1 Algebraic and Analytic Methods Topics of Discussion 1.3 Determinants, Linear Operators, and Spectral Expansions 1.5 Calculus of Two or More Variables

§1.4 Calculus of One Variable

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Keywords:: calculus, one variable
Referenced by:: Erratum (V1.2.0) §1.4
Permalink:: http://dlmf.nist.gov/1.4
See also:: Annotations for Ch.1

§1.4(i) Monotonicity

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Keywords:: decreasing, functions, increasing, monotonic, monotonicity, nondecreasing, nonincreasing, strictly decreasing, strictly increasing, strictly monotonic
Permalink:: http://dlmf.nist.gov/1.4.i
See also:: Annotations for §1.4 and Ch.1

If $f(x_{1})\leq f(x_{2})$ for every pair $x_{1}$ , $x_{2}$ in an interval such that $x_{1}<x_{2}$ , then is nondecreasing on . If the $\leq$ sign is replaced by , then is increasing (also called strictly increasing) on . 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

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Defines:: $C(\NVar{I})$ or $C(\NVar{a,b})$ : continuous on an interval or
Keywords:: at a point, continuous, continuous function, discontinuity, functions, limits, limits of functions, notation, of one variable, on an interval, on the left (or right), piecewise, piecewise continuous, removable, removable discontinuity, sectionally, simple discontinuity, singularity
Notes:: See Hardy (1952, Chapter 5) and Olver (1997b, p. 73).
Referenced by:: §1.17(i), §2.3(ii), Erratum (V1.0.19) for Notation
Permalink:: http://dlmf.nist.gov/1.4.ii
See also:: Annotations for §1.4 and Ch.1

A function is continuous on the right (or from above) at if

1.4.1 $f(c+)\equiv\lim_{x\to c+}f(x)=f(c),$

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Symbols:: $\equiv$ : equals by definition
Referenced by:: §1.4(ii)
Permalink:: http://dlmf.nist.gov/1.4.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(ii), §1.4 and Ch.1

that is, for every arbitrarily small positive constant $\epsilon$ there exists $\delta$ () such that

1.4.2 $\left|f(c+\alpha)-f(c)\right|<\epsilon,$

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Symbols:: $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.4.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(ii), §1.4 and Ch.1

for all $\alpha$ such that $0\leq\alpha<\delta$ . Similarly, it is continuous on the left (or from below) at if

1.4.3 $f(c-)\equiv\lim_{x\to c-}f(x)=f(c).$

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Symbols:: $\equiv$ : equals by definition
Referenced by:: §1.4(ii)
Permalink:: http://dlmf.nist.gov/1.4.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(ii), §1.4 and Ch.1

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

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

A removable singularity of at occurs when but is undefined. For example, $f(x)=(\sin x)/x$ with .

A simple discontinuity of at occurs when and exist, but $f(c+)\not=f(c-)$ . If is continuous on an interval save for a finite number of simple discontinuities, then is piecewise (or sectionally) continuous on . For an example, see Figure 1.4.1

Figure 1.4.1: Piecewise continuous function on

. Magnify

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Symbols:: $[\NVar{a},\NVar{b})$ : half-closed interval
Referenced by:: §1.4(ii)
Permalink:: http://dlmf.nist.gov/1.4.F1
Encodings:: pdf, png
See also:: Annotations for §1.4(ii), §1.4 and Ch.1

§1.4(iii) Derivatives

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Keywords:: definition, derivatives, notation
Notes:: See Hardy (1952, Chapter 6) or Rudin (1976, Chapter 5). For (1.4.13) see Riordan (1958, pp. 35–36) and Knuth (1968, p. 50).
Referenced by:: §1.2(iii), §1.4(vii), Erratum (V1.1.11) for Subsection 1.4(iii)
Permalink:: http://dlmf.nist.gov/1.4.iii
Addition (effective with 1.1.11):: A sentence was added just below (1.4.15) indicating that we assume that $g^{\prime}(x)\neq 0$ for all in some neighborhood of with $x\neq a$ .
Suggested 2023-08-21 by Svante Janson
See also:: Annotations for §1.4 and Ch.1

The derivative $f^{\prime}(x)$ of is defined by

1.4.4 $f^{\prime}(x)=\frac{\mathrm{d}f}{\mathrm{d}x}=\lim_{h\to 0}\frac{f(x+h)-f(x)}{% h}.$

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Defines:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to
Referenced by:: §1.8(ii)
Permalink:: http://dlmf.nist.gov/1.4.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(iii), §1.4 and Ch.1

When this limit exists is differentiable at .

1.4.5 $(f+g)^{\prime}(x)=f^{\prime}(x)+g^{\prime}(x),$

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A&S Ref:: 3.3.2
Permalink:: http://dlmf.nist.gov/1.4.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(iii), §1.4 and Ch.1

1.4.6 $(fg)^{\prime}(x)=f^{\prime}(x)g(x)+f(x)g^{\prime}(x),$

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A&S Ref:: 3.3.3
Permalink:: http://dlmf.nist.gov/1.4.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(iii), §1.4 and Ch.1

1.4.7 $\left(\frac{f}{g}\right)^{\prime}(x)=\frac{f^{\prime}(x)g(x)-f(x)g^{\prime}(x)% }{(g(x))^{2}}.$

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A&S Ref:: 3.3.4
Permalink:: http://dlmf.nist.gov/1.4.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(iii), §1.4 and Ch.1

Higher Derivatives

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Defines:: $C^{\NVar{n}}(\NVar{I})$ or $C^{\NVar{n}}(\NVar{a,b})$ : continuously differentiable times on an interval or
Keywords:: continuously differentiable, differentiable, differentiable functions, functions
See also:: Annotations for §1.4(iii), §1.4 and Ch.1

1.4.8 $f^{(2)}(x)=\frac{{\mathrm{d}}^{2}f}{{\mathrm{d}x}^{2}}=\frac{\mathrm{d}}{% \mathrm{d}x}\left(\frac{\mathrm{d}f}{\mathrm{d}x}\right),$

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to
Permalink:: http://dlmf.nist.gov/1.4.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1

1.4.9 $f^{(n)}=f^{(n)}(x)=\frac{\mathrm{d}}{\mathrm{d}x}f^{(n-1)}(x).$

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.4.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1

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

Chain Rule

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Keywords:: chain rule, derivatives, for derivatives
See also:: Annotations for §1.4(iii), §1.4 and Ch.1

For ,

1.4.10 $h^{\prime}(x)=f^{\prime}(g(x))g^{\prime}(x).$

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A&S Ref:: 3.3.5
Permalink:: http://dlmf.nist.gov/1.4.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1

Maxima and Minima

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Keywords:: local, maximum, minimum
See also:: Annotations for §1.4(iii), §1.4 and Ch.1

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

Mean Value Theorem

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Keywords:: derivatives, differentiable functions, mean value theorem, mean value theorems
See also:: Annotations for §1.4(iii), §1.4 and Ch.1

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

1.4.11 $f(b)-f(a)=(b-a)f^{\prime}(c).$

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Permalink:: http://dlmf.nist.gov/1.4.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1

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

Leibniz’s Formula

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Keywords:: Leibniz’s formula, Leibniz’s formula for derivatives, derivatives
See also:: Annotations for §1.4(iii), §1.4 and Ch.1

1.4.12 $(fg)^{(n)}=f^{(n)}g+\genfrac{(}{)}{0.0pt}{}{n}{1}f^{(n-1)}g^{\prime}+\dots+% \genfrac{(}{)}{0.0pt}{}{n}{k}f^{(n-k)}g^{(k)}+\dots+fg^{(n)}.$

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Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, : integer and : nonnegative integer
A&S Ref:: 3.3.8
Permalink:: http://dlmf.nist.gov/1.4.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1

Faà Di Bruno’s Formula

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Keywords:: Faà di Bruno’s formula, derivatives, for derivatives
See also:: Annotations for §1.4(iii), §1.4 and Ch.1

1.4.13 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}f(g(x))=\sum\left(\frac{n!}{m_{1}!m_% {2}!\cdots m_{n}!}\right)f^{(k)}(g(x))\*\left(\frac{g^{\prime}(x)}{1!}\right)^% {m_{1}}\left(\frac{g^{\prime\prime}(x)}{2!}\right)^{m_{2}}\dots\left(\frac{g^{% (n)}(x)}{n!}\right)^{m_{n}},$

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : factorial (as in ), : integer, : nonnegative integer and : nonnegative integer
Referenced by:: §1.4(iii)
Permalink:: http://dlmf.nist.gov/1.4.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1

where the sum is over all nonnegative integers $m_{1},m_{2},\dots,m_{n}$ that satisfy $m_{1}+2m_{2}+\dots+nm_{n}=n$ , and $k=m_{1}+m_{2}+\dots+m_{n}$ .

L’Hôpital’s Rule

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Keywords:: L’Hôpital’s rule, L’Hôpital’s rule for derivatives, derivatives
See also:: Annotations for §1.4(iii), §1.4 and Ch.1

1.4.14 $\lim\limits_{x\to a}f(x)=\lim\limits_{x\to a}g(x)=0\;\;\mbox{(or $\infty$)},$

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Permalink:: http://dlmf.nist.gov/1.4.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1

then

1.4.15 $\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f^{\prime}(x)}{g^{\prime}(x)}$

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A&S Ref:: 3.4.1
Referenced by:: §1.4(iii), Erratum (V1.1.11) for Subsection 1.4(iii)
Permalink:: http://dlmf.nist.gov/1.4.E15
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1

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

§1.4(iv) Indefinite Integrals

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Defines:: $\,\mathrm{d}\NVar{x}$ : differential of and $\int$ : integral
Keywords:: indefinite, integrals, integration
Notes:: See Hardy (1952, pp. 247–248, 258).
Permalink:: http://dlmf.nist.gov/1.4.iv
See also:: Annotations for §1.4 and Ch.1

If $F^{\prime}(x)=f(x)$ , then $\int f\,\mathrm{d}x=F(x)+C$ , where is a constant.

Integration by Parts

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Keywords:: by parts, integrals, integration, tables
See also:: Annotations for §1.4(iv), §1.4 and Ch.1

1.4.16 $\int fg\,\mathrm{d}x=\left(\int f\,\mathrm{d}x\right)g-\int\left(\int f\,% \mathrm{d}x\right)\frac{\mathrm{d}g}{\mathrm{d}x}\,\mathrm{d}x.$

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\,\mathrm{d}\NVar{x}$ : differential of and $\int$ : integral
A&S Ref:: 3.3.13
Permalink:: http://dlmf.nist.gov/1.4.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(iv), §1.4(iv), §1.4 and Ch.1

1.4.17 $\int x^{n}\,\mathrm{d}x=\begin{cases}\dfrac{x^{n+1}}{n+1}+C,&\quad n\not=-1,\\ \ln\left|x\right|+C,&\quad n=-1.\end{cases}$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, : nonnegative integer, : constant and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.4.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(iv), §1.4(iv), §1.4 and Ch.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 (2015), Gröbner and Hofreiter (1949, 1950), and Prudnikov et al. (1986a, b, 1990, 1992a, 1992b).

§1.4(v) Definite Integrals

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Keywords:: definite, integrals, integration
Notes:: See Hardy (1952, Chapters 6, 7). For (1.4.31) integrate by parts. For (1.4.34) see Olver (1997b, p. 28).
Referenced by:: §1.14(i), §1.14(ii), §1.16(i), §1.16(ii), §1.16(iv), §1.18(ii), §1.9(iii), §10.22(ii), §10.23(iii), §18.17(viii), §18.2(i), §19.2(ii), §2.3(i), §2.6(ii), §28.28(iii), §4.10(i), §9.10(vii), §9.12(vii), Erratum (V1.0.19) for Notation, Erratum (V1.2.0) §1.4
Permalink:: http://dlmf.nist.gov/1.4.v
Addition (effective with 1.2.0):: Equations (1.4.23_1), (1.4.23_2), (1.4.23_3) were added. A sentence was added about Riemann integrability after (1.4.18).
See also:: Annotations for §1.4 and Ch.1

Riemann Integrals

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Addition (effective with 1.2.0):: This paragraph was added. Just below (1.14.18) a sentence regarding Riemann integrability was added.
See also:: Annotations for §1.4(v), §1.4 and Ch.1

Suppose is defined on . Let $a=x_{0}<x_{1}<\cdots<x_{n}=b$ , and $\xi_{j}$ denote any point in $[x_{j},x_{j+1}]$ , $j=0,1,\dots,n-1$ . Then

1.4.18 $\int^{b}_{a}f(x)\,\mathrm{d}x=\lim\sum^{n-1}_{j=0}f(\xi_{j})(x_{j+1}-x_{j})$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : integer, : nonnegative integer and $\xi_{\NVar{j}}$ : point
Referenced by:: §1.4(v)
Permalink:: http://dlmf.nist.gov/1.4.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

as $\max(x_{j+1}-x_{j})\to 0$ . If the limit exists then is called Riemann integrable. Continuity, or piecewise continuity, of on is sufficient for the limit to exist.

1.4.19 $\int^{b}_{a}(cf(x)+dg(x))\,\mathrm{d}x=c\int^{b}_{a}f(x)\,\mathrm{d}x+d\int^{b% }_{a}g(x)\,\mathrm{d}x,$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of and $\int$ : integral
Permalink:: http://dlmf.nist.gov/1.4.E19
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

and constants.

1.4.20 $\int^{b}_{a}f(x)\,\mathrm{d}x=-\int^{a}_{b}f(x)\,\mathrm{d}x.$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of and $\int$ : integral
Permalink:: http://dlmf.nist.gov/1.4.E20
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

1.4.21 $\int^{b}_{a}f(x)\,\mathrm{d}x=\int^{c}_{a}f(x)\,\mathrm{d}x+\int^{b}_{c}f(x)\,% \mathrm{d}x.$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of and $\int$ : integral
Permalink:: http://dlmf.nist.gov/1.4.E21
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

Infinite Integrals

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Keywords:: absolute, convergence, infinite, integrals
See also:: Annotations for §1.4(v), §1.4 and Ch.1

1.4.22 $\int^{\infty}_{a}f(x)\,\mathrm{d}x=\lim_{b\to\infty}\int^{b}_{a}f(x)\,\mathrm{% d}x.$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of and $\int$ : integral
Referenced by:: §1.4(v), §1.5(v)
Permalink:: http://dlmf.nist.gov/1.4.E22
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

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

1.4.23 $\int^{b}_{a}f(x)\,\mathrm{d}x=\lim_{c\to b-}\int^{c}_{a}f(x)\,\mathrm{d}x.$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of and $\int$ : integral
Referenced by:: §1.4(v), §1.5(v)
Permalink:: http://dlmf.nist.gov/1.4.E23
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

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

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

Stieltjes, Lebesgue, and Lebesgue–Stieltjes integrals

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Keywords:: Lebesgue, Lebesgue-Stieltjes, Stieltjes, absolute, convergence, integrals
Addition (effective with 1.2.0):: This paragraph was added.
See also:: Annotations for §1.4(v), §1.4 and Ch.1

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

Absolutely Continuous Stieltjes Measure

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Keywords:: Absolutely continuous integration measure, Stieltjes, absolutely convergent, measure
Addition (effective with 1.2.0):: This paragraph was added.
See also:: Annotations for §1.4(v), §1.4 and Ch.1

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

1.4.23_1 $\alpha(d)-\alpha(c)=\int_{c}^{d}w(x)\,\mathrm{d}x,$ $[c,d]\subset I$ .

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Symbols:: $[\NVar{a},\NVar{b}]$ : closed interval, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\subset$ : is contained in and : weight function
Referenced by:: §1.4(v), Erratum (V1.2.0) §1.4
Permalink:: http://dlmf.nist.gov/1.4.E23_1
Encodings:: TeX, pMML, png
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

Then

1.4.23_2 $\int_{a}^{b}f(x)\,\mathrm{d}\alpha(x)=\int_{a}^{b}f(x)w(x)\,\mathrm{d}x,$

integrable with respect to $\,\mathrm{d}\alpha$ .

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and : weight function
Referenced by:: §1.4(v)
Permalink:: http://dlmf.nist.gov/1.4.E23_2
Encodings:: TeX, pMML, png
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

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

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

Stieltjes Measure with $\alpha(x)$ Discontinuous

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Keywords:: Steiltjes, Stieltjes, Stieltjes with jumps, integral, measure, measure with jumps
Addition (effective with 1.2.0):: This paragraph was added.
See also:: Annotations for §1.4(v), §1.4 and Ch.1

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 , 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 $\,\mathrm{d}\alpha(x)$ is $\delta\left(x-x_{n}\right)\,\mathrm{d}x$ , where $\delta\left(x-x_{n}\right)$ is the Dirac $\delta$ -function of §1.17, such that, for a continuous function on , $\int_{a}^{b}f(x)\,\mathrm{d}\alpha(x)=f(x_{n})$ 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 $\,\mathrm{d}\alpha(x)$ could represent a sum, an integral, or a combination of the two. Let $\,\mathrm{d}\alpha(x)=w(x)\,\mathrm{d}x+\sum_{n=1}^{N}w_{n}\delta\left(x-x_{n}% \right)\,\mathrm{d}x$ , $x_{n}\in(a,b)$ , $n=1,\dots N$ . Then for continuous on ,

1.4.23_3 $\int_{a}^{b}f(x)\,\mathrm{d}\alpha(x)=\int_{a}^{b}w(x)f(x)\,\mathrm{d}x+\sum_{% n=1}^{N}w_{n}f(x_{n}).$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : nonnegative integer and : weight function
Referenced by:: §1.4(v), Erratum (V1.2.0) §1.4
Permalink:: http://dlmf.nist.gov/1.4.E23_3
Encodings:: TeX, pMML, png
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

In the literature where 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.

Cauchy Principal Values

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Keywords:: Cauchy principal values, integrals
See also:: Annotations for §1.4(v), §1.4 and Ch.1

Let $c\in(a,b)$ and assume that $\int_{a}^{c-\epsilon}f(x)\,\mathrm{d}x$ and $\int_{c+\epsilon}^{b}f(x)\,\mathrm{d}x$ exist when $0<\epsilon<\min(c-a,b-c)$ , but not necessarily when $\epsilon=0$ . Then we define

1.4.24 $\pvint^{b}_{a}f(x)\,\mathrm{d}x=P\int^{b}_{a}f(x)\,\mathrm{d}x=\lim_{\epsilon% \to 0+}\left(\int^{c-\epsilon}_{a}f(x)\,\mathrm{d}x+\int^{b}_{c+\epsilon}f(x)% \,\mathrm{d}x\right),$

ⓘ

Defines:: $\pvint_{\NVar{a}}^{\NVar{b}}$ : Cauchy principal value
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of and $\int$ : integral
Referenced by:: §1.18(vi)
Permalink:: http://dlmf.nist.gov/1.4.E24
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

when this limit exists.

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

1.4.25 $\pvint^{\infty}_{-\infty}f(x)\,\mathrm{d}x=P\int^{\infty}_{-\infty}f(x)\,% \mathrm{d}x=\lim_{b\to\infty}\int^{b}_{-b}f(x)\,\mathrm{d}x,$

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and $\pvint_{\NVar{a}}^{\NVar{b}}$ : Cauchy principal value
Referenced by:: §1.14(i)
Permalink:: http://dlmf.nist.gov/1.4.E25
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

when this limit exists.

Fundamental Theorem of Calculus

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Keywords:: differentiation, fundamental theorem of calculus, integrals
See also:: Annotations for §1.4(v), §1.4 and Ch.1

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

1.4.26 $\int^{b}_{a}f(x)\,\mathrm{d}x=F(b)-F(a),$

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of and $\int$ : integral
Referenced by:: §18.17(i)
Permalink:: http://dlmf.nist.gov/1.4.E26
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

1.4.27 $\frac{\mathrm{d}}{\mathrm{d}x}\int^{x}_{a}f(t)\,\mathrm{d}t=f(x).$

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\,\mathrm{d}\NVar{x}$ : differential of and $\int$ : integral
Permalink:: http://dlmf.nist.gov/1.4.E27
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

Change of Variables

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Keywords:: change of variables, integrals
See also:: Annotations for §1.4(v), §1.4 and Ch.1

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

1.4.28 $\int^{b}_{a}f(\phi(x))\phi^{\prime}(x)\,\mathrm{d}x=\int^{\phi(b)}_{\phi(a)}f(% t)\,\mathrm{d}t.$

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and $\phi(\NVar{x})$ : function
Permalink:: http://dlmf.nist.gov/1.4.E28
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

First Mean Value Theorem

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Keywords:: first, integrals, mean value theorems
See also:: Annotations for §1.4(v), §1.4 and Ch.1

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

1.4.29 $\int^{b}_{a}f(x)\phi(x)\,\mathrm{d}x=f(c)\int^{b}_{a}\phi(x)\,\mathrm{d}x.$

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and $\phi(\NVar{x})$ : function
Permalink:: http://dlmf.nist.gov/1.4.E29
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

Second Mean Value Theorem

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Keywords:: integrals, mean value theorems, second
See also:: Annotations for §1.4(v), §1.4 and Ch.1

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

1.4.30 $\int^{b}_{a}f(x)\phi(x)\,\mathrm{d}x=f(a)\int^{c}_{a}\phi(x)\,\mathrm{d}x+f(b)% \int^{b}_{c}\phi(x)\,\mathrm{d}x.$

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and $\phi(\NVar{x})$ : function
Permalink:: http://dlmf.nist.gov/1.4.E30
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

Repeated Integrals

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Keywords:: integrals, repeated
See also:: Annotations for §1.4(v), §1.4 and Ch.1

If is continuous or piecewise continuous on , then

1.4.31 $\int_{a}^{b}\,\mathrm{d}x_{n}\int_{a}^{x_{n}}\,\mathrm{d}x_{n-1}\cdots\int_{a}% ^{x_{2}}\,\mathrm{d}x_{1}\int_{a}^{x_{1}}f(x)\,\mathrm{d}x=\frac{1}{n!}\int_{a% }^{b}(b-x)^{n}f(x)\,\mathrm{d}x.$

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , : factorial (as in ), $\int$ : integral and : nonnegative integer
Referenced by:: §1.15(vi), §1.4(v)
Permalink:: http://dlmf.nist.gov/1.4.E31
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

Square-Integrable Functions

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Keywords:: integrals, square-integrable, square-integrable function
See also:: Annotations for §1.4(v), §1.4 and Ch.1

A function is square-integrable if

1.4.32 $\|f\|^{2}_{2}\equiv\int^{b}_{a}{\left|f(x)\right|}^{2}\,\mathrm{d}x<\infty.$

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\equiv$ : equals by definition, $\int$ : integral and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.4.E32
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

Functions of Bounded Variation

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Keywords:: bounded, bounded variation, closure, functions, interval, of bounded variation, of interval, total, variation of real or complex functions
See also:: Annotations for §1.4(v), §1.4 and Ch.1

With , the total variation of on a finite or infinite interval is

1.4.33 $\mathcal{V}_{a,b}\left(f\right)=\sup\sum^{n}_{j=1}\left|f(x_{j})-f(x_{j-1})% \right|,$

ⓘ

Defines:: $\mathcal{V}\left(\NVar{f}\right)$ : total variation and $\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)$ : total variation
Symbols:: $\sup$ : least upper bound (supremum), : integer, : nonnegative integer and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.4.E33
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

where the supremum is over all sets of points $x_{0}<x_{1}<\cdots<x_{n}$ in the closure of , that is, with added when they are finite. If $\mathcal{V}_{a,b}\left(f\right)<\infty$ , then is of bounded variation on . In this case, $g(x)=\mathcal{V}_{a,x}\left(f\right)$ and $h(x)=\mathcal{V}_{a,x}\left(f\right)-f(x)$ are nondecreasing bounded functions and .

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

1.4.34 $\mathcal{V}_{a,b}\left(f\right)=\int^{b}_{a}\left|f^{\prime}(x)\right|\,% \mathrm{d}x,$

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)$ : total variation and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Referenced by:: §1.4(v), §1.4(v), Erratum (V1.0.28) for Equation (1.4.34)
Permalink:: http://dlmf.nist.gov/1.4.E34
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

whenever this integral exists.

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

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

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Keywords:: Taylor’s theorem, one variable
Notes:: See Hardy (1952, pp. 285–292, 327–328).
Permalink:: http://dlmf.nist.gov/1.4.vi
See also:: Annotations for §1.4 and Ch.1

If $f(x)\in C^{n+1}[a,b]$ , then

1.4.35 $f(x)=\sum^{n}_{k=0}\frac{f^{(k)}(a)}{k!}(x-a)^{k}+R_{n},$

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Symbols:: : factorial (as in ), : integer, : nonnegative integer and $R_{\NVar{n}}$ : remainder
A&S Ref:: 3.6.4
Permalink:: http://dlmf.nist.gov/1.4.E35
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(vi), §1.4 and Ch.1

1.4.36 $R_{n}=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1},$

ⓘ

Symbols:: : factorial (as in ), : nonnegative integer and $R_{\NVar{n}}$ : remainder
A&S Ref:: 3.6.5
Permalink:: http://dlmf.nist.gov/1.4.E36
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(vi), §1.4 and Ch.1

and

1.4.37 $R_{n}=\frac{1}{n!}\int^{x}_{a}(x-t)^{n}f^{(n+1)}(t)\,\mathrm{d}t.$

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , : factorial (as in ), $\int$ : integral, : nonnegative integer and $R_{\NVar{n}}$ : remainder
A&S Ref:: 3.6.3
Permalink:: http://dlmf.nist.gov/1.4.E37
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(vi), §1.4 and Ch.1

§1.4(vii) Maxima and Minima

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Keywords:: maximum, minimum
Notes:: See Hardy (1952, pp. 234–235).
Permalink:: http://dlmf.nist.gov/1.4.vii
See also:: Annotations for §1.4 and Ch.1

If is twice-differentiable, and if also $f^{\prime}(x_{0})=0$ and $f^{\prime\prime}(x_{0})<0$ (), then $x=x_{0}$ is a local maximum (minimum) (§1.4(iii)) of . The overall maximum (minimum) of on will either be at a local maximum (minimum) or at one of the end points or .

§1.4(viii) Convex Functions

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Keywords:: calculus, convex, convex functions, functions, one variable
Notes:: See Hardy et al. (1967, pp. 70–77).
Referenced by:: §1.7(iv), §5.5(iv)
Permalink:: http://dlmf.nist.gov/1.4.viii
See also:: Annotations for §1.4 and Ch.1

A function is convex on if

1.4.38 $f((1-t)c+td)\leq(1-t)f(c)+tf(d)$

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Permalink:: http://dlmf.nist.gov/1.4.E38
Encodings:: TeX, pMML, png
See also:: Annotations for §1.4(viii), §1.4 and Ch.1

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

If is twice differentiable, then is convex iff $f^{\prime\prime}(x)\geq 0$ on . A continuously differentiable function is convex iff the curve does not lie below its tangent at any point.