§1.4 Calculus of One Variable

If $f^{(n)}$ exists and is continuous on an interval $I$, then we write $f\in C^n(I)$. When $n\ge 1$, $f$ is continuously differentiable on $I$. When $n$ is unbounded, $f$ is infinitely differentiable on $I$ and we write $f\in C^{\mathrm{\infty }}(I)$.

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

A necessary condition that a differentiable function $f(x)$ has a local maximum (minimum) at $x=c$, that is, $f(x)\le f(c)$, ($f(x)\ge f(c)$) in a neighborhood $c-\delta \le x\le c+\delta$ ($\delta >0$) of $c$, is $f^{\prime }(c)=0$.

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

If $f^{\prime }(x)\ge 0$ ($\le 0$) ($=0$) for all $x\in (a,b)$, then $f$ is nondecreasing (nonincreasing) (constant) on $(a,b)$.

where the sum is over all nonnegative integers $m_1,m_2,\mathrm{\dots },m_n$ that satisfy $m_1+2m_2+\mathrm{\cdots }+n{m}_n=n$, and $k=m_1+m_2+\mathrm{\cdots }+m_n$.

If

then

when the last limit exists. We do assume that $g^{\prime }(x)\ne 0$ for all $x$ in some neighborhood of $a$ with $x\ne a$.

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 (2015), Gröbner and Hofreiter (1949, 1950), and Prudnikov et al. (1986a, b, 1990, 1992a, 1992b).

Suppose $f(x)$ is defined on $[a,b]$. Let , and ${\xi }_j$ denote any point in $[x_j,x_{j+1}]$, $j=0,1,\mathrm{\dots },n-1$. Then

as $\max (x_{j+1}-x_j)\to 0$. 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.

$c$ and $d$ constants.

Similarly for ${\int }_{-\mathrm{\infty }}^a$. Next, if $f(b)=\pm \mathrm{\infty }$, then

Similarly when $f(a)=\pm \mathrm{\infty }$.

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 $\left|f(x)\right|$, then the integrals are absolutely convergent. Absolute convergence also implies convergence.

A generalization of the Riemann integral is the Stieltjes integral ${\int }_a^{b}f(x)d\alpha (x)$, where $\alpha (x)$ is a nondecreasing function on the closure of $(a,b)$, which may be bounded, or unbounded, and $d\alpha (x)$ is the Stieltjes measure. See Riesz and Sz.-Nagy (1990, Ch. 3). Stieltjes integrability for $f$ with respect to $\alpha$ can be defined similarly as Riemann integrability in the case that $\alpha (x)$ 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 $C^{\mathrm{\infty }}$, 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 $x\in ℝ$. 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 $d\mu (x)$ and it is well defined for functions $f$ which are integrable with respect to that more general measure. See McDonald and Weiss (1999).

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

Then

In particular, absolute continuity occurs if the function $\alpha (x)$ is differentiable, ${\alpha }^{\prime }(x)=w(x)$ 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).

The utility of the generalization implicit in the Stieltjes measure appears when $\alpha (x)$ is not everywhere continuous, but has discontinuous jumps at specific values of $x$, say $x_n\in (a,b)$. See Riesz and Sz.-Nagy (1990, Ch. 3). If, for example, $\alpha (x)=H\left(x-x_n\right)$, the Heaviside unit step-function (1.16.14), then the corresponding measure $d\alpha (x)$ is $\delta \left(x-x_n\right)dx$, where $\delta \left(x-x_n\right)$ is the Dirac $\delta$-function of §1.17, such that, for $f(x)$ a continuous function on $(a,b)$, ${\int }_a^{b}f(x)d\alpha (x)=f(x_n)$ for $x_n\in (a,b)$ and $0$ otherwise. Delta distributions and Dirac $\delta$-functions are discussed in §§1.16(iii), 1.16(iv) and 1.17.

Definite integrals over the Stieltjes measure $d\alpha (x)$ could represent a sum, an integral, or a combination of the two. Let $d\alpha (x)=w(x)dx+{\sum }_{n=1}^N{w}_n\delta \left(x-x_n\right)dx$, $x_n\in (a,b)$, $n=1,\mathrm{\dots }N$. Then for $f(x)$ continuous on $(a,b)$,

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

Let $c\in (a,b)$ and assume that ${\int }_a^{c-ϵ}f(x)dx$ and ${\int }_{c+ϵ}^{b}f(x)dx$ exist when , but not necessarily when $ϵ=0$. Then we define

when this limit exists.

Similarly, assume that ${\int }_{-b}^{b}f(x)dx$ exists for all finite values of $b$ ($>0$), but not necessarily when $b=\mathrm{\infty }$. Then we define

when this limit exists.

For $F^{\prime }(x)=f(x)$ with $f(x)$ continuous,

If ${\varphi }^{\prime }(x)$ is continuous or piecewise continuous, then

For $f(x)$ continuous and $\varphi (x)\ge 0$ and integrable on $[a,b]$, there exists $c\in [a,b]$, such that

For $f(x)$ monotonic and $\varphi (x)$ integrable on $[a,b]$, there exists $c\in [a,b]$, such that

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

A function $f(x)$ is square-integrable if

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

where the supremum is over all sets of points in the closure of $(a,b)$, that is, $(a,b)$ with $a,b$ added when they are finite. If , then $f(x)$ is of bounded variation on $(a,b)$. In this case, $g(x)={𝒱}_{a,x}\left(f\right)$ and $h(x)={𝒱}_{a,x}\left(f\right)-f(x)$ are nondecreasing bounded functions and $f(x)=g(x)-h(x)$.

If $f(x)$ is continuous on the closure of $(a,b)$ and $f^{\prime }(x)$ is continuous on $(a,b)$, then

whenever this integral exists.

Lastly, whether or not the real numbers $a$ and $b$ satisfy , and whether or not they are finite, we define ${𝒱}_{a,b}\left(f\right)$ 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).