Chapter I.
The differential and integral calculus is based on two concepts of outstanding importance, apart from the concept of number, namely, the concepts of function and limit. While these concepts can be recognized here and there even in the mathematics of the ancients, it is only in modern mathematics that their essential character and significance have been fully clarified. We shall attempt here to explain these concepts as simply and clearly as possible.
1.1 The Continuum of Numbers
We shall consider the numbers and start with the natural numbers 1, 2, 3, ··· as given as well as the rules
(a + b) + c
= a + (b + c) - associative
law of addition,
a + b = b + a - commutative
law of addition,
(ab)c = a(bc)
- associative law of
multiplication,
ab = ba - commutative law of multiplication,
a(b + c) = ab + ac - distributive law of multiplication.
by which we calculate with them; we shall only briefly recall the way in which the concept of the positive integers (the natural numbers) has had to be extended.
1.1.1 The System of Rational Numbers and the Need for its Extension: In the domain of the natural numbers, the fundamental operations of addition and multiplication can always be performed without restriction, i.e., the sum and the product of two natural numbers are themselves always natural numbers. But the inverses of these operations, subtraction and division, cannot invariably be performed within the domain of natural numbers, whence mathematicians were long ago obliged to invent the number 0, the negative integers, and positive and negative fractions. The totality of all these numbers is usually called the class of rational numbers, since all of them are obtained from unity by means of the rational operations of calculation: Addition, multiplication, subtraction and division.
Numbers are usually represented graphically by means of the points on a straight line - the number axis - by taking an arbitrary point of the line as the origin or zero point and another arbitrary point as the point 1; the distance between these two points (the length of the unit interval) then serves as a scale by which we can assign a point on the line to every rational number, positive or negative. It is customary to mark off the positive numbers to the right and the negative numbers to the left of the origin (Fig. 1). If, as is usually done, we define the absolute value (also called the numerical value or modulus) |a| of a number a to be a itself when a ³ 0, and - a when a < 0, then |a| simply denotes the distance of the corresponding point on the number axis from the origin.
The symbol
³
means that either the sign > or the sign = shall
hold. A corresponding statement holds for the signs ± and 
which
will be used later on.
The geometrical representation of the rational
numbers by points on the number axis suggests an important
property which can be stated as follows: The set of rational numbers is
everywhere dense. This means that
in every interval of the number axis, no matter how small, there
are always rational numbers; in geometrical terms, in the segment
of the number axis between any two rational points, however close
together, there are points corresponding to rational numbers.
This density of the rational numbers at once becomes clear if we
start from the fact that the numbers 
··· become steadily
smaller and approach nearer and nearer to zero as n
increases. If we now divide the number axis into equal parts of
length 1/2n, beginning at the origin, the
end-points 1/2n, 2/2n,
3/2n, ··· of these intervals represent
rational numbers of the form m/2n, where we
still have the number n at our disposal. Now, if we are
given a fixed interval of the number axis, no matter how small,
we need only choose n so large that 1/2n
is less than the length of the interval; the intervals of the
above subdivision are then small enough for us to be sure that at
least one of the points of the sub-division m/2n
lies in the interval.
Yet, in spite of this property of density, the rational numbers are not sufficient to represent every point on the number axis. Even the Greek mathematicians recognized that, if a given line segment of unit length has been chosen, there are intervals, the lengths of which cannot be represented by rational numbers; these are the so-called segments incommensurable with the unit. For example, the hypotenuse l of a right-angled, isosceles triangle with sides of unit length is not commensurable with the length unit, because, by Pythagoras' Theorem, the square of this length must equal 2. Therefore, if l were a rational number, and consequently equal to p/q, where p and q are non-zero integers, we should have p² = 2q². We can assume that p and q have no common factors, for such common factors could be cancelled out to begin with. Since, according to the above equation, p² is an even number, p itself must be even, say p = 2p'. Substituting this expression for p yields 4p'² = 2q² or q² == 2p'², whence q² is even, and so is q. Hence p and q have the common factor 2, which contradicts our hypothesis that p and q do not have a common factor. Thus, the assumption that the hypotenuse can be represented by a fraction p/q leads to contradiction and is therefore false.
The above reasoning - a characteristic example of an indirect proof - shows that the symbol Ö2 cannot correspond to any rational number. Thus, if we insist that, after choice of a unit interval, every point of the number axis shall have a number corresponding to it, we are forced to extend the domain of rational numbers by the introduction of the new irrational numbers. This system of rational and irrational numbers, such that each point on the axis corresponds to just one number and each number corresponds to just one point on the axis, is called the system of real numbers.
Thus named to distinguish it from the system of complex numbers, obtained by yet another extension.
1.1.2 Real Numbers and Infinite Decimals: Our requirement that there shall correspond to each point of the axis one real number states nothing a priori about the possibility of calculating with these numbers in the same manner as with rational numbers. We establish our right to do this by showing that our requirement is equivalent to the following fact: The totality of all real numbers is represented by the totality of all finite and infinite decimals.
We first recall the fact, familiar from elementary mathematics, that every rational number can be represented by a terminating or by a recurring decimal; and conversely, that every such decimal represents a rational number. We shall now show that we can assign to every point of the number axis a uniquely determined decimal (usually an infinite one), so that we can represent the irrational points or irrational numbers by infinite decimals. (In accordance with the above remark, the irrational numbers must be represented by infinite non-recurring decimals, for example, 0.101101110···).
Let the points which correspond to the integers be marked on the number axis. By means of these points, the axis is subdivided into intervals or segments of length 1. In what follows, we shall say that a point of the line belongs to an interval, if it is an interior point or an end-point of the interval. Now let P be an arbitrary point of the number axis. Then this point belongs to one or, if it is a point of division, to two of the above intervals. If we agree that, in the second case, the right-hand point of the two intervals meeting at P is to be chosen, we have in all cases an interval with end-points g and g + 1 to which P belongs, where g is an integer. We subdivide this interval into ten equal sub-intervals by means of the points corresponding to the numbers
and we number these sub-intervals 0, 1, ··· , 9 in their natural order from the left to the right. The sub-interval with the number a then has the end-points g+a/10 and g+a/10 + 1/10. The point P must be contained in one of these sub-intervals. (If P is one of the new points of division, it belongs to two consecutive intervals; as before, we choose the one on the right hand side.) Let the interval thus determined be associated with the number a1. The end-points of this interval then correspond to the numbers g+a1/10 and g+a1/10+1/10. We again sub-divide this sub-interval into ten equal parts and determine that one to which P belongs; as before, if P belongs to two sub-intervals, we choose the one on the right hand side. Thus, we obtain an interval with the end-points g+a1/10+a2/10² and g+a1/10+a2/10²+1/10³, where a2 is one of the digits 0, 1, ··· , 9. We subdivide this sub-interval again and continue to repeat this process. After n steps, we arrive at a sub-interval, which contains P, has the length 1/10n and end-points corresponding to the numbers
where each a is one of the numbers 0, 1, ··· , 9, but
is simply the decimal fraction 0.a1a2···an. Hence, the end-points of the interval may also be written in the form
If we consider the above process repeated indefinitely, we obtain an infinite decimal 0.a1a2···, which has the following meaning: If we break off this decimal at any place, say, the n-th, the point P will lie in the interval of length 1/10n the end-points (approximating points) of which are
In particular, the point corresponding to the rational number g + 0.a1a2···an will lie arbitrarily near to the point P if only n is large enough; for this reason, the points g + 0.a1a2···an are called approximating points. We say that the infinite decimal g + 0.a1a2··· is the real number corresponding to the point P.
Thus, we emphasize the fundamental assumption that we can calculate in the usual way with real numbers, and hence with decimals. It is possible to prove this using only the properties of the integers as a starting-point. But this is no light task and, rather than allowing it to bar our progress at this early stage,we regard the fact that the ordinary rules of calculation apply to the real numbers to be an axiom, on which we shall base all of the differential and integral calculus.
We insert here a remark concerning the possibility arising in certain cases of choosing in the above scheme of expansion the interval in two ways. It follows from our construction that the points of division, arising in our repeated process of sub-division, and such points only can be represented by finite decimals g + 0.a1a2···an. Assume that such a point P first appears as a point of sub-division at the n-th stage of the sub-division. Then, according to the above process, we have chosen at the n-th stage the interval to the right of P. In the following stages, we must choose a sub-interval of this interval. But such an interval must have P as its left end-point. Therefore, in all further stages of the sub-division, we must choose the first sub-interval, which has the number 0. Thus, the infinite decimal corresponding to P is g + 0.a1a2···an000····. If, on the other hand, we had at the n-th stage chosen the left-hand interval containing P, then, at all later stages of sub-division, we should have had to choose the sub-interval furthest to the right, which has P as its right end-point. Such a sub-interval has the number 9. Thus, we should have obtained for P a decimal expansion in which all the digits from the (n + l)-th onwards are nines. The double possibility of choice in our construction therefore corresponds to the fact that, for example, the number ¼ has the two decimal expansions 0.25000··· and 0.24999···.
1.1.3 Expression of Numbers in Scales other than that of 10: In our representation of the real numbers, we gave the number 10 a special role, because each interval was subdivided into ten equal parts. The only reason for this is the widely spread use of the decimal system. We could just as well have taken p equal sub-intervals, where p is an arbitrary integer greater than 1. We should then have obtained an expression of the form
where each b is one of the numbers 0, 1, ··· , p - 1. Here we find again that the rational numbers, and only the rational numbers, have recurring or terminating expansions of this kind. For theoretical purposes, it is often convenient to choose p = 2. We then obtain the binary expansion of the real numbers
where each b is either* 0 or 1.
Even for numerical calculations, the decimal system is not the best. The sexagesimal system, with which the Babylonians calculated, has the advantage that a comparatively large proportion of the rational. numbers, the decimal expansions of which do not terminate, possess terminating sexagesimal expansions.
For numerical calculations, it is customary to express the whole number g, which, for the sake of for simplicity, we assume here to be positive, in the decimal system, that is, in the form
where each an is one of the digits 0, 1, ···, 9. Then, for g + 0.a1a2···, we write simply
Similarly, the positive whole number g can be written in one and only one way in the form
where each of the numbers bn is one of the numbers 0, 1, ··· , p - 1. Together with our previous expression, this yields: Every positive real number can be represented in the form
where bn and bn are whole numbers between 0 and p — 1. Thus, for example, the binary expansion of the fraction 21/4 is
1.1.4 Inequalities: Calculation with inequalities has a far larger role in higher than in elementary mathematics. We shall therefore briefly recall some of the simplest rules concerning them.
If a > b and c > d, then a + c > b + d, but not a - c > b - d. Moreover, if a>b, it follows that ac > bc, provided c is positive. On multiplication by a negative number, the sense of the inequality is reversed. If a>b>0 and c>d>0, it follows that ac>bd.
For the absolute values of numbers, the following inequalities hold:
The square of any real number is larger than or equal to zero, whence, if x and y are arbitrary real numbers
or
1.1.5 Schwarz's Inequality: Let a1, a2, ··· , an and b1, b2, ··· , bn be any real numbers. Substitute in the preceding inequality*
for i = 1, i = 2, ··· , i = n successively and add the resulting inequalities. We obtain on the right hand side the sum 2, because
If we divide both sides of the inequality by 2, we obtain
or, finally,
* Here and hereafter, the symbol Öx, where x > 0, denotes that positive number the square of which is x.
Since the expressions on both sides of this inequality are positive, we may take the square and then omit the modulus signs:
This is the Cauchy-Schwarz inequality.
Exercises 1.1: (more difficult examples are indicated by an asterisk)
1. Prove that the following numbers are
irrational: (a) Ö3, (b) Ön, where n is
not a perfect square, (c) 3Ö3, (d)* x
= Ö2 + 3Ö3, x = Ö3 + 3Ö2.
2*. In an ordinary system of rectangular co-ordinates, the points
for which both co-ordinates are integers axe called lattice
points. Prove that a triangle
the vertices of which are lattice points cannot be equilateral.
3. Prove the inequalities:
4. Show that, if a > 0, ax³+2bx+c³0 for all values of x, if and only if b²-ac£ 0. 5. Prove the inequalities:
6. Prove Schwarz's ineqality by considering the expression
collecting terms and applying Ex.
4.
7. Show that the equality sign in Schwarz's inequality holds if,
and only if, the a's and b's are proportional, that is, can
+ dbn = 0 for all n 's, where c, d are independent of n and not
both zero.
8. For n = 2, 3, state the geometrical interpretation of
Schwarz's inequality.
9. The numbers g1, g2 are direction cosines of a line; that is, g1²
+ g2² = 1. Similarly, h1² + h2²
= 1. Prove that the equation g1h1 + g2h2 = 1 implies the equations g1= h1, g2
= h2.
10. Prove the inequality
and state its geometrical interpretation.
1.2.1 Examples: (a) If an ideal gas is compressed in a vessel by means of a piston, the temperature being kept constant, the pressure p and the volume v are connected by the relation
where C is a constant. This formula - Boyle's Law - states nothing about the quantities v and p themselves; its meaning is: If p has a definite value, arbitrarily chosen in a certain range (the range being determined physically and not mathematically), then v can be determined, and conversely:
We then say that v is a function of p or, in the converse case, that p is a function of v.
(b) If we heat a metal rod, which at temperature 0º has the length lo, to the temperature q °, then its length l will be given, under the simplest physical assumptions, by the law
where b - the coefficient of thermal expansion - is a constant. Again, we say that l is a function of q.
(c) Let there be given in a triangle the lengths of two sides, say a and b. If we choose for the angle g between these two sides any arbitrary value less than 180°, the triangle is completely determined; in particular, the third side c is determined. In this case, we say that if a and b are given, c is a function of the angle g. As we know from trigonometry, this function is represented by the formula
1.2.2 Formulation of the Concept of Function: In order to give a general definition of the mathematical concept of function, we fix upon a definite interval of our number scale, say, the interval between the numbers a and b, and consider the totality of numbers x which belong to this interval, that is, which satisfy the relation
If we consider the symbol x as denoting any of the numbers in this interval, we call it a continuous variable in the interval.
If now there corresponds to each value of x in this interval a single definite value y, where x and y are connected by any law whatsoever, we say that y is a function of x and write symbolically
or some similar expression. We then call x the independent variable and y the dependent variable, or we call x the argument of the function y.
It should be noted that, for certain purposes, it makes a difference whether we include in the interval from a to b the end-points, as we have done above, or exclude them; in the latter case, the variable x is restricted by the inequalities
In order to avoid a misunderstanding, we may call the first kind of interval - including its end-points - a closed interval, the second kind an open interval. If only one end-point and not the other is included, as, for example, in a<x£b, we speak of an interval which is open at one end (in this case the end a). Finally,we may also consider open intervals which extend without bound in one direction or both. We then say that the variable x ranges over an infinite open interval and write symbolically
In the general definition of a function, which is defined in an interval, nothing is said about the nature of the relation, by which the dependent variable is determined when the independent variable is given. This relation may be as complicated as we please and in theoretical investigations this wide generality is an advantage. But in applications and, in particular, in the differential and integral calculus, the functions with which we have to deal are not of the widest generality; on the contrary, the laws of correspondence by which a value of y is assigned to each x are subject to certain simplifying restrictions.
1.2.3 Graphical Representation. Continuity. Monotonic Function: Natural restrictions of the general function concept are suggested when we consider the connection with geometry. In fact, the fundamental idea of analytical geometry is one of giving a curve, defined by some geometrical property, a characteristic analytical representation by regarding one of the rectangular co-ordinates, say y, as a function y = f(x) of the other co-ordinate x; for example, a parabola is represented by the function y = x², the circle with radius 1 about the origin by the two functions y = Ö(l - x²) and y = - Ö(l - x²). In the first example, we may think of the function as defined in the infinite interval -¥<x<¥; in the second example, we must restrict ourselves to the interval - l£x£ l, since outside this interval the function has no meaning (when x and y are real).
Conversely, if
instead of starting with a curve which is determined
geometrically, we consider a given function y = f(x),
we can represent the functional dependence of y on x
graphically by making use of a rectangular co-ordinate system in
the usual way (fig.2). If, for each abscissa x, we mark off the corresponding ordinate y
= f(x), we obtain the geometrical
representation of the function. The restriction which we now wish
to impose on the function concept is: The geometrical representation of the
function shall take the form of a reasonable geometrical curve. It is true that this implies a vague general idea
rather than a strict mathematical condition. But we shall soon
formulate conditions, such as continuity, differentiability, etc., which will ensure that the graph of a function
has the character of a curve capable of being visualized
geometrically. At any rate, we shall exclude a function such as
the following one: For
every rational value of x, the function y has
the value 1, for every irrational value of x, the value
0. This assigns a definite value of y
to each x, but in every interval of x, no
matter how small, the value of y jumps from 0 to 1 and
back an infinite number of times.
Unless the contrary is expressly stated, it will always be assumed that the law, which assigns a value of the function to each value of x, assigns just one value of y to each value of x, as, for example, y = x² or y = sin x. If we begin with a geometrically given curve, it may happen, as in the case of the circle x²+y²=1, that the whole course of the curve is not given by one single (single-valued) function, but requires several functions - in the case of the circle, the two functions y = Ö(l - x²) and y = - Ö(l - x²). The same is true for the hyperbola y²-x²=1, which is represented by the two functions y = Ö(l+x²) and y= -Ö(l+x²). Hence such curves do not determine the corresponding functions uniquely.
Consequently, it is sometimes said that the function corresponding to a curve is multi-valued. The separate functions representing a curve are then called the single-valued branches belonging to the curve. For the sake of clearness, we shall henceforth use the word function to mean a single-valued function. In conformity with this, the symbol Öx (for x ³ 0) will always denote the non-negative number, the square of which is x.
If a curve is the geometrical representation of one function, it will be cut by any parallel to the y-axis in at most one point, since there corresponds to each point x in the interval of definition just one value of y. Otherwise, as, for example, in the case of the circle, represented by the two functions
y = Ö(l - x²) and y = - Ö(l - x²),
such parallels to the y-axis
may intersect the curve in more than one point. The portions of a
curve corresponding to different single-valued branches are
sometimes so interlinked that the complete curve is a single 
figure which can be drawn with one stroke of the pen,
for example, the circle ( Fig. 3), or, on the other hand, the
branches may be completely separated, for example, the hyperbola
(Fig. 4).
Here follow some more examples of the graphical representation of functions.
y is proportional to x. The graph (Fig. 5) is a straight line through the origin of the co-ordinate system.
y is a linear function of x. The graph is a straight line through the point x = 0, y = b, which, if a ¹ 0, also passes through the point x=-b/a, y=0, and, if a=0, runs horizontally.
y is inversely proportional to x. In particular, if a = 1, so that
we find, for example, that
The graph (Fig. 6) is a curve - a rectangular hyperbola, symmetrical with respect to the bi-sectors of the angles between the co-ordinate axes.
This last function is obviously not defined for
the value x = 0, since division by zero has no meaning.
The exceptional point x = 0, in the neighbourhood of
which there occur arbitrarily large values of the function, both
positive and negative, is the simplest example of an infinite
discontinuity, a subject to
which we shall return later.
As is well known, this function is represented by a parabola (Fig. 7).
Similarly, the function y = x³ is represented by the so-called cubical parabola (Fig. 8).
The Curves just considered and their graphs exhibit a property which is of the greatest importance in the discussion of functions, namely, the property of continuity. We shall later analyze this concept in more detail; intuitively speaking, it amounts to: A small change in x causes only a small change in y and not a sudden jump in its value, that is, the graph is not broken off or else the change in y remains less than any arbitrarily chosen positive bound, provided that the change in x is correspondingly small.
A function which for all values of x in an interval has the same value y = a is called a constant; it is graphically represented by a horizontal straight line. A function y = f(x) such that throughout the interval, in which it is defined, an increase in the value of x always causes an increase in the value of y is said to be monotonic increasing; if, on the other hand, an increase in the value of x always causes a decrease in the value of y, the function is said to be monotonic decreasing. Such functions are represented graphically by curves, which in the corresponding interval always rise or always fall (from the left to the right.)( Fig. 9).
If the curve, represented by y = f(x), is symmetrical with respect to the y-axis, that is, if x= - a and x = a give the same value for the function, or
we say that the function is even. For example, the function y = x²
is even (Fig. 7). On the other hand, if the curve is symmetrical with respect to the origin, that is, if
it is an odd function; for example, the functions y = x and y=1/x³ (Fig. 8) and y= 1/x are odd.
1.2.4 Inverse Functions: Even in our first example, it was made evident that a formal relationship between two quantities may be regarded in two different ways, since it is possible either to consider the first variable to be a function of the second or the second one a function of the first vatiable.For example, if y=ax+b, where we assume that a ¹ 0, x is represented as a function of y by the equation x = (y - b)/a. Again, the functional relationship, represented by the equation y =x², can also be represented by the equation x = ± Öy, so that the function y=x² amounts to the same thing as the two functions x=Öy and x=-Öy. Thus, when an arbitrary function y = f(x) is given, we can attempt to determine x as a function of y, or, as we shall say, to replace the function y=f(x) by the inverse function x = f(y).
Geometrically speaking, this has the meaning: We consider the curve obtained by reflecting the graph of y = f(x) in the line bisecting the angle between the positive x-axis and the positive y-axis*(Fig. 10). This gives us at once a graphical representation of x as a function of y and thus represents the inverse function x = f (y).
* Instead of reflecting the graph in this way, we could first rotate the co-ordinate axes and the curve y = f(x) by a right angle and then reflect the graph in the x-axis.
However, these geometrical ideas show at once that a function y = f(x), defined in an interval, has not a single-valued inverse unless certain conditions are satisfied. If the graph of the function is cut in more than one point by a line y=c, parallel to the x-axis, the value y = c will correspond to more than one value of x, so that the function cannot have a single-valued inverse. This case cannot occur if y = f(x) is continuous and monotonic, because then Fig. 10 shows us that there corresponds to each value of y in the interval y1 y y3 just one value of x in the interval x1 x x3, and we infer from the figure that a function which is continuous and monotonic in an interval always has a single-valued inverse, and this inverse function is also continuous and monotonic.
1.3 More Detailed Study of the Elementary Functions
1.3.1 The Rational Functions: We now continue with a brief review of the elementary functions which the reader has already encountered in his previous studies. The simplest types of function are obtained by repeated application of
the elementary operations: Addition, multiplication, subtraction. If we apply these operations to an independent variable x and any real numbers, we obtain the rational, integral functions or polynomials:
The polynomials are the simplest and, in a sense, the basic functions of analysis.
If we now form the quotients of such functions, i.e., expressions of the form
we obtain the general or fractional rational functions, which are defined at all points where the denominator differs from zero.
The simplest rational, integral function is the linear function
It is represented graphically by a straight line. Every quadratic function of the form
is represented by a parabola. The curves which represent rational integral functions of the third degree
are occasionally called parabolas of the third order or cubical parabolas, and so on.
As examples, we have in Fig. 11 above the graphs of the function y = xn for the exponents 1, 2, 3, 4 . We see that for even values of n, the function y= xn satisfies the equation f(- x) = f(x), whence it is an even function, while for odd values of n it satisfies the condition f(- x) = -f(x} and is an odd function.
The simplest example of a rational function which is not a polynomial is y = 1/x (Fig.6); its graph is a rectangular hyperbola. Another example is the function y =1/x² (Fig. 12 above).
1.3.2 The Algebraic Functions: We are at once led away from the domain of the rational
functions by the problem of forming their inverses. The most
important example of this is the introduction of the function 
We start
with the function y = xn, which is
monotonic for x ³ 0. Hence it has a single-valued inverse, which we denote
by the symbol x = 
or, interchanging the letters used for
the dependent and independent variables,
In accordance with its definition, this root is
always non-negative. In the case of odd n, the function xn
is monotonic for all values of x, including negative values,
whence, for odd values of n, we can also define 
uniquely
for all values of x; in this case, is negative for
negative values of x.
More generally, we may consider
where R(x) is a rational function. We arrive at further functions of a similar type by applying rational operations to one or more of these special functions. Thus, for example, we may form the functions
These functions are special cases of algebraic functions. (The general concept of an algebraic function cannot be defined here; cf. Chapter X.)
1.3.3 The Trigonometric Functions: While the rational and algebraic functions just considered are defined directly in terms of the elementary, computational operations, geometry is the source, from which we first draw our knowledge of the other functions, the so-called transcendental functions. We shall here consider the elementary transcendental functions - the trigonometric functions, the exponential function and the logarithm.
The word transcendental does not mean anything particularly deep or mysterious; it merely suggests the fact that the definition of these functions by means of the elementary operations of calculation is not possible, "quod algebrae vires transcendit" (Latin for what exceeds the forces of algebra).
In all higher
analytical investigations, where there occur angles, it is
customary to measure these angles not in degrees, minutes and seconds, but in radians. We place the angle to be measured with its vertex at the centre of a circle of radius 1 and measure the
size of the angle by the length of the arc of the circumference
which the angle cuts out. Thus, an angle of 180° is the same as
an angle of p radians (has radian measure p ), an angle of 90° has radian measure p /2, an
angle of 45° a radian measure p /4), an angle of
360° a radian measure 2p. Conversely, an angle of 1 radian, expressed in
degrees, is
From here on, whenever we speak of an angle x, we shall mean an angle the radian measure of which is x.
After these preliminary remarks, we may briefly remind the reader of the meanings of the trigonometric functions sin x, cos x, tan x, cot x*. These are shown in Fig. 13, in which the angle x is measured from the arm OC (of length 1), angles being positive in the counter-clockwise direction.
At times, it is convenient to introduce the functions sec x = 1/cos x, cosec x = 1/sin x.
The rectangular co-ordinates of the point A yields at once the functions cos x and sin x. The graphs of the functions sin x, cos x, tan x, cot x are given in Figs. 14 and 15.
1.3.4 The Exponential Function and the Logarithm: In addition to the trigonometric functions, the exponential function with the positive base a,
and its inverse, the logarithm to the base a,
are also referred to as elementary transcendental functions. In elementary mathematics, it is customary to
disregard certain inherent difficulties in the definition of
these functions; we too shall postpone the exact discussion of
these functions until we have better methods at our disposal (3.6 et sequ.).
However, we will at least state here the basis of the
definitions. If x = p/q is a rational number
(where p and q are positive integers), then -
assuming the number a to be positive -
we define ax as 
, where the root, by
convention, is to be taken as positive. Since the rational values
of x are everywhere dense, it is natural to extend this
function ax so as to make it into a
continuous function, defined also for irrational values of x,
giving ax values when x is
irrational, which are continuous with the values already defined
when x is rational. This yields a continuous function y
= ax - the exponential function - which for all rational values of x gives the value of ax
found above. Meanwhile, we will take for granted the fact that
this extension is actually possible and can be carried out in
only one way; but it must be kept in mind that we still have to
prove that this is so (A1.2.5 and 3.6.5).
The function
can then be defined for y > 0 as the inverse of the exponential function.
1. Plot the graph of y = x³.
Without further calculation, find from this the graph of 
.
2. Sketch the these graphs and state whether the functions are
even or odd:
3. Sketch the graphs of the following functions and state whether they are (1) monotonic or not, (2) even or odd.
Which two of these functions are identical?
4. A body dropped from rest falls approximately 16 t² ft in t sec. If a ball falls from a window 25 ft. above ground, plot its height above the ground as a function of t for the first 4 sec. after it starts to fall.
1.4. Functions of an Integral Variable. Sequences of Numbers
Hitherto, we have considered the independent variable as a continuous variable, that is, as varying over a complete interval. However, there occur numerous cases in mathematics in which a quantity depends only on an integer, a number n which can take the values 1, 2, 3, ···; it is called a function of an integral variable. This idea will most easily be grasped by means of examples:
Example 1: The sum of the first n integers
is a function of n. Similarly, the sum of the first n squares
is a function * of the integer n.
* This last sum may easily be represented as a simple rational expression in n as follows: We start with the formula
write down this equation for the values n = 0, 1, 2, ··· , n and add them. We thus obtain
on substituting the formula for S1 just given, this becomes
so that
By a similar process, the functions
can be represented as rational functions of n.
Example 2: Other simple functions of integers are the factorials
and the binomial coefficients
for a fixed value of k.
Example 3: Every whole number n > 1, which is not a prime number, is divisible by more than two positive integers, while the prime numbers are only divisible by themselves and by 1. Obviously, we can consider the number T(n) of divisors of n as a function of the number n itself. For the first few numbers, this function is given by
Example 4: A function of this type, which is of great importance in the theory of numbers, is p(n), the number of primes which are less than the number n. Its detailed investigation is one of the most interesting and attractive problems in the theory of numbers. We mention here merely the principal result of these investigations: For large values of n, the number p(n) is given approximately by the function * n/log n, where we mean by log n the logarithm to the natural base e, to be defined later on.
* That is, the quotient of the number p (n) by the number n/log n differs arbitrarily little from 1, provided only that n is large enough.
As a rule, functions of an integral variable occur in the form of so-called sequences of numbers. By a sequence of numbers, we understand an ordered array of infinitely many numbers a1, a2, a3, ··· , an, ··· (not necessarily all different), determined by any law whatsoever. In other words, we are dealing simply with a function a of the integral variable n, the only difference being that we are using the subscript notation an instead of the symbol a(n).
1. Prove that
2. From the formula for
find a formula for
3. Prove the following properties of the binomial coefficients:
4. Evaluate the sums:
5. A sequence is called an arithmetic progression of the first order, if the differences of successive terms are constant, an arithmetic progression of the second order, if the differences of successive terms form an arithmetic progression of the first order and, in general, an arithmetic progression of order k, if the differences of successive terms form an arithmetic progression of order (k - 1).
The numbers 4, 6, 13, 27, 50, 84 are the first six terms of an arithmetic progression. What is its order? What is the eighth term?
6. Prove that the n-th term of an arithmetic progression of the second order can be written in the form an² + bn + c, where a, b, c are independent of n.
7*. Prove that the n-th term of an arithmetic progression of order k can be written in the form
where a, b, ··· , p, q
are independent of n.
Find the n-th term of the progression in 6.
1.5 The Concept of the Limit of a Sequence
Th concept on which the whole of analysis ultimately rests is that of the limit of a sequence. We shall first make the position clear by considering several examples.
1.5.1 an = 1/n: We consider the sequence
No number of this sequence is zero; but we see that the larger the number n, the closer to zero is the number an. Hence, if we mark off around the point 0 an interval as small as we please, then starting with a definite value of the subscript all the numbers an will fall into this interval. We express this state of affairs by saying that, as n increases, the numbers an tend to 0, or that they possess the limit 0, or that the sequence a1, a2, a3, ··· converges to 0.
If numbers are represented as points on a line, this means that the points l/n crowd closer and closer to the point 0 as n increases.
The situation is similar in the case of the sequence
Here too, the numbers an tend to zero as n increases; the only difference is that the numbers an are sometimes larger and sometimes smaller than the limit 0; as we say, they oscillate about the limit.
The convergence of the sequence to 0 is usually expressed symbolically by the equation
or occasionally by the abbreviation
1.5.2 a2m = 1/m; a2m-1 = 1/2m: In the preceding examples, the absolute value of the difference between an and the limit steadily becomes smaller as n increases. This is not necessarily always the case, as is shown by the sequence
that is, in general, for even values n = 2m, an = a2m, = 1/m, for odd values n = 2m - 1, an=a2m-1= l. This sequence also has a limit, namely zero, since every interval about the origin, no matter how small, will contain all the numbers an from a certain value of n onwards; but it is not true that every number lies nearer to the limit zero than the preceding one.
1.5.3 an = n/(n + 1): We consider the sequence
where the integral subscript n takes all the values 1, 2, 3, ··· . If we write an = 1 - 1/(n + 1), we see at once that, as n increases, the numbers an will approach closer and closer to the number 1, in the sense that, if we mark off any interval about the point 1, all the numbers an following a certain aN must fall into that interval. We write
The sequence
behaves in a similar manner. This sequence also
tends to a limit as n increases to the limit 1, in fact,
in symbols, 
We see this most readily, if we write
now we have only to show that the numbers rn tend to 0 as n increases. For all values of n greater than 2, we have n + 3 < 2n and n² + n + 1 > n². Hence we have for the remainder rn
which shows at once that rn tends to 0 as n increases. Our discussion yields at the same time an estimate of the amount by which the number an (for n > 2) can at most differ from the limit 1; this difference certainly cannot exceed 2/n.
This example illustrates the fact, which we should naturally expect, that for large values of n the terms with the highest indices in the numerator and denominator of the fraction for an predominate and that they determine the limit.
1.5.4 
: Let
p be any fixed positive number. We consider the sequence
a1, a2, a3,
··· , an, ···, where
We assert that
We can prove this very easily by using a lemma which we shall find also useful for other purposes.
If 1 + h is a positive number (that is, if h > - l) and n is an integer greater than 1, then
Assume that Inequality (1) already has been proved for a certain m > 1; multiply both sides by (1+h) and obtain
If we omit on the right hand side the positive term mh², the inequality remains valid. We thus obtain
However, this is our inequality for the index m + 1. Hence, if the inequality holds for the index m, is also holds for m + 1. Since it holds for m= 2, it also holds for m = 3, whence for m = 4, and so on, whence it holds for every index. This is a simple example of a proof by mathematical induction, a type of proof which is often useful.
Returning to our sequence, we distinguish
between the case p > 1 and the case p < 1
( if p = 1, then 
is also equal to 1 for every n
and our statement becomes trivial).
If p > 1, then will also be greater
than 1; let = 1 + hn, where hn
is a positive quantity depending on n and we find by
Inequality (1)
whence follows at once that
Thus, as n increases, the number hn must tend to 0, which proves that the numbers an converge to the limit 1, as stated. At the same time, we have a means for estimating how close any an is to the limit 1; the difference between an and 1 is certainly not greater than (p — l)/n.
If p < 1, then will likewise be less
than 1 and therefore may be taken equal to l/(l + hn),
where hn is a positive number. It
follows from this, using Inequality (1), that
(By decreasing the denominator, we increase the fraction. It follows that
whence
This shows that hn
tends to 0 as n increases. As the reciprocal of a
quantity tending to 1, itself tends to 1.
We consider the sequence an = a n, where a is fixed and n runs through the sequence of positive integers.
First, let a be a positive number less than 1. We may then put a = l/(l + h), where h is positive and Inequality (1) yields
Since the number h and, consequently, 1/h depends only on n and does not change as n increases, we see that as n increases a n tends to 0:
The same relationship holds when a is zero or negative, but greater than - 1. This is immediately obvious, since in any case
If a = 1, then a n will obviously be always equal to 1 and we shall have to regard the number 1 as the limit of a n.
If a > 1, we set a = 1 + h, where h is positive and see at once from our inequality that, as n increases, a n does not tend to any definite limit, but increases beyond all bounds. We express this state of affairs by saying that a n tends to infinity as n increases, or that a n becomes infinite; in symbols
Nevertheless, as we must explicitly emphasize, the symbol ¥ does not denote a number with which we can calculate as with any other number; equations or statements which express that a quantity is or becomes infinite never have the same sense as an equation between definite quantities. In spite of this, such modes of expression and the use of the symbol ¥ are extremely convenient, as we shall often see in the following pages.
If a = - 1, the values of a n will not tend to any limit, but, as n runs through the sequence of positive integers, it will take the values +1 and - 1 alternately. Similarly, if a < -1, the value of a n will increase numerically beyond all bounds, but its sign will be in sequence positive and. negative.
1.5.6. Geometrical Illustration
of the Limits of an
and :
If we consider the curves y
= xn and 
and restrict ourselves, for the sake of convenience,
to non-negative values of x, the preceding limits are
illustrated by Figs. 16 and. 17, respectively. In the case
of the curves y = xn, we see that in the interval 0 to 1 they approach
closer and closer to the x-axis as n
increases, while outside that interval they climb more and more
steeply and draw in closer and closer to a line parallel to the y-axis.
All the curves pass through the point with co-ordinates x=1,
y=1 and through the origin.
In the case of the functions 
the
curves approach closer and closer to the line parallel to the x-axis
and at a distance 1 above it. On the other hand, all the curves
must pass through the origin. Hence, in the limit, the curves
approach the broken line consisting of the part of the y-axis
between the points y = 0 and y = 1 and of
the parallel to the x-axis y = 1. Moreover, it is clear
that the two figures are closely related, as one would expect
from the fact that the functions 
are actually the
inverse functions of the n-th powers, from which we
infer that each figure is transformed into the other on
reflection in the line y = x.
1.5.7 The Geometric Series: An example of a limit which is more or less familiar from elementary mathematics is the geometric series
the number q is called the common ratio of the series. The value of this sum may, as is well known, be expressed in. the form
provided that q ¹ 1; we can derive this expression by multiplying the sum Sn by q and subtracting the equation thus obtained from the original equation, or we may verify the formula by division.
There arises now the question what happens to the sum Sn when n. increases indefinitely? The answer is: The sum Sn has a definite limit S if q lies between -1 and +1, these end values being excluded, and it is then true that
In order to verify this statement, we write the numbers Sn in the form
We have already shown that, provided |q| < 1, the quantity qn and with it qn/(1 - q) tends to 0 as n increases; hence, with the above assumption, the number Sn tends, as was stated, to the limit 1/(1 - q) as n increases.
The passage to the limit
is usually expressed by saying that when |q| < 1, the geometric series can be extended to infinity and that the sum of the infinite geometric series is the expression 1/(1 - q).
The sums Sn of the finite geometric series are also called the partial sums of the infinite geometric series 1 + q + q² + ···. (We must draw a sharp distinction between the sequence of numbers S1,S2,···, and the geometric series.)
The fact that the partial sums Sn of a geometric series tend to the limit S = 1/(1 - q) as n increases may also be expressed by saying that the infinite geometric series 1 + q + q² +··· converges to the sum S = 1/(1 - q) when |q| < 1.
1.5.8 
:
We shall show that the sequence of numbers
tends to 1 as n increases, i.e., that
Here we make use of a slight
artifice. Instead of the sequence 
, we first consider the
sequence 
.
When n > 1, the term bn
is also greater than 1. We can therefore set bn
= 1+ hn, where hn
is positive and depends on n. By Inequality (l), we
therefore have
so that
We now have
Obviously, the right hand side of this inequality tends to 1, and so does an.
We assert that
In order to prove this formula, we need only write this expression in the form
we see at once that this expression tends to 0 as n increases.
1.5.10 
: Let
a be a number greater than 1. We assert that as n
increases the sequence of the numbers an=n/an
tends to the limit 0.
As in the case of 
above, we consider the
sequence
We set 
Here h > 0,
since a and hence Öa is greater than 1. By Inequality (1), we have
so that
Hence
1.Prove that
Find an N such that for n > N
the difference between 
is (a) less than 1/10, (b)
less than 1/1,000, (c) lees than 1/1,000,000.
2. Find the limits of the following expressions as n ® ¥:
3. Prove that 
4. Prove that
Find an n such that n²/2n
< 1/1,000 whenever n > N.
5. Find numbers N1, N2, N3 such that
6. Do the same thing for the sequence 
7. Prove that 
8. Prove that 
9. Let an = 10n/n!.
(a) To what limit does an
converge?
(b) Is the sequence monotonic?
(c) Is it monotonic from a certain n onwards?
(d) Give an estimate of the difference between an
and the limit.
(e) From what value of n onwards is this difference less
than 1/100?
10. Prove that 
11. Prove that 
12. Prove that 
13. Prove that 
14.* Prove that 
15. Prove that if a and b £ a are positive, the sequence 
converges to a. Similarly, for any k fixed
positive numbers a1, a2,
··· , ak, prove that 
converges and find its limit.
16. Prove that the sequence 
converges. Find its limit.
17.* If n(n) is the number
of distinct prime factors of n, prove that 
