2.1.1 Polynomials

The polynomial function has the general form

(2.1)

where a_i(i = 0, 1, 2, …, n) are constants, n is a non-negative integer (i.e. including zero), and the symbol Σ means that a sum is to be taken of all terms labelled by the indices 0, 1, 2, …, n. The value of n defines the order (or degree) of the polynomial. The expression x³ − 3x² − 6x + 8 plotted in Figure 1.1 is therefore a polynomial of order 3.

The roots of polynomials are defined as the solutions of the equation

(2.2)

and correspond to the points where a graph of P_n(x) crosses the x-axis. For first-order polynomials, (2.2) is a linear equation of the form

(2.3)

where a and b are constants. This has one root, which is trivially given by x = −b/a.

For second-order polynomials, (2.2) leads to a quadratic equation of the form

(2.4)

where a, b and c are numerical constants. When the coefficients are simple, for example integers, one can sometimes spot that the quadratic form factorises, i.e.

(2.5)

where α and β are real numbers. The solutions are then clearly x₁ = α and x₂ = β. For example, 2x² − 7x + 3 = (2x − 1) (x − 3) and so the roots of this polynomial are . In some circumstances a solution can be ‘lost’ if care is not taken. For example, the equation x² − 5x = 0 factorises to x(x − 5) = 0, with solutions x = 0 and x = 5. However, had we divided both sides of the original equation by x, we would have only found the solution x = 5. So when manipulating the original equation, particularly when using division to simplify it, one should always check that no solutions are thereby being omitted.

If there is no obvious factorisation, the general solution of the quadratic is obtained by a process known as ‘completing the square’, as follows. By taking the constant c onto the right-hand side of the equation and then adding b²/4a to both sides, the left-hand side may be written as a perfect square. We have,

(2.6a)

and hence

(2.6b)

We thus see that for b² > 4ac there are two solutions, which coincide when b² = 4ac. If we denote these two solutions by α and β, then from (2.6b) one easily confirms that the polynomial can be written in the factorised form (2.5) and that

(2.7)

These results are sometimes useful because a quadratic equation may be written

(2.8)

so that, for example, the polynomial with roots 2.1 and 3.2 is x² − 5.30x + 6.72. If b² − 4ac < 0, the argument of the square root in (2.6b) is negative, and (2.4) has no solutions for real x. For example, the quadratic equation x² − 3x + k = 0 has two roots if 9 − 4k ≥ 0, i.e. k ≤ 9/4, but no real roots if 9 − 4k < 0.

Exact solutions for cubic and quartic equations exist, so that the roots of third-order and fourth-order polynomials may be determined exactly. However, the solutions are algebraically very complicated and we will not pursue them further. Except in special cases, the roots of higher-order polynomials are found by approximate methods. However, one can establish some important general results. To obtain these we initially consider the result obtained by dividing a polynomial P_n(x) of order n by a factor (x − a) using long division, until only a constant remainder is left. For example, on dividing x⁴ − 2x³ + 3x² − 4x + 5 by (x − 1) one obtains

so that

More generally, dividing any polynomial of order n by (x − a) leads to an expression of the form

(2.9)

where the quotient Q(x) is a polynomial of order (n − 1) and the remainder R = P_n(a). This result is called the remainder theorem and implies that if a = α is a root of P_n(x), then R = 0 and (2.9) reduces to the partially factorised form

(2.10a)

where P_{n − 1}(x) is a polynomial of order n − 1. This is called the factor theorem. Furthermore, repeating the process for all m roots α₁, α₂, …, α_m gives

(2.10b)

and since the highest power on the left is xⁿ, there are at most n real roots. Thus a polynomial P_n(x) of order n has at most n real roots.¹ Beyond this, one can only say that the number of roots is odd or even, corresponding to whether the order of the polynomial is odd or even, respectively, provided that if two or more factors in (2.10b) are equal, we still count them separately. This is most easily seen by considering a graph of the polynomial, in which the roots correspond to the values at which the curve intercepts the x-axis, as illustrated in Figure 1.1. The results then follow by considering the asymptotic behaviour of the polynomial as x → ±∞, which is dominated by the term a_nxⁿ in (2.1). Hence if n is even, the polynomial has the same sign in the limits x → ±∞, and since it is continuous, it must either not cross the x-axis at all, corresponding to no roots, or cross it an even number of times, corresponding to an even number of roots. A similar argument shows that a polynomial whose order is odd must have at least one root and there can only be an odd number of roots.

We now return to the problem of finding the roots. As noted above, the general solution for third and fourth order polynomials is very complicated, and for higher orders no general exact solution is known. However, for simple cases it may still be possible to find exact solutions of higher-order polynomials by spotting factors and using the factor theorem. For example, consider the fourth-order polynomial

(2.11)

By inspection, f(1) = 0, so (x − 1) is a factor. To find the quotient Q(x) we need to carry out a long division, which yields (2x³ + x² − 7x − 6), so that

(2.12)

We now repeat the process by finding factors (if they exist) of the cubic. The final result is

(2.13)

so the solutions are .

Not all polynomials factorise, and in practice equations involving higher order polynomials are solved by approximate methods, either graphical or numerical. In the former, the function is plotted and the points where f(x) crosses the x-axis are found. In Figure 2.1 the function x³ − 3x² − 4x + 7 is plotted and it is seen that it crosses the x-axis at the values x ≈ −1.7, 1.1 and 3.6, which are therefore the approximate solutions of the equation x³ − 3x² − 4x + 7 = 0. It is worth reiterating that a polynomial of order n does not necessarily have n real roots, and so a graph of the function will not necessarily cut the x-axis at n points.

Although only approximate, the graphical solutions are still useful, as numerical methods for finding accurate roots often rely on knowing approximate solutions as starting values. One simple technique is the so-called bisection method, which can be applied to any continuous function. In this method one starts by finding two values of x, say x₁ and x₂, that straddle the position of a zero. Thus f(x₁) and f(x₂) will have opposite signs and so f(x₁)f(x₂) < 0. Now let and calculate f(x_m). If f(x₁)f(x_m) < 0, then the root lies between x₁ and x_m. In this case, the mid-point of x₁ and x_m is found and the calculation repeated. If, however, f(x₁)f(x_m) > 0, then the root lies between x₂ and x_m, and in this case, the mid-point of x₂ and x_m is found and the calculation repeated. This iterative method can be rapidly implemented on a computer, and when applied to any of the roots, the range of values of x that produces a value of f(x) as close to zero as desired may be found.

2.1.2 Rational functions and partial fractions

Given two polynomials P(x) and Q(x), we can form the rational function f(x), defined by f(x) ≡ P(x)/Q(x). These are generalisations of numerical fractions and, by analogy with those, the rational expression P/Q is said to be proper if the order of the numerator is less than the order of the denominator. Otherwise it is called an improper fractional expression. Examples are:

(2.14)

In contrast to polynomials, rational functions are not in general continuous, but can have discontinuities corresponding to the roots of the denominator function Q(x), that is, where Q(x) = 0, and so are undefined at those points. For example, the rational function

(2.15)

has discontinuities at x = ±1, where the denominator vanishes, as shown in Figure 1.6b.

Rational expressions where the denominator is itself the product of polynomials may often usefully be decomposed into a sum of simpler terms called partial fractions. Assume for the moment that the initial expression is a proper fraction. There are several possible forms this can take and we will look at each in turn, before illustrating them with specific examples.

1. The first form is

   (2.16)

   where a, b, …, n are constants, and because the fraction is proper, P(x) is a polynomial of lower order than the product of factors in the denominator. In this case, we may write the identity

   (2.17)

   where A, B, … N are constants. By putting the terms on the right-hand side over a common denominator, (2.17) may be written

   (2.18)

   Because this is an identity, it is true for all values of x. Thus we can choose any values of x to evaluate it. So, in particular, if we choose x = a, x = b, … in turn, in each case all the terms on the right-hand side are zero except one and we can solve for the coefficients A, B, etc.

2. A second common occurrence is where one of the factors in the denominator is a quadratic of the form αx² + βx + γ that cannot be factored further, since β² < 4αγ. In this case, we may write

   (2.19)

   and again, by using the fact that this is an identity, we can take both sides over a common denominator and equate the coefficients of the same powers of x on both sides. Then by choosing suitable values for x, values of the constants A, B, …, N, and A′ and B′ can be found.
3. The third type is when there are repeated factors (ax + b)ⁿ in the denominator. These will give rise to partial fractions of the form

   (2.20)

   that again have to be added to any other terms of the type (2.19) and the sum treated in an analogous way to the previous cases, that is, take both sides over a common denominator and using the fact that the expression is an identity, equate coefficients of the same powers of x on both sides, and then choose suitable values of x to determine the coefficients.

In all the above cases the original fractional function was proper. If the fraction is improper, then an initial long division must be made to write it as the sum of a polynomial and a proper fraction. The latter is then decomposed into partial fractions as above.

2.1.3 Algebraic and transcendental functions

Polynomials and rational functions are the simplest examples of a broader class of functions, called algebraic functions. An algebraic function is any function y that can be defined by an equation of the form

where P⁽ⁱ⁾(x) (i = 0, 1, …, n) are given polynomials of any order. This definition is implicit, and for any x the function can be evaluated by first evaluating the polynomials P⁽ⁱ⁾(x), and then finding the roots of the resulting polynomial in y.

For n = 1, one easily sees that the above definition reduces to a rational function, or a polynomial in the case of P⁽⁰⁾ = 1. More generally, it implies that any algebraic function can be defined in terms of a finite number of the basic operations of algebra (i.e. addition, subtraction, multiplication and division).

In contrast, functions that are not of the above form cannot be defined by a finite sequence of basic algebraic operations. Such functions are called transcendental functions and are somewhat analogous to irrational numbers, which cannot be evaluated from integers by a finite sequence of the operations of arithmetic. The functions to be discussed in the next two subsections – trigonometric functions, logarithms and exponential functions – are all examples of transcendental functions.

2.2.1 Angles and polar co-ordinates

Trigonometry is the study of angles, and before turning to the trigonometric functions themselves, it will be useful to consider angles and their use as co-ordinates. In doing so, we will make reference to Figure 2.2, which shows the angle of intersection θ between two lines OA, OB, together with a circle of radius r whose centre lies at the point of intersection.

One unit of angle is the degree, which is defined to be a part of a complete rotation and is denoted 1°. Thus a right-angle corresponds to 90°. In scientific work it is more usual to work in terms of the radian, which is defined as the angle when the length l of the arc P₀P₁ is equal to the radius r. Since the circumference of a circle of radius r is 2πr and corresponds to an angle of 2π radians, it follows that the length of an arc of a circle of radius r that subtends an angle θ at the centre of the circle is

(2.21)

It also follows that 2π radians = 360 degrees, so that a right angle is π/2 radians and 1 radian ≈ 57.3°. In addition, the area of the corresponding sector shown in Figure 2.2 is

(2.22)

We stress that, like many other equations in this book, (2.21) and (2.22) are only valid if the angles are expressed in radians. Unless stated otherwise it will be assumed from now on that all angles are expressed in radians.

Angles can also be used as co-ordinates, provided we adopt a convention to specify their sign. This is illustrated in Figure 2.3, where the position of the point P can be specified by the Cartesian co-ordinates (x, y) used in Section 1.3.1, or the plane polar co-ordinates (r, θ). Here, r > 0 is the distance of P from the origin O, with

(2.23)

by Pythagoras' theorem, and θ is the angle between the line OP and the x-axis measured in a counter-clockwise sense. Thus in Figure 2.4a the point P corresponds to θ = −π/4, since OP is at an angle π/4 to the x-axis when measured in a clockwise direction. However, the polar angle is not unique, and P also corresponds to θ = 7π/4, since OP is at an angle 7π/4 to the x-axis when measured in the counter-clockwise direction, as shown in Figure 2.4b. In general, the points (r, θ) and (r, θ + 2nπ) correspond to the same point in the plane for any integer n. This is illustrated for the case n = 1 in Figures 2.4c and 2.4d.

The ambiguity in the value of the polar angle corresponding to a given point can be removed by restricting the range of θ to 0 < θ < 2π. However, this is not always convenient. Consider, for example, a particle moving in a circular orbit of constant radius r with constant speed , as shown in Figure 2.5.

Assuming that θ = 0 at time t = 0, the motion is described in polar co-ordinates by the simple equations

(2.24)

where we have deduced the equation for θ from (2.21) together with the fact that the particle traverses a length of arc in time t. The angle increases indefinitely as t increases and θ = 2nπ + φ, with 0 < φ < 2π, corresponds to the particle arriving at the point (r, φ) after n complete revolutions since t = 0.

2.2.2 Sine and cosine

For angles less than , the sine and cosine functions (written ‘sine’ and ‘cosine’) are defined in terms of the sides of a right-angled triangle by

and applying this to the triangle in Figure 2.3 we obtain

(2.25)

where x and y are the projections of OP onto the x-axis and y-axis, respectively, and r is the length of OP. However, if we consider a point P rotating in a counter-clockwise direction about the origin at (0, 0), as shown in Figure 2.5, then (2.25) allows us to extend the definitions of sine and cosine to all angles provided the signs of x and y are taken into account. For example, in the fourth quadrant , we see from Figure 2.4(a) that x > 0 while y < 0, so that cos θ > 0 and sin θ < 0. More generally, as θ increases in Figure 2.5, one sees that x and y oscillate between r and −r, and hence cos θ and sin θ oscillate between −1 and +1, with a period of 2π corresponding to a single revolution. In other words,

(2.26)

and they are periodic with a period of 2π, that is, the form of the function repeats at intervals of 2π, so that

(2.27)

In addition, together with (2.23), the definitions (2.25) imply the important relation

(2.28)

for all values of θ.

The graphical forms of the sine and cosine functions follow from the definitions (2.25), together with the construction in Figure 2.5. They are shown in Figures 2.6a and 2.6b, respectively, and have a number of other general features which, in view of the enormous importance of these functions in physical science, are worth emphasising.

1. If we replace θ by − θ in Figure 2.5, then y → −y, but x remains unchanged. Hence from (2.25), sin θ and cos θ are odd and even functions, respectively:

   (2.29)

2. One sees from (2.28) and Figure 2.5 that y, and hence sin θ, vanishes when θ = 0 and π. Similarly y = r and − r, and hence sin θ = 1 and − 1 at , respectively. Together with (2.27) this implies the results

   (2.30a)

   for any integer n. The corresponding results for the cosine function are

   (2.30b)

3. By inspection, it can be seen that the forms of the sine and cosine curves in Figure 2.6 are the same but displaced by a distance along the θ-axis. So we deduce that

   (2.31)

For the first quadrant, , this result follows from the construction of Figure 2.7, where P and P′ correspond to polar angles θ and . From this diagram, one easily sees that the triangles OAP and OBP′ are similar triangles and hence, since OP = OP′( = r), they are congruent, that is, identical if superimposed. Thus OC = BP′ = AC, and the result (2.31) follows. A similar construction works in the other three quadrants in 0 < θ < 2π, establishing the result for all angles.

2.2.3 More trigonometric functions

Sine and cosine are not the only important circular functions, but the others can be defined in terms of them. In particular, we define the tangent and cotangent, written as ‘tan’ and ‘cot’, respectively, as

(2.32a)

and the secant and cosecant, written as ‘sec’ and ‘cosec’, by

(2.32b)

which, together with (2.28), lead to the relations

(2.33)

The behaviours of these functions follow from the behaviour of sine and cosine shown in Figures 2.6a and 2.6b. The functions tan θ and cotθ are plotted in Figures 2.8a and 2.8b, respectively. Like the sine and cosine functions, they are periodic, but with a period of π rather than 2π. However, unlike those functions, tan θ is unbounded and is discontinuous at the points , for n = 0, ± 1, ± 2, …, where cos θ vanishes. Similarly, cotθ is discontinuous at the points where sin θ vanishes. The remaining circular functions may also be deduced from (2.28) and are shown in Figure 2.9.

In Section 1.3.1 we defined inverse functions. In the case of the circular functions this must be done with care, because it is clear from Figures 2.6, 2.8 and 2.9 that there are an infinite number of angles for a given value of sine, cosine or tangent. To obtain a single-valued function, we would therefore have to formally restrict the angular range of θ. The corresponding inverse circular functions for sine, cosine and tangent are shown for convenient choices in Figures 2.10a–2.10c, respectively. Using the notation of Section 1.3, it would be natural to refer to these as sin ^{− 1}, cos^{− 1} andtan ^{− 1}, but to avoid ambiguity with , etc. it is probably better to always use their alternative explicit names arcsin, arccos and arctan. An example of their use is furnished by the relation between the Cartesian co-ordinates (x,y) and the polar co-ordinates (r, θ) of Figure 2.3. From (2.25) we see that x and y are given in terms of r and θ by

(2.34)

from which the relations r² = x² + y² and directly follow. Hence r and θ are given in terms of x and y by the relations

(2.35)

respectively.

2.2.4 Trigonometric identities and equations

Equation (2.33) is an example of a trigonometric identity. Here we list some of the most important identities, before commenting on their derivation and giving examples of their use. They are:

(2.36a)

(2.36b)

(2.36c)

(2.36d)

(2.36e)

(2.36f)

Specific useful cases that follow directly from (2.36a) (2.36b) and (2.36f) are the ‘double-angle’ formulas obtained by setting φ = θ:

(2.37a)

and

(2.37b)

The analogous ‘half-angle’ formulas are

(2.37c)

and

(2.37d)

These identities can be proved by simple geometrical methods. To illustrate this we will prove (2.36a) by referring to Figure 2.11. From triangle ABC, we have

(2.38a)

since DC = EF. But from the triangles BDE and AEF,

(2.38b)

so that

(2.38c)

Also, from triangle ABE,

(2.38d)

so finally, using these relations in (2.38c), we have

(2.38e)

This derivation establishes (2.38e) for acute angles only, since this is what we have assumed in Figure 2.11. However, the proof can be extended to all angles. The result (2.36a) with a minus sign follows by letting φ → −φ and using the odd and even properties of the sine and cosine functions (2.29).

The rest of the formulas (2.36) follow from (2.30a) using our previous results. For example, to derive (2.36b) we write

using (2.30b) and (2.36a). Equation (2.36b) then follows, since

using (2.36a) and (2.29). The other identities follow in a similar way, and can be used to derive many more results, and solve trigonometric equations, as the following examples illustrate.

2.2.5 Sine and cosine rules

The trigonometric functions enable the discussion of co-ordinate geometry to be extended in a number of ways. One is to solve triangles, that is, to determine completely the lengths of their sides and the magnitude of all three angles. If two angles and one side, or two sides and a non-included angle are given, this can be done using the sine rule,

(2.39)

where the definitions of the angles A, B and C and lengths of the sides a, b and c are specified in Figure 2.12.

Alternatively, if the three sides, or two sides and the included angle are known, we can use the cosine rule,

(2.40a)

together with its permutations

(2.40b)

In what follows we shall prove these rules then illustrate their use by examples.

The sine and cosine rules can both be derived from the construction of Figure 2.13. To obtain the sine rule, we infer the length AP by using the definition of sine in the triangles PAB and PAC to give

(2.41a)

from which the second equality in (2.39) immediately follows. Applying the same argument to the triangles BCP′ and ACP′ gives

(2.41b)

where we have used (2.36a) and (2.30a) to show that sin (π − A) = sin A. The first equality in (2.39) then follows directly, completing the derivation of the sine rule.

To prove the cosine rule (2.40a), we apply Pythagoras' theorem to the triangle P′BC to obtain

where we have multiplied out the brackets and used (2.31). Since cos (π − A) = −cos A, from (2.36b) and (2.30b), this gives

and the cosine rule (2.40a). In a similar way, applying Pythagoras' theorem to the triangle PAC gives

thus establishing the first part of (2.40b). The second equation in (2.40b) follows by the same argument applied to the triangle PAB.

Finally, before giving examples of the application of the sine and cosine rules, we note that in proving them we have assumed that one of the angles A is obtuse.² The corresponding proofs for the case of three acute angles are very similar and are left as exercises for the reader.

2.3.1 The laws of logarithms

In Section 1.1.2 we met expressions of the form a^b, where the index, or power, b was a rational number. For a > 0, this definition can be extended to irrational numbers, since, for example, 2^π can be evaluated to arbitrary precision by exploiting the fact that π itself, like any irrational number, can be approximated to arbitrary accuracy by a rational number.³ Since we wish to include irrational numbers in our discussion, we restrict ourselves to the case a > 0, when the expression c = a^b is called an exponential expression with a the base and b the index. Conversely, b is called the logarithm of c to base a and is written b = log _ac. To summarise,

(2.42)

where the symbol ⇔ means ‘implies and is implied by’, that is, the expressions on either side of the symbol are equivalent. Graphs of log _ax for various integer values of a are shown in Figure 2.15.

Logarithms obey a number of laws that are easily derived from the basic result (2.42). If we set

(2.43a)

then

(2.43b)

and hence, from the result on indices (1.5), A′ + B′ = C, i.e.

(2.44a)

and in general

(2.44b)

By setting A = B = C… in (2.44b), it follows that

(2.44c)

a result that holds also for fractional and negative values of n. In a similar way to the proof of (2.44b), we can show that

(2.44d)

Finally, setting A = B in (2.44d) gives log _a1 = 0 for any base a. The results (2.44) are referred to as the laws of logarithms.

These relations may be used to simplify expressions and solve equations involving logarithms, as we shall illustrate by examples below. They may also be used to derive the general formula for changing a logarithm from base a to base b. Thus, if log _ac = x, then c = a^x. So log _bc = xlog _ba and

(2.45)

which, for the special case b = c, implies

(2.46)

Because the decimal system is so widespread, logarithms to base 10 are called common logarithms and the base is usually omitted. For example, log 7 = 0.845. In the binary system, it would be equally appropriate to use base 2, when

by (2.46). However, it is usual instead to choose the irrational number e = 2.71828… as the base, for reasons to be explained in the next section.

2.3.2 Exponential function

We next consider the exponential function a^x, where again a > 0, but now the exponent is a real variable x. The resulting function is plotted for the values in Figure 2.16. As can be seen, a^x increases rapidly for large positive x if a > 1, but decreases for all a < 1. In addition, a^x = 1 for x = 0 for all x, and the behaviours for positive and negative x are related by

(2.47)

so that, for example, the curves 2^{− x} and are identical.

Perhaps the most important property of the exponential function is that it is proportional to its own gradient. To see this, consider the line AB joining the function a^x at x and x + d, as shown in Figure 2.17. As can be seen, the gradient of this line becomes a better and better approximation to the gradient at x itself as d gets increasingly smaller. Hence, since

(2.48)

by (1.45) and (1.17), we immediately obtain the desired result

(2.49)

where the constant of proportionality

(2.50)

and the notation means ‘take the limit of the term in the brackets as d approaches zero’.⁴

At this point, we note that the constant k depends on the base a, and we define the Euler number e such that k = 1 for a = e. To find this number, for any given d, we choose a value

(2.51)

so that (2.48) gives

for any given d. Since as d approaches zero, the slope (AB) approaches the slope of the curve, this implies that

(2.52)

as required, if

(2.53)

The number e can now be estimated with increasing accuracy by choosing smaller and smaller d, the values d = 0.1, 0.01, 0.001, 0.0001, … giving 2.594, 2.705, 2.717, 2.718, … A better method for evaluating Euler's number will be given in Section 5.3.4, and to 6 significant figures

(2.54)

The corresponding behaviour of e^x and of the closely related function e^{− x} = (e^x)^{− 1} is shown in Figure 2.18.

Because of the property (2.52), the number e is almost always chosen as a base in physical science work and the corresponding function exp (x), defined by

(2.55)

is called the natural exponential function, or more usually, but imprecisely, just the exponential function. The corresponding inverse function

(2.56)

is referred to as the natural logarithmic function, or simply the natural logarithm. Since it is the inverse of e^x, its behaviour can be inferred from the plot of e^x, and is shown in Figure 2.18. Finally, from (2.45), natural and common logarithms are related by

2.3.3 Hyperbolic functions

Given the natural logarithms, we can define hyperbolic functions as follows.

(2.57)

These are called the hyperbolic sine, hyperbolic cosine and hyperbolic tangent, respectively, and are shown in Figures 2.19 and 2.20. Their inverses are defined as

We will show in Section 6.4.1 that the hyperbolic functions are related to the circular functions, hence the origin of their names. The word ‘hyperbolic’ appears because they are also related to the equation for a hyperbola. This follows from the first of the identities

(2.58a)

and

(2.58b)

which may be checked using the definitions (2.57). Equations (2.57) and (2.58a) imply that if x = cosh θ and y = sinh θ, then the point (x, y) lies on the branch of the rectangular hyperbola x² − y² = 1 for which x + y > 0. (Hyperbolas are discussed in Section 2.4.)

By analogy with the circular functions, we have

(2.59)

and we can also form the inverse hyperbolic functions by ‘inverting’ (2.59). For example, let

(2.60)

This can be written as a quadratic in the variable z = e^x, treating y as if it were a constant, leading to the solution . Hence

and

(2.61)

In a similar way we find that

(2.62)

where unlike in the case of sinh ^{− 1}x, both signs of the square root lead to positive values for e^x. However, because

(2.63a)

the result for cosh ^{− 1}x may also be written

(2.63b)

which shows explicitly that the two values are equal in magnitude but with opposite signs. Finally,

(2.64)

Just as for the inverse trigonometric functions, both the ‘arc’ and ‘−1’ notations used in (2.61b), (2.62) and (2.64) are in common use.

The hyperbolic functions satisfy a number of identities. By analogy with (2.36), they are

(2.65a)

(2.65b)

(2.65c)

(2.65d)

(2.65e)

(2.65f)

Specific useful cases that follow directly from (2.65) by setting x = y are the double-argument identities,

(2.66a)

(2.66b)

and

(2.66c)