + i sin m θ = (cos θ + i sin θ)^{m} = cos^{m} θ (1 + i tan θ)^{m} .]

5. We proved (Ex. lxxxi. 6), by direct multiplication of series, that $f (m, z) = \sum (n m) z^{n}$ , where $∣ z ∣ < 1$ , satisfies the functional equation

f (m, z) f (m^{'}, z) = f (m + m^{'}, z) .

Deduce, by an argument similar to that of § 216, and without assuming the general result of p. 423, that if $m$ is real and rational then

f (m, z) = exp {m lo g (1 + z)} .

6. If $z$ and $μ$ are real, and $- 1 < z < 1$ , then

\sum (n i μ) z^{n} = cos {μ lo g (1 + z)} + i sin {μ lo g (1 + z)} .

Misc.Miscellaneous Examples on Chapter X

1. Show that the real part of $i^{l o g (1 + i)}$ is

e^{(4 k + 1) π^{2} /8} cos {\frac{1}{4} (4 k + 1) π lo g 2},

where $k$ is any integer.

2. If $a cos θ + b sin θ + c = 0$ , where $a$ , $b$ , $c$ are real and $c^{2} > a^{2} + b^{2}$ , then

θ = mπ + α \pm i lo g \frac{∣ c ∣ + c ^{2} - a ^{2} - b ^{2}}{a ^{2} + b ^{2}},

where $m$ is any odd or any even integer, according as $c$ is positive or negative, and $α$ is an angle whose cosine and sine are $a / a^{2} + b^{2}$ and $b / a^{2} + b^{2}$ .

3. Prove that if $θ$ is real and $sin θ sin ϕ = 1$ then

ϕ = (k + \frac{1}{2}) π \pm i lo g cot \frac{1}{2} (k π + θ),

where $k$ is any even or any odd integer, according as $sin θ$ is positive or negative.

4. Show that if $x$ is real then

\frac{d}{d x} exp {(a + ib) x} = (a + ib) exp {(a + ib) x}, \int exp {(a + ib) x} d x = \frac{exp ( a + ib ) x}{a + ib} .

Deduce the results of Ex. lxxxvii. 3.

5. Show that if $a > 0$ then $\int_{0}^{\infty} exp {- (a + ib) x} d x = \frac{1}{a + ib}$ , and deduce the results of Ex. lxxxvii. 5.

6. Show that if $(x / a)^{2} + (y / b)^{2} = 1$ is the equation of an ellipse, and $f (x, y)$ denotes the terms of highest degree in the equation of any other algebraic curve, then the sum of the eccentric angles of the points of intersection of the ellipse and the curve differs by a multiple of $2 π$ from

- i {lo g f (a, ib) - lo g f (a, - ib)} .

[The eccentric angles are given by $f (a cos α, b sin α) + \dots = 0$ or by

f {\frac{1}{2} a (u + \frac{1}{u}), - \frac{1}{2} ib (u - \frac{1}{u})} + \dots = 0,

where $u = exp i α$ ; and $\sum α$ is equal to one of the values of $- i Log P$ , where $P$ is the product of the roots of this equation.]

7. Determine the number and approximate positions of the roots of the equation $tan z = a z$ , where $a$ is real.

[We know already (Ex. xvii. 4) that the equation has infinitely many real roots. Now let $z = x + i y$ , and equate real and imaginary parts. We obtain

sin 2 x / (cos 2 x + cosh 2 y) = a x, sinh 2 y / (cos 2 x + cosh 2 y) = a y,

so that, unless $x$ or $y$ is zero, we have

(sin 2 x) /2 x = (sinh 2 y) /2 y .

This is impossible, the left-hand side being numerically less, and the right-hand side numerically greater than unity. Thus $x = 0$ or $y = 0$ . If $y = 0$ we come back to the real roots of the equation. If $x = 0$ then $tanh y = a y$ . It is easy to see that this equation has no real root other than zero if $a ≦ 0$ or $a ≧ 1$ , and two such roots if $0 < a < 1$ . Thus there are two purely imaginary roots if $0 < a < 1$ ; otherwise all the roots are real.]

8. The equation $tan z = a z + b$ , where $a$ and $b$ are real and $b$ is not equal to zero, has no complex roots if $a ≦ 0$ . If $a > 0$ then the real parts of all the complex roots are numerically greater than $∣ b /2 a ∣$ .

9. The equation $tan z = a / z$ , where $a$ is real, has no complex roots, but has two purely imaginary roots if $a < 0$ .

10. The equation $tan z = a tanh cz$ , where $a$ and $c$ are real, has an infinity of real and of purely imaginary roots, but no complex roots.

11. Show that if $x$ is real then

e^{a x} cos b x = 0 \sum \infty \frac{x ^{n}}{n !} {a^{n} - (2 n) a^{n - 2} b^{2} + (4 n) a^{n - 4} b^{4} - \dots},

where there are $\frac{1}{2} (n + 1)$ or $\frac{1}{2} (n + 2)$ terms inside the large brackets. Find a similar series for $e^{a x} sin b x$ .

12. If $n ϕ (z, n) \to z$ as $n \to \infty$ , then ${1 + ϕ (z, n)}^{n} \to exp z$ .

13. If $ϕ (t)$ is a complex function of the real variable $t$ , then

\frac{d}{d t} lo g ϕ (t) = \frac{ϕ ^{'} ( t )}{ϕ ( t )} .

[Use the formulae

ϕ = ψ + i χ, lo g ϕ = \frac{1}{2} lo g (ψ^{2} + χ^{2}) + i arc tan (χ / ψ) .]

14. Transformations. In Ch. III (Exs. xxi. 21 et seq., and Misc. Exs. 22 et seq.) we considered some simple examples of the geometrical relations between figures in the planes of two variables $z$ , $Z$ connected by a relation $z = f (Z)$ . We shall now consider some cases in which the relation involves logarithmic, exponential, or circular functions.

Suppose firstly that

z = exp (π Z / a), Z = (a / π) Log z

where $a$ is positive. To one value of $Z$ corresponds one of $z$ , but to one of $z$ infinitely many of $Z$ . If $x$ , $y$ , $r$ , $θ$ are the coordinates of $z$ and $X$ , $Y$ , $R$ , $Θ$ those of $Z$ , we have the relations

x X = e^{π X / a} cos (π Y / a), = (a / π) lo g r, y Y = e^{π X / a} sin (π Y / a), = (a θ / π) + 2 k a,

where $k$ is any integer. If we suppose that $- π < θ ≦ π$ , and that $Log z$ has its principal value $lo g z$ , then $k = 0$ , and $Z$ is confined to a strip of its plane parallel to the axis $O X$ and extending to a distance $a$ from it on each side, one point of this strip corresponding to one of the whole $z$ -plane, and conversely. By taking a value of $Log z$ other than the principal value we obtain a similar relation between the $z$ -plane and another strip of breadth $2 a$ in the $Z$ -plane.

To the lines in the $Z$ -plane for which $X$ and $Y$ are constant correspond the circles and radii vectores in the $z$ -plane for which $r$ and $θ$ are constant. To one of the latter lines corresponds the whole of a parallel to $O X$ , but to a circle for which $r$ is constant corresponds only a part, of length $2 a$ , of a parallel to $O Y$ . To make $Z$ describe the whole of the latter line we must make $z$ move continually round and round the circle.

15. Show that to a straight line in the $Z$ -plane corresponds an equiangular spiral in the $z$ -plane.

16. Discuss similarly the transformation $z = c cosh (π Z / a)$ , showing in particular that the whole $z$ -plane corresponds to any one of an infinite number of strips in the $Z$ -plane, each parallel to the axis $O X$ and of breadth $2 a$ . Show also that to the line $X = X_{0}$ corresponds the ellipse

{\frac{x}{c cosh ( π X _{0} / a )}}^{2} + {\frac{y}{c sinh ( π X _{0} / a )}}^{2} = 1,

and that for different values of $X_{0}$ these ellipses form a confocal system; and that the lines $Y = Y_{0}$ correspond to the associated system of confocal hyperbolas. Trace the variation of $z$ as $Z$ describes the whole of a line $X = X_{0}$ or $Y = Y_{0}$ . How does $Z$ vary as $z$ describes the degenerate ellipse and hyperbola formed by the segment between the foci of the confocal system and the remaining segments of the axis of $x$ ?

17. Verify that the results of Ex. 16 are in agreement with those of Ex. 14 and those of Ch. III, Misc. Ex. 25. [The transformation $z = c cosh (π Z / a)$ may be regarded as compounded from the transformations

z = c z_{1}, z_{1} = \frac{1}{2} {z_{2} + (1/ z_{2})}, z_{2} = exp (π Z / a) .]

18. Discuss similarly the transformation $z = c tanh (π Z / a)$ , showing that to the lines $X = X_{0}$ correspond the coaxal circles

{x - c coth (2 π X_{0} / a)}^{2} + y^{2} = c^{2} cosech^{2} (2 π X_{0} / a),

and to the lines $Y = Y_{0}$ the orthogonal system of coaxal circles.

19. The Stereographic and Mercator’s Projections. The points of a unit sphere whose centre is the origin are projected from the south pole (whose coordinates are $0$ , $0$ , $- 1$ ) on to the tangent plane at the north pole. The coordinates of a point on the sphere are $ξ$ , $η$ , $ζ$ , and Cartesian axes $O X$ , $O Y$ are taken on the tangent plane, parallel to the axes of $ξ$ and $η$ . Show that the coordinates of the projection of the point are

x = 2 ξ / (1 + ζ), y = 2 η / (1 + ζ),

and that $x + i y = 2 tan \frac{1}{2} θ Cis ϕ$ , where $ϕ$ is the longitude (measured from the plane $η = 0$ ) and $θ$ the north polar distance of the point on the sphere.

This projection gives a map of the sphere on the tangent plane, generally known as the Stereographic Projection. If now we introduce a new complex variable

Z = X + iY = - i lo g \frac{1}{2} z = - i lo g \frac{1}{2} (x + i y)

so that $X = ϕ$ , $Y = lo g cot \frac{1}{2} θ$ , we obtain another map in the plane of $Z$ , usually called Mercator’s Projection. In this map parallels of latitude and longitude are represented by straight lines parallel to the axes of $X$ and $Y$ respectively.

20. Discuss the transformation given by the equation

z = Log (\frac{Z - a}{Z - b}),

showing that the straight lines for which $x$ and $y$ are constant correspond to two orthogonal systems of coaxal circles in the $Z$ -plane.

21. Discuss the transformation

z = Log {\frac{Z - a + Z - b}{b - a}},

showing that the straight lines for which $x$ and $y$ are constant correspond to sets of confocal ellipses and hyperbolas whose foci are the points $Z = a$ and $Z = b$ .

[We have

Z - a + Z - b Z - a - Z - b = b - a exp (= b - a exp (- x + i y), x - i y);

and it will be found that

∣ Z - a ∣ + ∣ Z - b ∣ = ∣ b - a ∣ cosh 2 x, ∣ Z - a ∣ - ∣ Z - b ∣ = ∣ b - a ∣ cos 2 y .]

22. The transformation $z = Z^{i}$ . If $z = Z^{i}$ , where the imaginary power has its principal value, we have

exp (lo g r + i θ) = z = exp (i lo g Z) = exp (i lo g R - Θ),

so that $lo g r = - Θ$ , $θ = lo g R + 2 k π$ , where $k$ is an integer. As all values of $k$ give the same point $z$ , we shall suppose that $k = 0$ , so that

lo g r = - Θ, θ = lo g R . (1)

The whole plane of $Z$ is covered when $R$ varies through all positive values and $Θ$ from $- π$ to $π$ : then $r$ has the range $exp (- π)$ to $exp π$ and $θ$ ranges through all real values. Thus the $Z$ -plane corresponds to the ring bounded by the circles $r = exp (- π)$ , $r = exp π$ ; but this ring is covered infinitely often. If however $θ$ is allowed to vary only between $- π$ and $π$ , so that the ring is covered only once, then $R$ can vary only from $exp (- π)$ to $exp π$ , so that the variation of $Z$ is restricted to a ring similar in all respects to that within which $z$ varies. Each ring, moreover, must be regarded as having a barrier along the negative real axis which $z$ (or $Z$ ) must not cross, as its amplitude must not transgress the limits $- π$ and $π$ .

We thus obtain a correspondence between two rings, given by the pair of equations

z = Z^{i}, Z = z^{- i},

where each power has its principal value. To circles whose centre is the origin in one plane correspond straight lines through the origin in the other.

23. Trace the variation of $z$ when $Z$ , starting at the point $exp π$ , moves round the larger circle in the positive direction to the point $- exp π$ , along the barrier, round the smaller circle in the negative direction, back along the barrier, and round the remainder of the larger circle to its original position.

24. Suppose each plane to be divided up into an infinite series of rings by circles of radii

\dots, e^{- (2 n + 1) π}, \dots, e^{- π}, e^{π}, e^{3 π}, \dots, e^{(2 n + 1) π}, \dots .

Show how to make any ring in one plane correspond to any ring in the other, by taking suitable values of the powers in the equations $z = Z^{i}$ , $Z = z^{- i}$ .

25. If $z = Z^{i}$ , any value of the power being taken, and $Z$ moves along an equiangular spiral whose pole is the origin in its plane, then $z$ moves along an equiangular spiral whose pole is the origin in its plane.

26. How does $Z = z^{ai}$ , where $a$ is real, behave as $z$ approaches the origin along the real axis? [ $Z$ moves round and round a circle whose centre is the origin (the unit circle if $z^{ai}$ has its principal value), and the real and imaginary parts of $Z$ both oscillate finitely.]

27. Discuss the same question for $Z = z^{a + bi}$ , where $a$ and $b$ are any real numbers.

28. Show that the region of convergence of a series of the type $- \infty \sum \infty a_{n} z^{nai}$ , where $a$ is real, is an angle, i.e. a region bounded by inequalities of the type $θ_{0} < am z < θ_{1}$ [The angle may reduce to a line, or cover the whole plane.]

29. Level Curves. If $f (z)$ is a function of the complex variable $z$ , we call the curves for which $∣ f (z) ∣$ is constant the level curves of $f (z)$ . Sketch the forms of the level curves of

z - a (z - a) / (z - b) (concentric circles), (coaxal circles), (z - a) (z - b) exp z (Cartesian ovals), (straight lines) .

30. Sketch the forms of the level curves of $(z - a) (z - b) (z - c)$ , $(1 + z 3 + z^{2}) / z$ . [Some of the level curves of the latter function are drawn in Fig. 59, the curves marked i–vii corresponding to the values

.10, 2 - 3 = .27, .40, 1.00, 2.00, 2 + 3 = 3.73, 4.53

of $∣ f (z) ∣$ . The reader will probably find but little difficulty in arriving at a general idea of the forms of the level curves of any given rational function; but to enter into details would carry us into the general theory of functions of a complex variable.]

31. Sketch the forms of the level curves of (i) $z exp z$ , (ii) $sin z$ . [See Fig. 60, which represents the level curves of $sin z$ . The curves marked i–viii correspond to $k = .35$ , $.50$ , $.71$ , $1.00$ , $1.41$ , $2.00$ , $2.83$ , $4.00$ .]

32. Sketch the forms of the level curves of $exp z - c$ , where $c$ is a real constant. [Fig. 61 shows the level curves of $∣ exp z - 1∣$ , the curves i–vii corresponding to the values of $k$ given by $lo g k = - 1.00$ , $- .20$ , $- .05$ , $0.00$ , $.05$ , $.20$ , $1.00$ .]

33. The level curves of $sin z - c$ , where $c$ is a positive constant, are sketched in Figs. 62, 63. [The nature of the curves differs according as to whether $c < 1$ or $c > 1$ . In Fig. 62 we have taken $c = .5$ , and the curves i–viii correspond to $k = .29$ , $.37$ , $.50$ , $.87$ , $1.50$ , $2.60$ , $4.50$ , $7.79$ . In Fig. 63 we have taken $c = 2$ , and the curves i–vii correspond to $k = .58$ , $1.00$ , $1.73$ , $3.00$ , $5.20$ , $9.00$ , $15.59$ . If $c = 1$ then the curves are the same as those of Fig. 60, except that the origin and scale are different.]

34. Prove that if $0 < θ < π$ then

cos θ sin θ + \frac{1}{3} cos 3 θ + \frac{1}{3} sin 3 θ + \frac{1}{5} cos 5 θ + \frac{1}{5} sin 5 θ + \dots = \frac{1}{4} lo g cot^{2} \frac{1}{2} θ, + \dots = \frac{1}{4} π,

and determine the sums of the series for all other values of $θ$ for which they are convergent. [Use the equation

z + \frac{1}{3} z^{3} + \frac{1}{5} z^{5} + \dots = \frac{1}{2} lo g (\frac{1 + z}{1 - z})

where $z = cos θ + i sin θ$ . When $θ$ is increased by $π$ the sum of each series simply changes its sign. It follows that the first formula holds for all values of $θ$ save multiples of $π$ (for which the series diverges), while the sum of the second series is $\frac{1}{4} π$ if $2 k π < θ < (2 k + 1) π$ , $- \frac{1}{4} π$ if $(2 k + 1) π < θ < (2 k + 2) π$ , and $0$ if $θ$ is a multiple of $π$ .]

35. Prove that if $0 < θ < \frac{1}{2} π$ then

cos θ sin θ - \frac{1}{3} cos 3 θ - \frac{1}{3} sin 3 θ + \frac{1}{5} cos 5 θ + \frac{1}{5} sin 5 θ - \dots = \frac{1}{4} π, - \dots = \frac{1}{4} lo g (sec θ + tan θ)^{2};

and determine the sums of the series for all other values of $θ$ for which they are convergent.

36. Prove that

cos θ cos α + \frac{1}{2} cos 2 θ cos 2 α + \frac{1}{3} cos 3 θ cos 3 α + \dots = - \frac{1}{4} lo g {4 (cos θ - cos α)^{2}},

unless $θ - α$ or $θ + α$ is a multiple of $2 π$ .

37. Prove that if neither $a$ nor $b$ is real then

\int_{0}^{\infty} \frac{d x}{( x - a ) ( x - b )} = - \frac{lo g ( - a ) - lo g ( - b )}{a - b},

each logarithm having its principal value. Verify the result when $a = c i$ , $b = - c i$ , where $c$ is positive. Discuss also the cases in which $a$ or $b$ or both are real and negative.

38. Prove that if $α$ and $β$ are real, and $β > 0$ , then

\int_{0}^{\infty} \frac{d}{x ^{2} - ( α + i β ) ^{2}} = \frac{π i}{2 ( α + i β )} .

What is the value of the integral when $β < 0$ ?

39. Prove that, if the roots of $A x^{2} + 2 B x + C = 0$ have their imaginary parts of opposite signs, then

\int_{- \infty}^{\infty} \frac{d x}{A x ^{2} + 2 B x + C} = \frac{π i}{B ^{2} - A C},

the sign of $B^{2} - A C$ being so chosen that the real part of ${B^{2} - A C} / A i$ is positive.

Chapter XThe General Theory of the Logarithmic, Exponential, and Circular Functions

217Functions of a complex variable

Footnotes

218

219Real and complex curvilinear integrals

220The definition of Log⁡ζ\operatorname{Log} \zetaLogζ

221The values of Log⁡ζ\operatorname{Log} \zetaLogζ

222The exponential function

223The value of exp⁡ζ\exp \zetaexpζ

224The functional equation satisfied by exp⁡ζ\exp \zetaexpζ

225The general power aζa^{\zeta}aζ

226The general value of aζa^{\zeta}aζ

227The exponential values of the sine and cosine

228Definition of sin⁡ζ\sin\zetasinζ and cos⁡ζ\cos\zetacosζ for all values of ζ\zetaζ

229The generalised hyperbolic functions

230Formulae for cos⁡(ξ+iη)\cos(\xi + i\eta)cos(ξ+iη), sin⁡(ξ+iη)\sin(\xi + i\eta)sin(ξ+iη), etc

231The connection between the logarithmic and the inverse trigonometrical functions

232The power series for exp⁡z\exp zexpz

Footnotes

233The power series for cos⁡z\cos zcosz and sin⁡z\sin zsinz

234The logarithmic series

Footnotes

235

Footnotes

236Some applications of the logarithmic series. The exponential limit

237The general form of the Binomial Theorem

Misc.Miscellaneous Examples on Chapter X

220The definition of $\operatorname{Log} \zeta$

221The values of $\operatorname{Log} \zeta$

223The value of $\exp \zeta$

224The functional equation satisfied by $\exp \zeta$

225The general power $a^{\zeta}$

226The general value of $a^{\zeta}$

228Definition of $\sin\zeta$ and $\cos\zeta$ for all values of $\zeta$

230Formulae for $\cos(\xi + i\eta)$ , $\sin(\xi + i\eta)$ , etc

232The power series for $\exp z$

233The power series for $\cos z$ and $\sin z$