Chapter X
The General Theory of the Logarithmic, Exponential, and Circular Functions

217. Functions of a complex variable. In Ch. III we defined the complex variable

z = x + iy,108
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) + ig(x,y),
of the two real variables x and y. For example
x iy,xy,z = x2 + y2, am z = arctan(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.

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 (§§ 2627) 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

log x =1xdt 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.

219. Real and complex curvilinear integrals. Let AB 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 t0 to t1, the point (x,y) moves along the curve, in the same direction, from A to B.

Then we define the curvilinear integral

C{g(x,y)dx + h(x,y)dy}, (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

t0t1 {g(φ,ψ)φ + h(φ,ψ)ψ}dt.
We call C the path of integration.

Let us suppose now that

z = x + iy = φ(t) + iψ(t),
so that z describes the curve C in Argand’s diagram as t varies. Further let us suppose that
f(z) = u + iv
is a polynomial in z or rational function of z.

Then we define

Cf(z)dz (2)

as meaning

C(u + iv)(dx + idy),
which is itself defined as meaning
C(udx vdy) + iC(vdx + udy).

220. The definition of Log ζ. Now let ζ = ξ + iη be any complex number. We define  Log ζ, the general logarithm of ζ, by the equation

Log ζ =Cdz 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


pict

Fig. 54.

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

Log ξ =1ξdt t ,
so that Log ξ is equal to  log 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  log 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  log 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 log ζ log ξ. Log ξ is (possibly at any rate) a many valued function, and log 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.

221. The values of Log ζ. 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 =Cdz z =Cdx + idy x + iy =t0t1 1 x + iy dx dt + idy dtdt,

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

dx dt + idy dt = cos θdr dt r sin θdθ dt + i sin θdr dt + r cos θdθ dt = (cos θ + i sin θ) dr dt + irdθ dt ;

so that

Log ζ =t0t1 1 rdr dtdt + it0t1 dθ dtdt = [log r] + i[θ],
where [log r] denotes the difference between the values of  log r at the points corresponding to t = t1 and t = t0, and [θ] has a similar meaning.

It is clear that

[log r] = log ρ log 1 = log 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


pict

Fig. 55.

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

[θ] = (2kπ + φ) 2kπ = φ,
and, when the path of integration is a straight line, Log ζ = log ρ + iφ.

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  log ζ. Hence it will be convenient in general to denote the principal value of  Log ζ by  log ζ. Thus

log ζ = log ρ + iφ,
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


pict

Fig. 56.

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 2kπ, first decreases to the value

2kπ XOP
and then increases again, being equal to 2kπ at Q, and finally to 2kπ + 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 = log ζ = log ρ + iφ.

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 2kπ, it will have increased


pict

Fig. 57.

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

Log ζ = log ρ + i(4π + 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] = 2kπ + φ and

Log ζ = log ρ + i(2kπ + 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 2kπ + φ are the different values of  am ζ, we conclude that every value of  log 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

log ζ + i am ζ = log ρ + i(2kπ + 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 = log ζ = log ρ + iφ.

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 amz, and θ is equal to one or other of them: also r = z. The values of Logz are still the values of logz+ iamz, viz. 

log(z) + (2k + 1)πi,
where k is an integer. The values log(z) + πi and log(z) πi 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 Logz, as convenience dictates. We shall choose the value log(z) + π i corresponding to the first path.

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

3. The functional equation satisfied by Logz. The function Logz satisfies the equation

Logz1z2 = Logz1 + Logz2, (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

z1 = r1(cosθ1 + isinθ1),z2 = r2(cosθ2 + isinθ2),
and applying the formula of p. 1364. It is however not true that

logz1z2 = logz1 + logz2 (2)

in all circumstances. If, e.g.,

z1 = z2 = 1 2(1 + i3) = cos 2 3π + isin 2 3π,
then logz1 = logz2 = 2 3πi, and logz1 + logz2 = 4 3πi, which is one of the values of Logz1z2, but not the principal value. In fact logz1z2 = 2 3πi.

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 Logzm = mLogz, 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) = Logz is completely true. It is also true that log(1/z) = logz, except when z is real and negative.

6. The equation

log z a z b = log(z a) log(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 OX and extending to infinity in the negative direction.

7. The equation

log a z b z = log 1 a z log 1 b z
is true if z lies outside the triangle formed by the three points Oab.

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

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

πf(x) = pπ + (q p)I(logx),
is equal to p when x is positive and to q when x is negative.

10. The function f(x) defined by

πf(x) = pπ + (q p)I{log(x 1)}+ (r q)I(logx)
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) logz (ii) any value of Logz (a) real or (b) purely imaginary?

12. If z = x + iy then LogLogz = logR + i(Θ + 2kπ), where

R2 = (logr)2 + (θ + 2kπ)2
and Θ is the least positive angle determined by the equations
cosΘ : sinΘ : 1 :: logr : θ + 2kπ : (logr)2 + (θ + 2kπ)2.
Plot roughly the doubly infinite set of values of LogLog(1 + i3), indicating which of them are values of logLog(1 + i3) and which of Loglog(1 + i3).

222. The exponential function. In Ch. IX we defined a function ey of the real variable y as the inverse of the function y = log 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

z1 = r1(cos θ1 + i sin θ1),z2 = r2(cos θ2 + i sin θ2)
are both values of  exp ζ. Then
ζ = Log z1 = Log z2,
and so
log r1 + i(θ1 + 2mπ) = log r2 + i(θ2 + 2nπ),
where m and n are integers. This involves
log r1 = log r2,θ1 + 2mπ = θ2 + 2nπ.
Thus r1 = r2, and θ1 and θ2 differ by a multiple of 2π. Hence z1 = z2.

COROLLARY. If ζ is real then exp ζ = eζ, the real exponential function of ζ defined in Ch. IX.

For if z = eζ then log z = ζ, i.e. one of the values of  Log z is ζ. Hence z = exp ζ.

223. The value of exp ζ. Let ζ = ξ + iη and

z = exp ζ = r(cos θ + i sin θ).
Then
ξ + iη = Log z = log r + i(θ + 2mπ),
where m is an integer. Hence ξ = log r, η = θ + 2mπ, or
r = eξ,θ = η 2mπ;
and accordingly
exp(ξ + iη) = eξ(cos η + i sin η).

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

224. The functional equation satisfied by exp ζ. Let ζ1 = ξ1 + iη1, ζ2 = ξ2 + iη2. Then

exp ζ1 × exp ζ2 = eξ1 (cos η1 + i sin η1) × eξ2 (cos η2 + i sin η2) = eξ1+ξ2 {cos(η1 + η2) + i sin(η1 + η2)} = exp(ζ1 + ζ2).

The exponential function therefore satisfies the functional relation f(ζ1 + ζ2) = f(ζ1)f(ζ2), an equation which we have proved already (§ 205) to be true for real values of ζ1 and ζ2.

225. The general power aζ. It might seem natural, as exp ζ = eζ 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 eζ: we shall find then that eζ represents a function with infinitely many values of which exp ζ is only one.

We have already defined the meaning of the symbol aζ 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 aζ 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

aζ = eζ log 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

31/2,(1)1/3,(3 + 1 2i)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)2,2i,(3 + 2i)2+3i.
We shall now give a general definition of aζ 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 aζ is defined by the equation

aζ = 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 log a, is real: and exp(ζ log a) = eζ log 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 = eτ(cos ψ + i sin ψ), then

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

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 ap/q found in § 48. Hence our new definition is also consistent with that of Ch. III.

226. The general value of aζ. Let

ζ = ξ + iη,a = σ(cos ψ + i sin ψ)
where π < ψ π, so that, in the notation of § 225, σ = eτ or τ = log σ.

Then

ζ Log a = (ξ + iη){log σ + i(ψ + 2mπ)} = L + iM,
where
L = ξ log σ η(ψ + 2mπ),M = η log σ + ξ(ψ + 2mπ);
and
aζ = exp(ζ Log a) = eL(cos M + i sin M).
Thus the general value of aζ is
eξ log ση(ψ+2mπ)[cos{η log σ + ξ(ψ + 2mπ)} + i sin{η log σ + ξ(ψ + 2mπ)}].

In general aζ is an infinitely many-valued function. For

aζ = eξ log ση(ψ+2mπ)
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 aζ are the same. But any two values differ unless their amplitudes are the same or differ by a multiple of 2π. This requires that ξ(ψ + 2mπ) and ξ(ψ + 2nπ), where m and n are different integers, shall differ, if at all, by a multiple of 2π. But if
ξ(ψ + 2mπ) ξ(ψ + 2nπ) = 2kπ,
then ξ = k/(m n) is rational. We conclude that aζ 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, aζ has but a finite number of values.

The principal value of aζ = exp(ζLoga) is obtained by giving Loga its principal value, i.e. by supposing m = 0 in the general formula. Thus the principal value of aζ is

eξ log σηψ{cos(ηlogσ + ξψ) + isin(ηlogσ + ξψ)}.

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 aζ is eζ log 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 (cosψ + isinψ)ζ is cosζψ + isinζψ. This is a further generalisation of De Moivre’s Theorem (§§4549).

Examples XCIV. 1. Find all the values of ii. [By definition

ii = exp(iLogi).
But
i = cos 1 2π + isin 1 2π,Logi = (2k + 1 2)πi,
where k is any integer. Hence
ii = exp{(2k + 1 2)π}= e(2k+1 2)π.
All the values of ii are therefore real and positive.]

2. Find all the values of (1 + i)i, i1+i, (1 + i)1+i.

3. The values of aζ, 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 aζ = r(cosθ + isinθ) we have

r = eξ log ση(ψ+2mπ),θ = ηlogσ + ξ(ψ + 2mπ);
and all the points lie on the spiral r = σ(ξ2+η2)/ξ eηθ/ξ.]

4. The function eζ. If we write e for a in the general formula, so that logσ = 1, ψ = 0, we obtain

eζ = eξ2mπη{cos(η + 2mπξ) + isin(η + 2mπξ)}.
The principal value of eζ is eξ(cosη + isinη), which is equal to expζ (§223). In particular, if ζ is real, so that η = 0, we obtain
eζ(cos2mπζ + isin2mπζ)
as the general and eζ as the principal value, eζ denoting here the positive value of the exponential defined in Ch. IX.

5. Show that Logeζ = (1 + 2mπi)ζ + 2nπi, where m and n are any integers, and that in general Logaζ has a double infinity of values.

6. The equation 1/aζ = aζ is completely true (Ex. XCIII. 3): it is also true of the principal values.

7. The equation aζ ×bζ = (ab)ζ is completely true but not always true of the principal values.

8. The equation aζ ×aζ= aζ+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 aζ ×aζ, viz.

exp{ζ(loga + 2mπi) + ζ(loga + 2nπi)},
is not as a rule a value of aζ+ζ unless m = n.]

9. What are the corresponding results as regards the equations

Logaζ = ζLoga,(aζ)ζ= (aζ)ζ = aζζ?

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

11. The necessary and sufficient conditions that all the values of aζ should be real are that 2ξ and {ηloga+ ξama}/π, where ama 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 xi + xi, where x > 0, is

e(mn)π2{cosh2(m + n)π + cos(2logx)}.

13. Explain the fallacy in the following argument: since e2mπi = e2nπi = 1, where m and n are any integers, therefore, raising each side to the power i we obtain e2mπ = e2nπ.

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

xx = exp(xLogx) = exp(xlogx)Cis2mπx,
the first factor being real. The principal value, for which m = 0, is always real.

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

If x = ξ < 0 then

xx = exp{ξLog(ξ)}= exp(ξlogξ)Cis{(2m + 1)πξ}.
The only case in which any value is real is that in which ξ = p/(2q + 1), when m = q gives the real value
exp(ξlogξ)Cis(pπ) = (1)pξξ.
The cases of reality are illustrated by the examples
(1 3)1/3 = 1 33,(1 2)1 2 = ±12,(23)2 3 = 9 43,(13)1 3 = 33.]

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

Thus if a = e then ζ = Logez, according to the first definition, if the principal value of eζ is equal to z, or if expζ = z; and so Logez is identical with Logz. But, according to the second definition, ζ = Logez if

eζ = exp(ζLoge) = z,ζLoge = Logz,
or ζ = (Logz)/(Loge), any values of the logarithms being taken. Thus
ζ = Logez = logz+ (amz + 2mπ)i 1 + 2nπi ,
so that ζ is a doubly infinitely many-valued function of z. And generally, according to this definition, Logaz = (Logz)/(Loga).

16. Loge1 = 2mπi/(1 + 2nπi), Loge(1) = (2m + 1)πi/(1 + 2nπi), where m and n are any integers.

227. The exponential values of the sine and cosine. From the formula

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

cos η = 1 2{exp(iη) + exp(iη)}, sin η = 1 2i{exp(iη) exp(iη)}.

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

228. Definition of sin ζ and  cos ζ for all values of ζ. We saw in the last section that, when ζ is real,

cos ζ = 1 2{exp(iζ) + exp(iζ)}, (1a) sin ζ = 1 2i{exp(iζ) exp(iζ)}. (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 = sin ζ cos ζ, cot ζ = cos ζ sin ζ, sec ζ = 1 cos ζ, cosec ζ = 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(iζ) = t, we have

cos ζ = 1 2{t + (1/t)}, sin ζ = 1 2i{t (1/t)}, cos 2ζ + sin 2ζ = 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(iζ) = t, exp(iζ) = t, we have

cos(ζ + ζ) = 1 2 tt + 1 tt = 1 4 t + 1 t t + 1 t + t 1 t t1 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 + 1 2π) = sin ζ, sin(ζ + 1 2π) = 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.

229. The generalised hyperbolic functions. In Ex. LXXXVII. 19, we defined coshζ and sinhζ, for real values of ζ, by the equations

coshζ = 1 2{expζ + exp(ζ)},sinhζ = 1 2{expζ 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 coshζ and sinhζ for all values of ζ real or complex. The reader will easily verify the following relations:

cosiζ = coshζ,siniζ = isinhζ,coshiζ = cosζ,sinhiζ = isinζ.

We have seen that any elementary trigonometrical formula, such as the formula cos2ζ = cos2ζ sin2ζ, remains true when ζ is allowed to assume complex values. It remains true therefore if we write cosiζ for cosζ, siniζ for sinζ and cos2iζ for cos2ζ; or, in other words, if we write coshζ for cosζ, isinhζ for sinζ, and cosh2ζ for cos2ζ. Hence

cosh2ζ = cosh2ζ + sinh2ζ.
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.

230. Formulae for cos(ξ + iη), sin(ξ + iη), etc. It follows from the addition formulae that

cos(ξ + iη) = cosξcosiη sinξsiniη = cosξcoshη isinξsinhη, sin(ξ + iη) = sinξcosiη + cosξsiniη = sinξcoshη + icosξsinhη.

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 cosζ and sinζ are (i) real (ii) purely imaginary. [For example cosζ is real when η = 0 or when ξ is any multiple of π.]

2.

cos(ξ + iη) = cos2ξ + sinh2η = 1 2(cosh2η + cos2ξ), sin(ξ + iη) = sin2ξ + sinh2η = 1 2(cosh2η cos2ξ).

[Use (e.g.) the equation cos(ξ + iη)= cos(ξ + iη)cos(ξ iη).]

3. tan(ξ + iη) = sin2ξ + isinh2η cosh2η + cos2ξ , cot(ξ + iη) = sin2ξ isinh2η cosh2η cos2ξ .

[For example

tan(ξ + iη) = sin(ξ + iη)cos(ξ iη) cos(ξ + iη)cos(ξ iη) = sin2ξ + sin2iη cos2ξ + cos2iη,
which leads at once to the result given.]

4.

sec(ξ + iη) = cosξcoshη + isinξsinhη 1 2(cosh2η + cos2ξ) , cosec(ξ + iη) = sinξcoshη icosξsinhη 1 2(cosh2η cos2ξ) .

5. If cos(ξ + iη)= 1 then sin2ξ = sinh2η, and if sin(ξ + iη)= 1 then cos2ξ = sinh2η.

6. If cos(ξ + iη)= 1, then

sin{amcos(ξ + iη)}= ±sin2ξ = ±sinh2η.

7. Prove that Logcos(ξ + iη) = A + iB, where

A = 1 2 log{1 2(cosh2η + cos2ξ)}
and B is any angle such that
cosB cosξcoshη = sinB sinξsinhη = 1 1 2(cosh2η + cos2ξ).
Find a similar formula for Logsin(ξ + iη).

8. Solution of the equation cosζ = a, where a is real. Putting ζ = ξ + iη, and equating real and imaginary parts, we obtain

cosξcoshη = a,sinξsinhη = 0.
Hence either η = 0 or ξ is a multiple of π. If (i) η = 0 then cosξ = a, which is impossible unless 1 a 1. This hypothesis leads to the solution
ζ = 2kπ ±arccosa, where arccosa lies between 0 and 1 2π. If (ii) ξ = mπ then coshη = (1)ma, so that either a 1 and m is even, or a 1 and m is odd. If a = ±1 then η = 0, and we are led back to our first case. If a> 1 then coshη = a, and we are led to the solutions

ζ = 2kπ ±ilog{a + a2 1}(a > 1), ζ = (2k + 1)π ±ilog{a + a2 1}(a < 1).

For example, the general solution of cosζ = 5 3 is ζ = (2k + 1)π ±ilog3.

9. Solve sinζ = α, where α is real.

10. Solution of cosζ = α + iβ, where β0. We may suppose β > 0, since the results when β < 0 may be deduced by merely changing the sign of i. In this case

cosξcoshη = α,sinξsinhη = β, (1)

and

(α/coshη)2 + (β/sinhη)2 = 1.

If we put cosh2η = x we find that

x2 (1 + α2 + β2)x + α2 = 0
or x = (A1 ±A2)2, where
A1 = 1 2(α + 1)2 + β2,A2 = 1 2(α 1)2 + β2.
Suppose α > 0. Then A1 > A2 > 0 and coshη = A1 ±A2. Also
cosξ = α/(coshη) = A1 A2,
and since coshη > cosξ we must take
coshη = A1 + A2,cosξ = A1 A2.
The general solutions of these equations are

ξ = 2kπ ±arccosM,η = ±log{L + L2 1}, (2)

where L = A1 + A2, M = A1 A2, and arccosM lies between 0 and 1 2π.

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

cosξcoshη = α,sinξsinhη = 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 sinξ is the same as the ambiguous sign in the first of the equations (2), and the sign of sinhη 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 = 2kπ ±[arccosM ilog{L + L2 1}].

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

12. If β = 0 then L = 1 2α + 1+ 1 2α 1 and M = 1 2α + 11 2α 1. Verify that the results thus obtained agree with those of Ex. 8.

13. Showthatifα and β are positive then the general solution of sinζ = α + iβ is

ζ = kπ + (1)k[arcsinM + ilog{L + L2 1}], where arcsinM lies between 0 and 1 2π. Obtain the solution in the other possible cases.

14. Solve tanζ = α, where α is real. [All the roots are real.]

15. Show that the general solution of tanζ = α + iβ, where β0, is

ζ = kπ + 1 2θ + 1 4ilog α2+(1+β)2 α2+(1β)2 ,
where θ is the numerically least angle such that
cosθ : sinθ : 1 :: 1 α2 β2 : 2α : (1 α2 β2)2 + 4α2.

16. If z = ξexp(1 4πi), where ξ is real, and c is also real, then the modulus of cos2πz cos2πc is

[1 2{1 + cos4πc + cos(2πξ2) + cosh(2πξ2) 4cos2πccos(πξ2)cosh(πξ2)}] .

17. Prove that

expexp(ξ + iη)= exp(expξcosη), R{coscos(ξ + iη)} = cos(cosξcoshη)cosh(sinξsinhη), I{sinsin(ξ + iη)} = cos(sinξcoshη)sinh(cosξsinhη).

18. Prove that expζ tends to if ζ moves away towards infinity along any straight line through the origin making an angle less than 1 2π with OX, and to 0 if ζ moves away along a similar line making an angle greater than 1 2π with OX.

19. Prove that cosζ and sinζtend to if ζ moves away towards infinity along any straight line through the origin other than either half of the real axis.

20. Prove that tanζ 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.

231. The 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,), where αβ, … are constants, often assumes different forms according to the values of αβ, …; sometimes it can be expressed by means of logarithms, and sometimes by means of inverse trigonometrical functions. Thus, for example,

dx x2 + α = 1 αarctan x α (1)

if α > 0, but

dx x2 + α = 1 2 αlog 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 expiζ, and that the logarithm is the inverse of the exponential function.

Let us consider more particularly the equation

dx x2 α2 = 1 2αlog x α x + α,
which holds when α is real and (x α)/(x + α) is positive. If we could write iα instead of α in this equation, we should be led to the formula

arctan x α = 1 2ilog x iα x + iα + 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 ±iα) = 1 2 log(x2 + α2) ±i(φ + 2kπ),
where k is an integer and φ is the numerically least angle such that cosφ = x/x2 + α2 and sinφ = α/x2 + α2. Thus
1 2iLog x iα x + iα = φ lπ,
where l is an integer, and this does in fact differ by a constant from any value of arctan(x/α).

The standard formula connecting the logarithmic and inverse circular functions is

arctanx = 1 2iLog 1 + ix 1 ix, (4)

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

1 2iLog cosy + isiny cosy isiny = 1 2iLog(exp2iy) = y + kπ,
where k is any integer, so that the equation (4) is ‘completely’ true (Ex. XCIII. 3). The reader should also verify the formulae

arccosx = iLog{x ±i1 x2},arcsinx = iLog{ix ±1 x2}, (5)

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

Example. Solving the equation

cosu = x = 1 2{y + (1/y)},
where y = exp(iu), with respect to y, we obtain y = x ±i1 x2. Thus:
u = iLogy = iLog{x ±i1 x2},
which is equivalent to the first of the equations (5). Obtain the remaining equations (4) and (5) by similar reasoning.

232. The power series for exp z.109 We saw in § 212 that when z is real

exp z = 1 + z + z2 2! + . (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 = iy, where y is real, and F(z) = f(y). Then

f(y)f(k) = f(y + k);
and so
f(y + k) f(y) k = f(y) f(k) 1 k .

But

f(k) 1 k = i 1 + ik 2! + (ik)2 3! + ;
and so, if k < 1,
f(k) 1 k i < 1 2! + 1 3! + k < (e 2)k.
Hence {f(k) 1}/k i as k 0, and so

f(y) = lim k0f(y + k) f(y) k = if(y). (3)

Now

f(y) = F(iy) = 1 + (iy) + (iy)2 2! + = φ(y) + iψ(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) + iψ(y)}{φ(y) iψ(y)} = F(iy)F(iy) = 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
K k ={ cos(Y + K) cos Y k }{ cos(Y + K) cos Y K }.
Of the two quotients on the right-hand side the first tends to a limit when k 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 )dY dy .
But we have seen already that
f(y) = if(y) = sin Y + i cos Y.
Hence
dY dy = 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 2π, and f(y) = cos y + i sin y. Thus F(iy) = cos y + i sin y for all real values of y. And, if x also is real, we have

F(x + iy) = F(x)F(iy) = exp x(cos y + i sin y) = exp(x + iy),
or
exp z = 1 + z + z2 2! + ,
for all values of z.

233. The 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 z2 2! + z4 4! , sin z = z z3 3! + z5 5!
for all values of z. These results were proved for real values of z in Ex. LVI. 1.

Examples XCVI. 1. Calculate cosi and sini to two places of decimals by means of the power series for cosz and sinz.

2. Prove that coszcoshz and sinzsinhz.

3. Prove that if z< 1 then cosz< 2 and sinz< 6 5z.

4. Since sin2z = 2sinzcosz we have

(2z) (2z)3 3! + (2z)5 5! = 2 z z3 3! + 1 z2 2! + .
Prove by multiplying the two series on the right-hand side (§195) and equating coefficients (§194) that
2n + 1 1 + 2n + 1 3 + + 2n + 1 2n + 1 = 22n.
Verify the result by means of the binomial theorem. Derive similar identities from the equations
cos2z + sin2z = 1,cos2z = 2cos2z 1 = 1 2sin2z.

5. Show that

exp{(1 + i)z}= 021 2n exp(14nπi)zn n! .

6. Expand coszcoshz in powers of z. [We have

coszcoshz + isinzsinhz = cos{(1 i)z}= 1 2[exp{(1 + i)z}+ exp{(1 + i)z}] = 1 2 021 2n{1 + (1)n}exp(14nπi)zn n! ,

and similarly

coszcoshz isinzsinhz = cos(1 + i)z = 1 2 021 2n{1 + (1)n}exp(14nπi)zn n! .
Hence
coszcoshz = 1 2 021 2n{1 + (1)n}cos 1 4nπzn n! = 1 22z4 4! + 24z8 8! .]

7. Expand sinzsinhz, coszsinhz, and sinzcoshz in powers of z.

8. Expand sin2z and sin3z in powers of z. [Use the formulae

sin2z = 1 2(1 cos2z),sin3z = 1 4(3sinz sin3z),.
It is clear that the same method may be used to expand cosnz and sinnz, where n is any integer.]

9. Sum the series

C = 1 + cosz 1! + cos2z 2! + cos3z 3! + ,S = sinz 1! + sin2z 2! + sin3z 3! + .

[Here

C + iS = 1 + exp(iz) 1! + exp(2iz) 2! + = exp{exp(iz)} = exp(cosz){cos(sinz) + isin(sinz)},

and similarly

C iS = exp{exp(iz)}= exp(cosz){cos(sinz) isin(sinz)}.
Hence
C = exp(cosz)cos(sinz),S = exp(cosz)sin(sinz).]

10. Sum

1 + acosz 1! + a2 cos2z 2! + ,asinz 1! + a2 sin2z 2! + .

11. Sum

1 cos2z 2! + cos4z 4! ,cosz 1! cos3z 3! +
and the corresponding series involving sines.

12. Show that

1 + cos4z 4! + cos8z 8! + = 1 2{cos(cosz)cosh(sinz) + cos(sinz)cosh(cosz)}.

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.

234. The logarithmic series. We found in § 213 that

log(1 + z) = z 1 2z2 + 1 3z3 (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 log(1 + z) is the principal value of Log(1 + z), and that

log(1 + z) =Cdu 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) = ζr,
so that r 1, and
u = 1 + ζt,
then u will describe C as t increases from 0 to r. And

Cdu u =0r ζdt 1 + ζt =0r ζ ζ2t + ζ3t2 + (1)m1ζmtm1 + (1)mζm+1tm 1 + ζt dt = ζr (ζr)2 2 + (ζr)3 3 + (1)m1(ζr)m m + Rm = z z2 2 + z3 3 + (1)m1zm m + Rm,  (2) 

where

Rm = (1)mζm+10r tmdt 1 + ζt. (3)

It follows from (1) of § 164 that

Rm0r tmdt 1 + ζt. (4)

Now 1 + ζt or u is never less than ϖ, the perpendicular from O on to the line C.110 Hence

Rm 1 ϖ0rtmdt = rm+1 (m + 1)ϖ 1 (m + 1)ϖ,
and so Rm 0 as m . It follows from (2) that

log(1 + z) = z 1 2z2 + 1 3z3 . (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

log 1 1 z = log(1 z) = z + 1 2z2 + 1 3z3 + . (6)

235. Now

log(1 + z) = log{(1 + r cos θ) + ir sin θ} = 1 2 log(1 + 2r cos θ + r2) + i arctan r sin θ 1+r cos θ .

That value of the inverse tangent must be taken which lies between 1 2π and 1 2π. 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 = 1.111

Since zm = rm(cos mθ + i sin mθ), we obtain, on equating the real and imaginary parts in equation (5) of § 234,

1 2 log(1 + 2r cos θ + r2) = r cos θ 1 2r2 cos 2θ + 1 3r3 cos 3θ , arctan r sin θ 1 + r cos θ = r sin θ 1 2r2 sin 2θ + 1 3r3 sin 3θ .

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

log(1 + z) = log(1 + Cis θ) = 1 2 log(2 + 2 cos θ) + i arctan sin θ 1+cos θ = 1 2 log(4 cos 21 2θ) + 1 2iθ,

if π < θ < π, and so

cos θ 1 2 cos 2θ + 1 3 cos 3θ = 1 2 log(4 cos 21 2θ), sin θ 1 2 sin 2θ + 1 3 sin 3θ = 1 2θ.

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 2π. Thus the sum of the cosine series is 1 2 log(4 cos 21 2θ) 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 1 2(θ 2kπ) if (2k 1)π < θ < (2k + 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 = (2k + 1)π.


pict

Fig. 58.

If we write iz and  iz for z in (5), and subtract, we obtain

1 2ilog 1 + iz 1 iz = z 1 3z3 + 1 5z5 .
If z is real and numerically less than unity, we are led, by the results of §231, to the formula
arctanz = z 1 3z3 + 1 5z5 , already proved in a different manner in §214.

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

logc = loga b acosC b2 2a2 cos2C .

[Use the formula logc = 1 2 log(a2 + b2 2abcosC).]

2. Prove that if 1 < r < 1 and 1 2π < θ < 1 2π then

rsin2θ 1 2r2 sin4θ + 1 3r3 sin6θ = θ arctan 1r 1+r tanθ, the inverse tangent lying between 1 2π and 1 2π. Determine the sum of the series for all other values of θ.

3. Prove, by considering the expansions of log(1 + iz) and log(1 iz) in powers of z, that if 1 < r < 1 then

4rsinθ + 1 2r2 cos2θ 1 3r3 sin3θ 1 4r4 cos4θ + = 1 2 log(1 + 2rsinθ + r2), rcosθ + 1 2r2 sin2θ 1 3r3 cos3θ 1 4r4 sin4θ + = arctan rcosθ 1 rsinθ, 2rsinθ 1 3r3 sin3θ + = 1 4 log 1+2r sin θ+r2 12r sin θ+r2 , rcosθ 1 3r3 cos3θ + = 1 2 arctan 2r cos θ 1r2 ,

the inverse tangents lying between 1 2π and 1 2π.

4. Prove that

cosθcosθ 1 2 cos2θcos2θ + 1 3 cos3θcos3θ = 1 2 log(1 + 3cos2θ), sinθsinθ 1 2 sin2θsin2θ + 1 3 sin3θsin3θ = arccot(1 + cotθ + cot2θ),

the inverse cotangent lying between 1 2π and 1 2π; and find similar expressions for the sums of the series

cosθsinθ 1 2 cos2θsin2θ + ,sinθcosθ 1 2 sin2θcos2θ + .

236. Some applications of the logarithmic series. The exponential limit. Let z be any complex number, and h a real number small enough to ensure that hz < 1. Then

log(1 + hz) = hz 1 2(hz)2 + 1 3(hz)3 ,
and so
log(1 + hz) h = z + φ(h,z),
where

φ(h,z) = 1 2hz2 + 1 3h2z3 1 4h3z4 + , φ(h,z) < hz2(1 + hz + h2z2 + ) = hz2 1 hz,

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

lim h0 log(1 + hz) h = z. (1)

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

lim nn log 1 + z n = z,
and so

lim n1 + z nn = lim n exp n log 1 + 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

log(1 + tz + hz) log(1 + tz) h = 1 h log 1 + hz 1 + tz
which tends to the limit z/(1 + tz) as h 0. Hence

d dt{log(1 + tz)} = z 1 + tz. (3)

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

d dt(exp φ) = d dt{(cos χ + i sin χ) exp ψ} = {(cos χ + i sin χ)ψ + ( sin χ + i cos χ)χ} exp ψ = (ψ + iχ)(cos χ + i sin χ) exp ψ = (ψ + iχ) exp(ψ + iχ) = φ exp φ,

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

d dt(1 + tz)m = d dt exp{m log(1 + tz)} = mz 1 + tz exp{m log(1 + tz)} = mz(1 + tz)m1.  (4)

Here both (1 + tz)m and (1 + tz)m1 have their principal values.

237. The general form of the Binomial Theorem. We have proved already (§ 215) that the sum of the series

1 + m 1 z + m 2 z2 +
is (1 + z)m = exp{m log(1 + z)}, for all real values of m and all real values of z between 1 and 1. If an is the coefficient of zn then
an+1 an = m n n + 1 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 log(1 + z)}, i.e. the principal value of (1 + z)m.

It follows from § 236 that if t is real then

d dt(1 + tz)m = mz(1 + tz)m1,
z and m having any real or complex values and each side having its principal value. Hence, if φ(t) = (1 + tz)m, we have
φ(n)(t) = m(m 1)(m n + 1)zn(1 + tz)mn.
This formula still holds if t = 0, so that
φn(0) n! = m n zn.

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

φ(1) = φ(0) + φ(0) + φ(0) 2! + + φ(n1)(0) (n 1)! + Rn,
where
Rn = 1 (n 1)!01(1 t)n1φ(n)(t)dt.
But if z = r(cos θ + i sin θ) then
1 + tz = 1 + 2tr cos θ + t2r2 1 tr,
and therefore

Rn < m(m 1)(m n + 1) (n 1)! rn01 (1 t)n1 (1 tr)nmdt < m(m 1)(m n + 1) (n 1)! (1 θ)n1rn (1 θr)nm ,

where 0 < θ < 1; so that (cf. § 163)

Rn < Km(m 1)(m n + 1) (n 1)! rn = ρ n,
say. But
ρn+1 ρn = m n n r r,
and so (Ex. XXVII. 6) ρn 0, and therefore Rn 0, as n . Hence we arrive at the following theorem.

THEOREM. The sum of the binomial series

1 + m 1 z + m 2 z2 +
is exp{m log(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

log(1 + z) = 1 2 log(1 + 2rcosθ + r2) + iarctan r sin θ 1+r cos θ , we obtain

0m n zn = exp{1 2mlog(1 + 2rcosθ + r2)}Cis marctan r sin θ 1+r cos θ = (1 + 2rcosθ + r2)1 2m Cis marctan rsinθ 1 + rcosθ,

all the inverse tangents lying between 1 2π and 1 2π. In particular, if we suppose θ = 1 2π, z = ir, and equate the real and imaginary parts, we obtain

1 m 2 r2 + m 4 r4 = (1 + r2)1 2m cos(marctanr), m 1 r m 3 r3 + m 5 r5 = (1 + r2)1 2m sin(marctanr).

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 1 3 2 4r2 + 1 3 5 7 2 4 6 8r4 = 1 + r2 + 1 2(1 + r2) , 1 2r 1 3 5 2 4 6r3 + 1 3 5 7 9 2 4 6 8 10r5 = 1 + r2 1 2(1 + r2) .

[Take m = 1 2 in the last two formulae of Ex. 1.]

4. Prove that if 1 4π < θ < 1 4π then

cosmθ = cosmθ 1 m 2 tan2θ + m 4 tan4θ , sinmθ = cosmθ m 1 tanθ m 3 tan3θ + ,

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

cosmθ + isinmθ = (cosθ + isinθ)m = cosmθ(1 + itanθ)m.]

5. We proved (Ex. LXXXI. 6), by direct multiplication of series, that f(m,z) = m n zn, 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. 1454, that if m is real and rational then
f(m,z) = exp{mlog(1 + z)}.

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

iμ n zn = cos{μlog(1 + z)}+ isin{μlog(1 + z)}.

Miscellaneous Examples on Chapter X.

1. Show that the real part of ilog(1+i) is

e(4k+1)π2/8 cos{1 4(4k + 1)πlog2},
where k is any integer.

2. If acosθ + bsinθ + c = 0, where abc are real and c2 > a2 + b2, then

θ = mπ + α ±ilog c+ c2 a2 b2 a2 + b2 ,
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/a2 + b2 and b/a2 + b2.

3. Prove that if θ is real and sinθsinφ = 1 then

φ = (k + 1 2)π ±ilogcot 1 2(kπ + θ),
where k is any even or any odd integer, according as sinθ is positive or negative.

4. Show that if x is real then

d dxexp{(a + ib)x}= (a + ib)exp{(a + ib)x}, exp{(a + ib)x}dx = exp(a + ib)x a + ib .

Deduce the results of Ex. LXXXVII. 3.

5. Show that if a > 0 then 0exp{(a + ib)x}dx = 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 2π from

i{logf(a,ib) logf(a,ib)}.

[The eccentric angles are given by f(acosα,bsinα) + = 0 or by

f 1 2a u + 1 u , 1 2ib u 1 u + = 0,
where u = expiα; and α is equal to one of the values of  iLogP, where P is the product of the roots of this equation.]

7. Determine the number and approximate positions of the roots of the equation tanz = az, where a is real.

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

sin2x/(cos2x + cosh2y) = ax,sinh2y/(cos2x + cosh2y) = ay,
so that, unless x or y is zero, we have
(sin2x)/2x = (sinh2y)/2y.
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 tanhy = ay. 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 tanz = az + 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/2a.

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

10. The equation tanz = atanhcz, 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

eax cosbx = 0xn n! an n 2an2b2 + n 4an4b4 ,
where there are 1 2(n + 1) or 1 2(n + 2) terms inside the large brackets. Find a similar series for eax sinbx.

12. If nφ(z,n) z as n , then {1 + φ(z,n)}n expz.

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

d dtlogφ(t) = φ(t) φ(t) .

[Use the formulae

φ = ψ + iχ,logφ = 1 2 log(ψ2 + χ2) + iarctan(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 zZ 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(πZ/a),Z = (a/π)Logz
where a is positive. To one value of Z corresponds one of z, but to one of z infinitely many of Z. If xy, rθ are the coordinates of z and XY , RΘ those of Z, we have the relations

x = eπX/a cos(πY/a),y = eπX/a sin(πY/a), X = (a/π)logr, Y = (aθ/π) + 2ka,

where k is any integer. If we suppose that π < θ π, and that Logz has its principal value logz, then k = 0, and Z is confined to a strip of its plane parallel to the axis OX 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 Logz other than the principal value we obtain a similar relation between the z-plane and another strip of breadth 2a 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 OX, but to a circle for which r is constant corresponds only a part, of length 2a, of a parallel to OY . 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 = ccosh(πZ/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 OX and of breadth 2a. Show also that to the line X = X0 corresponds the ellipse

x ccosh(πX0/a)2 + y csinh(πX0/a)2 = 1,
and that for different values of X0 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 = X0 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 = ccosh(πZ/a) may be regarded as compounded from the transformations

z = cz1,z1 = 1 2{z2 + (1/z2)},z2 = exp(πZ/a).]

18. Discuss similarly the transformation z = ctanh(πZ/a), showing that to the lines X = X0 correspond the coaxal circles

{x ccoth(2πX0/a)}2 + y2 = c2 cosech2(2πX 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 00 1) on to the tangent plane at the north pole. The coordinates of a point on the sphere are ξηζ, and Cartesian axes OXOY 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 = 2ξ/(1 + ζ),y = 2η/(1 + ζ),
and that x + iy = 2tan 1 2θCisφ, 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 = ilog 1 2z = ilog 1 2(x + iy)
so that X = φ, Y = logcot 1 2θ, 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 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 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 = b aexp( x + iy), Z a Z b = b aexp( x iy);

and it will be found that

Z a+ Z b= b acosh2x,Z aZ b= b acos2y.]

22. The transformation z = Zi. If z = Zi, where the imaginary power has its principal value, we have

exp(logr + iθ) = z = exp(ilogZ) = exp(ilogR Θ),
so that logr = Θ, θ = logR + 2kπ, where k is an integer. As all values of k give the same point z, we shall suppose that k = 0, so that

logr = Θ,θ = logR. (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 expπ and θ ranges through all real values. Thus the Z-plane corresponds to the ring bounded by the circles r = exp(π), r = expπ; 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 expπ, 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 = Zi,Z = zi,
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 expπ, moves round the larger circle in the positive direction to the point  expπ, 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

,e(2n+1)π,,eπ,eπ,e3π,,e(2n+1)π,.
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 = Zi, Z = zi.

25. If z = Zi, 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 = zai, 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 zai has its principal value), and the real and imaginary parts of Z both oscillate finitely.]

27. Discuss the same question for Z = za+bi, where a and b are any real numbers.

28. Show that the region of convergence of a series of the type anznai, where a is real, is an angle, i.e. a region bounded by inequalities of the type θ0 < amz < θ1 [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 (concentric circles),(z a)(z b) (Cartesian ovals), (z a)/(z b) (coaxal circles), expz (straight lines).

30. Sketch the forms of the level curves of (z a)(z b)(z c), (1 + z3 + z2)/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.]


pict

Fig. 59.


pict
Fig. 60.
pict
Fig. 61.

31. Sketch the forms of the level curves of (i) zexpz, (ii) sinz. [See Fig. 60, which represents the level curves of sinz. The curves marked I–VIII correspond to k = .35, .50, .71, 1.00, 1.41, 2.00, 2.834.00.]

32. Sketch the forms of the level curves of expz c, where c is a real constant. [Fig. 61 shows the level curves of expz 1, the curves I–VII corresponding to the values of k given by logk = 1.00, .20, .05, 0.00, .05, .201.00.]

33. The level curves of sinz 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, 4.507.79. 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, 9.0015.59. If c = 1 then the curves are the same as those of Fig. 60, except that the origin and scale are different.]


pict
Fig. 62.
pict
Fig. 63.

34. Prove that if 0 < θ < π then

cosθ + 1 3 cos3θ + 1 5 cos5θ + = 1 4 logcot21 2θ, sinθ + 1 3 sin3θ + 1 5 sin5θ + = 1 4π,

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

z + 1 3z3 + 1 5z5 + = 1 2 log 1+z 1z
where z = cosθ + isinθ. 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 1 4π if 2kπ < θ < (2k + 1)π, 1 4π if (2k + 1)π < θ < (2k + 2)π, and 0 if θ is a multiple of π.]

35. Prove that if 0 < θ < 1 2π then

cosθ 1 3 cos3θ + 1 5 cos5θ = 1 4π, sinθ 1 3 sin3θ + 1 5 sin5θ = 1 4 log(secθ + tanθ)2;

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

36. Prove that

cosθcosα + 1 2 cos2θcos2α + 1 3 cos3θcos3α + = 1 4 log{4(cosθ cosα)2},
unless θ α or θ + α is a multiple of 2π.

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

0 dx (x a)(x b) = log(a) log(b) a b ,
each logarithm having its principal value. Verify the result when a = ci, b = ci, 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

0 d x2 (α + iβ)2 = πi 2(α + iβ).
What is the value of the integral when β < 0?

39. Prove that, if the roots of Ax2 + 2Bx + C = 0 have their imaginary parts of opposite signs, then

dx Ax2 + 2Bx + C = πi B2 AC,
the sign of B2 AC being so chosen that the real part of {B2 AC}/Ai is positive.