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1 Algebraic and Analytic MethodsTopics of Discussion

§1.4 Calculus of One Variable

  1. §1.4(i) Monotonicity
  2. §1.4(ii) Continuity
  3. §1.4(iii) Derivatives
  4. §1.4(iv) Indefinite Integrals
  5. §1.4(v) Definite Integrals
  6. §1.4(vi) Taylor’s Theorem for Real Variables
  7. §1.4(vii) Maxima and Minima
  8. §1.4(viii) Convex Functions

§1.4(i) Monotonicity

If f(x1)f(x2) for every pair x1, x2 in an interval I such that x1<x2, then f(x) is nondecreasing on I. If the sign is replaced by <, then f(x) is increasing (also called strictly increasing) on I. Similarly for nonincreasing and decreasing (strictly decreasing) functions. Each of the preceding four cases is classified as monotonic; sometimes strictly monotonic is used for the strictly increasing or strictly decreasing cases.

§1.4(ii) Continuity

A function f(x) is continuous on the right (or from above) at x=c if

1.4.1 f(c+)limxc+f(x)=f(c),

that is, for every arbitrarily small positive constant ϵ there exists δ (>0) such that

1.4.2 |f(c+α)f(c)|<ϵ,

for all α such that 0α<δ. Similarly, it is continuous on the left (or from below) at x=c if

1.4.3 f(c)limxcf(x)=f(c).

And f(x) is continuous at c when both (1.4.1) and (1.4.3) apply.

If f(x) is continuous at each point c(a,b), then f(x) is continuous on the interval (a,b) and we write fC(a,b). If also f(x) is continuous on the right at x=a, and continuous on the left at x=b, then f(x) is continuous on the interval [a,b], and we write f(x)C[a,b].

A removable singularity of f(x) at x=c occurs when f(c+)=f(c) but f(c) is undefined. For example, f(x)=(sinx)/x with c=0.

A simple discontinuity of f(x) at x=c occurs when f(c+) and f(c) exist, but f(c+)f(c). If f(x) is continuous on an interval I save for a finite number of simple discontinuities, then f(x) is piecewise (or sectionally) continuous on I. For an example, see Figure 1.4.1

See accompanying text
Figure 1.4.1: Piecewise continuous function on [a,b). Magnify

§1.4(iii) Derivatives

The derivative f(x) of f(x) is defined by

1.4.4 f(x)=dfdx=limh0f(x+h)f(x)h.

When this limit exists f is differentiable at x.

1.4.5 (f+g)(x)=f(x)+g(x),
1.4.6 (fg)(x)=f(x)g(x)+f(x)g(x),
1.4.7 (fg)(x)=f(x)g(x)f(x)g(x)(g(x))2.

Higher Derivatives

1.4.8 f(2)(x)=d2fdx2=ddx(dfdx),
1.4.9 f(n)=f(n)(x)=ddxf(n1)(x).

If f(n) exists and is continuous on an interval I, then we write fCn(I). When n1, f is continuously differentiable on I. When n is unbounded, f is infinitely differentiable on I and we write fC(I).

Chain Rule

For h(x)=f(g(x)),

1.4.10 h(x)=f(g(x))g(x).

Maxima and Minima

A necessary condition that a differentiable function f(x) has a local maximum (minimum) at x=c, that is, f(x)f(c), (f(x)f(c)) in a neighborhood cδxc+δ (δ>0) of c, is f(c)=0.

Mean Value Theorem

If f(x) is continuous on [a,b] and differentiable on (a,b), then there exists a point c(a,b) such that

1.4.11 f(b)f(a)=(ba)f(c).

If f(x)0 (0) (=0) for all x(a,b), then f is nondecreasing (nonincreasing) (constant) on (a,b).

Leibniz’s Formula

1.4.12 (fg)(n)=f(n)g+(n1)f(n1)g++(nk)f(nk)g(k)++fg(n).

Faà Di Bruno’s Formula

1.4.13 dndxnf(g(x))=(n!m1!m2!mn!)f(k)(g(x))(g(x)1!)m1(g′′(x)2!)m2(g(n)(x)n!)mn,

where the sum is over all nonnegative integers m1,m2,,mn that satisfy m1+2m2++nmn=n, and k=m1+m2++mn.

L’Hôpital’s Rule


1.4.14 limxaf(x)=limxag(x)=0(or ),


1.4.15 limxaf(x)g(x)=limxaf(x)g(x)

when the last limit exists. We do assume that g(x)0 for all x in some neighborhood of a with xa.

§1.4(iv) Indefinite Integrals

If F(x)=f(x), then fdx=F(x)+C, where C is a constant.

Integration by Parts

1.4.16 fgdx=(fdx)g(fdx)dgdxdx.
1.4.17 xndx={xn+1n+1+C,n1,ln|x|+C,n=1.

For the function ln see §4.2(i).

See §§4.10, 4.26(ii), 4.26(iv), 4.40(ii), and 4.40(iv) for indefinite integrals involving the elementary functions.

For extensive tables of integrals, see Apelblat (1983), Bierens de Haan (1867), Gradshteyn and Ryzhik (2000), Gröbner and Hofreiter (1949, 1950), and Prudnikov et al. (1986a, b, 1990, 1992a, 1992b).

§1.4(v) Definite Integrals

Riemann Integrals

Suppose f(x) is defined on [a,b]. Let a=x0<x1<<xn=b, and ξj denote any point in [xj,xj+1], j=0,1,,n1. Then

1.4.18 abf(x)dx=limj=0n1f(ξj)(xj+1xj)

as max(xj+1xj)0. If the limit exists then f is called Riemann integrable. Continuity, or piecewise continuity, of f(x) on [a,b] is sufficient for the limit to exist.

1.4.19 ab(cf(x)+dg(x))dx=cabf(x)dx+dabg(x)dx,

c and d constants.

1.4.20 abf(x)dx=baf(x)dx.
1.4.21 abf(x)dx=acf(x)dx+cbf(x)dx.

Infinite Integrals

1.4.22 af(x)dx=limbabf(x)dx.

Similarly for a. Next, if f(b)=±, then

1.4.23 abf(x)dx=limcbacf(x)dx.

Similarly when f(a)=±.

When the limits in (1.4.22) and (1.4.23) exist, the integrals are said to be convergent. If the limits exist with f(x) replaced by |f(x)|, then the integrals are absolutely convergent. Absolute convergence also implies convergence.

Stieltjes, Lebesgue, and Lebesgue–Stieltjes integrals

A generalization of the Riemann integral is the Stieltjes integral abf(x)dα(x), where α(x) is a nondecreasing function on the closure of (a,b), which may be bounded, or unbounded, and dα(x) is the Stieltjes measure. See Riesz and Sz.-Nagy (1990, Ch. 3). Stieltjes integrability for f with respect to α can be defined similarly as Riemann integrability in the case that α(x) is differentiable with respect to x; a generalization follows below.

For the functions discussed in the following DLMF chapters these two integration measures are adequate, as these special functions are analytic functions of their variables, and thus C, and well defined for all values of these variables; possible exceptions being at boundary points.

A more general concept of integrability of a function on a bounded or unbounded interval is Lebesgue integrability, which allows discussion of functions which may not be well defined everywhere (especially on sets of measure zero) for x. see Rudin (1966), and often used in more abstract mathematical discussions. Similarly the Stieltjes integral can be generalized to a Lebesgue–Stieltjes integral with respect to the Lebesgue-Stieltjes measure dμ(x) and it is well defined for functions f which are integrable with respect to that more general measure. See McDonald and Weiss (1999).

Absolutely Continuous Stieltjes Measure

For α(x) nondecreasing on the closure I of an interval (a,b), the measure dα is absolutely continuous if α(x) is continuous and there exists a weight function w(x)0, Riemann (or Lebesgue) integrable on finite subintervals of I, such that

1.4.23_1 α(d)α(c)=cdw(x)dx,


1.4.23_2 abf(x)dα(x)=abf(x)w(x)dx,
f integrable with respect to dα.

In particular, absolute continuity occurs if the function α(x) is differentiable, α(x)=w(x) with w(x) continuous.

For historical reasons, w(x) is also sometimes referred to as a density, as, for example, the mass per unit length at point x, see Shohat and Tamarkin (1970, p vii).

Stieltjes Measure with α(x) Discontinuous

The utility of the generalization implicit in the Stieltjes measure appears when α(x) is not everywhere continuous, but has discontinuous jumps at specific values of x, say xn(a,b). See Riesz and Sz.-Nagy (1990, Ch. 3). If, for example, α(x)=H(xxn), the Heaviside unit step-function (1.16.14), then the corresponding measure dα(x) is δ(xxn)dx, where δ(xxn) is the Dirac δ-function of §1.17, such that, for f(x) a continuous function on (a,b), abf(x)dα(x)=f(xn) for xn(a,b) and 0 otherwise. Delta distributions and Dirac δ-functions are discussed in §§1.16(iii), 1.16(iv) and 1.17.

Definite integrals over the Stieltjes measure dα(x) could represent a sum, an integral, or a combination of the two. Let dα(x)=w(x)dx+n=1Nwnδ(xxn)dx, xn(a,b), n=1,N. Then for f(x) continuous on (a,b),

1.4.23_3 abf(x)dα(x)=abw(x)f(x)dx+n=1Nwnf(xn).

In the literature where w(x) is considered to be a mass density, the xn are often referred to as mass points, wn being the mass at that point. Ismail (2005, p 5) refers to these xn as isolated atoms.

Cauchy Principal Values

Let c(a,b) and assume that acϵf(x)dx and c+ϵbf(x)dx exist when 0<ϵ<min(ca,bc), but not necessarily when ϵ=0. Then we define

1.4.24 abf(x)dx=𝑃abf(x)dx=limϵ0+(acϵf(x)dx+c+ϵbf(x)dx),

when this limit exists.

Similarly, assume that bbf(x)dx exists for all finite values of b (>0), but not necessarily when b=. Then we define

1.4.25 f(x)dx=𝑃f(x)dx=limbbbf(x)dx,

when this limit exists.

Fundamental Theorem of Calculus

For F(x)=f(x) with f(x) continuous,

1.4.26 abf(x)dx=F(b)F(a),
1.4.27 ddxaxf(t)dt=f(x).

Change of Variables

If ϕ(x) is continuous or piecewise continuous, then

1.4.28 abf(ϕ(x))ϕ(x)dx=ϕ(a)ϕ(b)f(t)dt.

First Mean Value Theorem

For f(x) continuous and ϕ(x)0 and integrable on [a,b], there exists c[a,b], such that

1.4.29 abf(x)ϕ(x)dx=f(c)abϕ(x)dx.

Second Mean Value Theorem

For f(x) monotonic and ϕ(x) integrable on [a,b], there exists c[a,b], such that

1.4.30 abf(x)ϕ(x)dx=f(a)acϕ(x)dx+f(b)cbϕ(x)dx.

Repeated Integrals

If f(x) is continuous or piecewise continuous on [a,b], then

1.4.31 abdxnaxndxn1ax2dx1ax1f(x)dx=1n!ab(bx)nf(x)dx.

Square-Integrable Functions

A function f(x) is square-integrable if

1.4.32 f22ab|f(x)|2dx<.

Functions of Bounded Variation

With a<b, the total variation of f(x) on a finite or infinite interval (a,b) is

1.4.33 𝒱a,b(f)=supj=1n|f(xj)f(xj1)|,

where the supremum is over all sets of points x0<x1<<xn in the closure of (a,b), that is, (a,b) with a,b added when they are finite. If 𝒱a,b(f)<, then f(x) is of bounded variation on (a,b). In this case, g(x)=𝒱a,x(f) and h(x)=𝒱a,x(f)f(x) are nondecreasing bounded functions and f(x)=g(x)h(x).

If f(x) is continuous on the closure of (a,b) and f(x) is continuous on (a,b), then

1.4.34 𝒱a,b(f)=ab|f(x)|dx,

whenever this integral exists.

Lastly, whether or not the real numbers a and b satisfy a<b, and whether or not they are finite, we define 𝒱a,b(f) by (1.4.34) whenever this integral exists. This definition also applies when f(x) is a complex function of the real variable x. For further information on total variation see Olver (1997b, pp. 27–29).

§1.4(vi) Taylor’s Theorem for Real Variables

If f(x)Cn+1[a,b], then

1.4.35 f(x)=k=0nf(k)(a)k!(xa)k+Rn,
1.4.36 Rn=f(n+1)(c)(n+1)!(xa)n+1,


1.4.37 Rn=1n!ax(xt)nf(n+1)(t)dt.

§1.4(vii) Maxima and Minima

If f(x) is twice-differentiable, and if also f(x0)=0 and f′′(x0)<0 (>0), then x=x0 is a local maximum (minimum) (§1.4(iii)) of f(x). The overall maximum (minimum) of f(x) on [a,b] will either be at a local maximum (minimum) or at one of the end points a or b.

§1.4(viii) Convex Functions

A function f(x) is convex on (a,b) if

1.4.38 f((1t)c+td)(1t)f(c)+tf(d)

for any c,d(a,b), and t[0,1]. See Figure 1.4.2. A similar definition applies to closed intervals [a,b].

If f(x) is twice differentiable, then f(x) is convex iff f′′(x)0 on (a,b). A continuously differentiable function is convex iff the curve does not lie below its tangent at any point.

See accompanying text
Figure 1.4.2: Convex function f(x). g(t)=f((1t)c+td), l(t)=(1t)f(c)+tf(d), c,d(a,b), 0t1. Magnify