13. Analytic Function.—If ƒ(x) and its first n differential
coefficients, denoted by ƒ′(x), ƒ″(x), ... ƒ(n) (x), are continuous
in the interval between a and a + h, then
| ƒ(a + h) = ƒ(a) + hƒ′(a) + | h2 | ƒ″(a) + ... + | hn−1 | ƒ(n−1)(a) + Rn, |
| 2! | (n − 1)! |
where Rn may have various forms, some of which are given in the article Infinitesimal Calculus. This result is known as “Taylor’s theorem.”
When Taylor’s theorem leads to a representation of the function by means of an infinite series, the function is said to be “analytic” (cf. § 21).
14. Ordinary Function.—The idea of a curve representing a continuous function in an interval is that of a line which has the following properties: (1) the co-ordinates of a point of the curve are a value x of the argument and the corresponding value y of the function; (2) at every point the curve has a definite tangent; (3) the interval can be divided into a finite number of partial intervals within each of which the function is monotonous; (4) the property of monotony within partial intervals is retained after interchange of the axes of co-ordinates x and y. According to condition (2) y is a continuous and differentiable function of x, but this condition does not include conditions (3) and (4): there are continuous partially monotonous functions which are not differentiable, there are continuous differentiable functions which are not monotonous in any interval however small; and there are continuous, differentiable and monotonous functions which do not satisfy condition (4) (cf. § 24). A function which can be represented by a curve, in the sense explained above, is said to be “ordinary,” and the curve is the graph of the function (§2). All analytic functions are ordinary, but not all ordinary functions are analytic.
15. Integrable Function.—The idea of integration is twofold. We may seek the function which has a given function as its differential coefficient, or we may generalize the question of finding the area of a curve. The first inquiry leads directly to the indefinite integral, the second directly to the definite integral. Following the second method we define “the definite integral of the function ƒ(x) through the interval between a and b” to be the limit of the sum
when the interval is divided into ultimately indefinitely small partial intervals by points x1, x2, ... xn−1. Here x′r denotes any point in the rth partial interval, x0 is put for a, and xn for b. It can be shown that the limit in question is finite and independent of the mode of division into partial intervals, and of the choice of the points such as x′r , provided (1) the function is defined for all points of the interval, and does not tend to become infinite at any of them; (2) for any one mode of division of the interval into ultimately indefinitely small partial intervals, the sum of the products of the oscillation of the function in each partial interval and the difference of the end-values of that partial interval has limit zero when n is increased indefinitely. When these conditions are satisfied the function is said to be “integrable” in the interval. The numbers a and b which limit the interval are usually called the “lower and upper limits.” We shall call them the “nearer and further end-values.” The above definition of integration was introduced by Riemann in his memoir on trigonometric series (1854). A still more general definition has been given by Lebesgue. As the more general definition cannot be made intelligible without the introduction of some rather recondite notions belonging to the theory of aggregates, we shall, in what follows, adhere to Riemann’s definition.
We have the following theorems:—
1. Any continuous function is integrable.
2. Any function with restricted oscillation is integrable.
3. A discontinuous function is integrable if it does not tend to become infinite, and if the points at which the oscillation of the function exceeds a given number σ, however small, can be enclosed in partial intervals the sum of whose breadths can be diminished indefinitely.
These partial intervals must be a set chosen out of some complete set obtained by the process used in the definition of integration.
4. The sum or product of two integrable functions is integrable.
As regards integrable functions we have the following theorems:
1. If S and I are the superior and inferior limits (or greatest and least values) of ƒ(x) in the interval between a and b, ∫ ba ƒ(x)dx is intermediate between S(b − a) and I(b − a).
2. The integral is a continuous function of each of the end-values.
3. If the further end-value b is variable, and if ∫ xa ƒ(x)dx = F(x), then if ƒ(x) is continuous at b, F(x) is differentiable at b, and F′(b) = ƒ(b).
4. In case ƒ(x) is continuous throughout the interval F(x) is continuous and differentiable throughout the interval, and F′(x) = ƒ(x) throughout the interval.
5. In case ƒ′(x) is continuous throughout the interval between a and b,
6. In case ƒ(x) is discontinuous at one or more points of the interval between a and b, in which it is integrable,
is a function of x, of which the four derivates at any point of the interval are equal to the limits of indefiniteness of ƒ(x) at the point.
7. It may be that there exist functions which are differentiable throughout an interval in which their differential coefficients are not integrable; if, however, F(x) is a function whose differential coefficient, F′(x), is integrable in an interval, then
where a is a fixed point, and x a variable point, of the interval. Similarly, if any one of the four derivates of a function is integrable in an interval, all are integrable, and the integral of either differs from the original function by a constant only.
The theorems (4), (6), (7) show that there is some discrepancy between the indefinite integral considered as the function which has a given function as its differential coefficient, and as a definite integral with a variable end-value.
We have also two theorems concerning the integral of the product of two integrable functions ƒ(x) and φ(x); these are known as “the first and second theorems of the mean.” The first theorem of the mean is that, if φ(x) is one-signed throughout the interval between a and b, there is a number M intermediate between the superior and inferior limits, or greatest and least values, of ƒ(x) in the interval, which has the property expressed by the equation
The second theorem of the mean is that, if ƒ(x) is monotonous throughout the interval, there is a number ξ between a and b which has the property expressed by the equation
(See Fourier’s Series.)
16. Improper Definite Integrals.—We may extend the idea of integration to cases of functions which are not defined at some point, or which tend to become infinite in the neighbourhood of some point, and to cases where the domain of the argument extends to infinite values. If c is a point in the interval between a and b at which ƒ(x) is not defined, we impose a restriction on the points x′r of the definition: none of them is to be the point c. This comes to the same thing as defining ƒ(x)dx to be
where, to fix ideas, b is taken > a, and ε and ε′ are positive. The same definition applies to the case where ƒ(x) becomes infinite, or tends to become infinite, at c, provided both the limits exist. This definition may be otherwise expressed by saying that a partial interval containing the point c is omitted from the interval of integration, and a limit taken by diminishing the breadth of this partial interval indefinitely; in this form it applies to the cases where c is a or b.
Again, when the interval of integration is unlimited to the right, or extends to positively infinite values, we have as a definition
