Cauchy–Schwarz inequalities for sums and integrals

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31: 19.25 Relations to Other Functions

… ►

§19.25(i) Legendre’s Integrals as Symmetric Integrals

… ►with Cauchy principal value … ►If ${\alpha }^2>c$, then the Cauchy principal value is … ►

§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals

… ►

§19.25(iii) Symmetric Integrals as Legendre’s Integrals

…

32: 18.17 Integrals

… ►provided that $\mathrm{\ell }+m+n$ is even and the sum of any two of $\mathrm{\ell },m,n$ is not less than the third; otherwise the integral is zero. … ►

18.17.42 $${⨍}_{-1}^1{T}_n\left(y\right)\frac{{(1-y^2)}^{-\frac{1}{2}}}{y-x}dy=\pi U_{n-1}\left(x\right),$$

ⓘ

Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , ${⨍}_a^b$: Cauchy principal value, $y$: real variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.13.3

Referenced by:

§18.17(viii)

Permalink:

http://dlmf.nist.gov/18.17.E42

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Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

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18.17.43 $${⨍}_{-1}^1{U}_{n-1}\left(y\right)\frac{{(1-y^2)}^{\frac{1}{2}}}{y-x}dy=-\pi T_n\left(x\right).$$

ⓘ

Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , ${⨍}_a^b$: Cauchy principal value, $y$: real variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.13.4

Referenced by:

§18.17(viii)

Permalink:

http://dlmf.nist.gov/18.17.E43

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Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

►These integrals are Cauchy principal values (§1.4(v)). … ►provided that $\mathrm{\ell }+m+n$ is even and the sum of any two of $\mathrm{\ell },m,n$ is not less than the third; otherwise the integral is zero. …

33: 2.3 Integrals of a Real Variable

… ►

2.3.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-xt}q(t)dt\sim \sum _{s=0}^{\mathrm{\infty }}\frac{q^{(s)}(0)}{x^{s+1}},$$ $x\to +\mathrm{\infty }$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $q(t)$: infinitely differentiable function

Keywords:

Laplace transform

Referenced by:

§2.3(i)

Permalink:

http://dlmf.nist.gov/2.3.E2

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Annotations for §2.3(i), §2.3 and Ch.2

… ►is finite and bounded for $n=0,1,2,\mathrm{\dots }$, then the $n$th error term (that is, the difference between the integral and $n$th partial sum in (2.3.2)) is bounded in absolute value by $|q^{(n)}(0)/(x^n(x-{\sigma }_n))|$ when $x$ exceeds both $0$ and ${\sigma }_n$. … ►

2.3.4 $${\int }_a^b{\mathrm{e}}^{\mathrm{i}xt}q(t)dt\sim {\mathrm{e}}^{\mathrm{i}ax}\sum _{s=0}^{\mathrm{\infty }}{q}^{(s)}(a){\left(\frac{\mathrm{i}}{x}\right)}^{s+1}-{\mathrm{e}}^{\mathrm{i}bx}\sum _{s=0}^{\mathrm{\infty }}{q}^{(s)}(b){\left(\frac{\mathrm{i}}{x}\right)}^{s+1},$$ $x\to +\mathrm{\infty }$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $q(t)$: infinitely differentiable function, $a$: left endpoint and $b$: right endpoint

Permalink:

http://dlmf.nist.gov/2.3.E4

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Annotations for §2.3(i), §2.3 and Ch.2

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2.3.5 $${\int }_a^{\mathrm{\infty }}{\mathrm{e}}^{\mathrm{i}xt}q(t)dt\sim {\mathrm{e}}^{\mathrm{i}ax}\sum _{s=0}^{\mathrm{\infty }}{q}^{(s)}(a){\left(\frac{\mathrm{i}}{x}\right)}^{s+1},$$ $x\to +\mathrm{\infty }$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $q(t)$: infinitely differentiable function and $a$: left endpoint

Permalink:

http://dlmf.nist.gov/2.3.E5

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Annotations for §2.3(i), §2.3 and Ch.2

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2.3.8 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-xt}q(t)dt\sim \sum _{s=0}^{\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{s+\lambda }{\mu }\right)\frac{a_s}{x^{(s+\lambda )/\mu }},$$ $x\to +\mathrm{\infty }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $a$: positive constant, $q(t)$: function, $\lambda$: positive constant and $\mu$: positive constant

Keywords:

Laplace transform

Referenced by:

§2.3(ii), §2.4(i)

Permalink:

http://dlmf.nist.gov/2.3.E8

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Annotations for §2.3(ii), §2.3 and Ch.2

…

34: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►

1.18.16 $$\underset{m\to \mathrm{\infty }}{lim}{\int }_a^b{\left|f(x)-\sum _{n=0}^m{c}_n{\varphi }_n(x)\right|}^{2}dx=0.$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral, $m$: nonnegative integer, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.18.E16

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Annotations for §1.18(ii), §1.18 and Ch.1

… ►Should an eigenvalue correspond to more than a single linearly independent eigenfunction, namely a multiplicity greater than one, all such eigenfunctions will always be implied as being part of any sums or integrals over the spectrum. … ►

1.18.35 $$(F(T)f)(x)={\int }_a^b\left(\sum _{n=0}^{\mathrm{\infty }}F({\lambda }_n){\varphi }_{{\lambda }_n}(x)\overline{{\varphi }_{{\lambda }_n}(y)}\right)f(y)dy.$$

ⓘ

Symbols:

$\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\int$: integral and $n$: nonnegative integer

Referenced by:

§1.18(vi)

Permalink:

http://dlmf.nist.gov/1.18.E35

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Annotations for §1.18(v), §1.18(v), §1.18 and Ch.1

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1.18.54 $$\underset{ϵ\to 0+}{lim}{\int }_X\frac{f(y)}{x\pm \mathrm{i}ϵ-y}dy=P{⨍}_X\frac{f(y)}{x-y}dy\mp \mathrm{i}\pi f(x).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and ${⨍}_a^b$: Cauchy principal value

Permalink:

http://dlmf.nist.gov/1.18.E54

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Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

… ►In what follows, integrals over the continuous parts of the spectrum will be denoted by ${𝝈}_c$, and sums over the discrete spectrum by ${𝝈}_p$, with $𝝈={𝝈}_c\cup {𝝈}_p$ denoting the full spectrum. …

35: 6.15 Sums

§6.15 Sums

►

6.15.1 $$\sum _{n=1}^{\mathrm{\infty }}\mathrm{Ci}\left(\pi n\right)=\frac{1}{2}(\ln 2-\gamma ),$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E1

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Annotations for §6.15 and Ch.6

►

6.15.2 $$\sum _{n=1}^{\mathrm{\infty }}\frac{\mathrm{si}\left(\pi n\right)}{n}=\frac{1}{2}\pi (\ln \pi -1),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{si}\left(z\right)$: sine integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E2

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Annotations for §6.15 and Ch.6

►

6.15.3 $$\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\mathrm{Ci}\left(2\pi n\right)=1-\ln 2-\frac{1}{2}\gamma ,$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E3

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See also:

Annotations for §6.15 and Ch.6

►

6.15.4 $$\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{\mathrm{si}\left(2\pi n\right)}{n}=\pi (\frac{3}{2}\ln 2-1).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{si}\left(z\right)$: sine integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E4

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Annotations for §6.15 and Ch.6

…

36: 36.2 Catastrophes and Canonical Integrals

§36.2 Catastrophes and Canonical Integrals

►

§36.2(i) Definitions

… ►

Canonical Integrals

… ► … ►

§36.2(iii) Symmetries

…

37: 9.12 Scorer Functions

… ►

9.12.17 $$\mathrm{Hi}\left(z\right)=\frac{3^{-2/3}}{\pi }\sum _{k=0}^{\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{k+1}{3}\right)\frac{{(3^{1/3}z)}^k}{k!},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable

Proof sketch:

Expand the integral in (9.12.20) in powers of $zt$ and integrate term-by-term by means of (5.2.1).

Referenced by:

(9.12.15), (9.12.18)

Permalink:

http://dlmf.nist.gov/9.12.E17

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Annotations for §9.12(vi), §9.12 and Ch.9

… ►

9.12.23 $$\mathrm{Gi}\left(x\right)=\frac{4x^2}{3^{3/2}{\pi }^2}{⨍}_0^{\mathrm{\infty }}\frac{K_{1/3}\left(t\right)}{{\zeta }^2-t^2}dt,$$ $x>0$,

ⓘ

Symbols:

$\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, ${⨍}_a^b$: Cauchy principal value, $x$: real variable and $\zeta$: variable

Source:

Gordon (1970, Appendix A)

Permalink:

http://dlmf.nist.gov/9.12.E23

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Annotations for §9.12(vii), §9.12 and Ch.9

►where the last integral is a Cauchy principal value (§1.4(v)). … ►

9.12.30 $${\int }_0^z\mathrm{Gi}\left(t\right)dt\sim \frac{1}{\pi }\ln z+\frac{2\gamma +\ln 3}{3\pi }-\frac{1}{\pi }\sum _{k=1}^{\mathrm{\infty }}\frac{(3k-1)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}z|\le \frac{1}{3}\pi -\delta$.

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

Integrate (9.12.25) and obtain the constant term by combining (9.12.12) and (9.12.31).(this equation first appeared in Rothman (1954a). As noted in this reference these results were derived by the author of the present DLMF chapter, but the proof was not included.)

Permalink:

http://dlmf.nist.gov/9.12.E30

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Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

►

9.12.31 $${\int }_0^z\mathrm{Hi}\left(-t\right)dt\sim \frac{1}{\pi }\ln z+\frac{2\gamma +\ln 3}{3\pi }+\frac{1}{\pi }\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k-1}\frac{(3k-1)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$,

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $k$: nonnegative integer, $x$: real variable, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

Except for the constant term, this be verified by termwise integration of (9.12.27). To evaluate the constant term replace $z$ by $-x$ ($\le 0$) in (9.12.20) and integrate (§1.5(v)) to obtain $\pi {\int }_0^x\mathrm{Hi}\left(-t\right)dt={\int }_0^{\mathrm{\infty }}(1-{\mathrm{e}}^{-xt}){\mathrm{e}}^{-\frac{1}{3}{t}^3}{t}^{-1}dt$. Next, integrate the right-hand side of this equation by parts—integrating the factor $t^{-1}$ and differentiating the rest. As $x\to \mathrm{\infty }$ the asymptotic expansions of ${\int }_0^{\mathrm{\infty }}x{\mathrm{e}}^{-xt}{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ and ${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-xt}{t}^2{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ follow from (2.3.9). Also, ${\int }_0^{\mathrm{\infty }}{t}^2{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ can be found by replacing $\frac{1}{3}{t}^3$ by $t$ and referring to the first of (5.9.18). (This equation first appeared in Rothman (1954a). As noted in this reference these results were derived by the author of the present DLMF chapter, but the proof was not included.)

Referenced by:

(9.12.30)

Permalink:

http://dlmf.nist.gov/9.12.E31

Encodings:

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See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

…

38: 19.9 Inequalities

… ►

§19.9(i) Complete Integrals

… ►

§19.9(ii) Incomplete Integrals

… ►Simple inequalities for incomplete integrals follow directly from the defining integrals (§19.2(ii)) together with (19.6.12): … ►Sharper inequalities for $F(\varphi ,k)$ are: … ►Other inequalities for $F(\varphi ,k)$ can be obtained from inequalities for $R_F(x,y,z)$ given in Carlson (1966, (2.15)) and Carlson (1970) via (19.25.5).

39: 8.21 Generalized Sine and Cosine Integrals

… ►

8.21.14 $$\mathrm{Si}(a,z)=z^a\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{z}^{2k+1}}{(2k+a+1)(2k+1)!},$$ $a\ne -1,-3,-5,\mathrm{\dots }$,

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\mathrm{Si}(a,z)$: generalized sine integral, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Referenced by:

§8.21(vi)

Permalink:

http://dlmf.nist.gov/8.21.E14

Encodings:

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Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8

►

8.21.15 $$\mathrm{Ci}(a,z)=z^a\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{z}^{2k}}{(2k+a)(2k)!},$$ $a\ne 0,-2,-4,\mathrm{\dots }$.

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\mathrm{Ci}(a,z)$: generalized cosine integral, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Referenced by:

§8.21(vi)

Permalink:

http://dlmf.nist.gov/8.21.E15

Encodings:

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Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8

… ►

8.21.16 $$\mathrm{Si}(a,z)=z^a\sum _{k=0}^{\mathrm{\infty }}\frac{\left(2k+\frac{3}{2}\right){\left(1-\frac{1}{2}a\right)}_k}{{\left(\frac{1}{2}+\frac{1}{2}a\right)}_{k+1}}{𝗃}_{2k+1}\left(z\right),$$ $a\ne -1,-3,-5,\mathrm{\dots }$,

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{Si}(a,z)$: generalized sine integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Referenced by:

§8.21(vi)

Permalink:

http://dlmf.nist.gov/8.21.E16

Encodings:

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Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8

►

8.21.17 $$\mathrm{Ci}(a,z)=z^a\sum _{k=0}^{\mathrm{\infty }}\frac{\left(2k+\frac{1}{2}\right){\left(\frac{1}{2}-\frac{1}{2}a\right)}_k}{{\left(\frac{1}{2}a\right)}_{k+1}}{𝗃}_{2k}\left(z\right),$$ $a\ne 0,-2,-4,\mathrm{\dots }$.

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{Ci}(a,z)$: generalized cosine integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Referenced by:

§8.21(vi)

Permalink:

http://dlmf.nist.gov/8.21.E17

Encodings:

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Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8

…

40: 6.6 Power Series

… ►

6.6.1 $$\mathrm{Ei}\left(x\right)=\gamma +\ln x+\sum _{n=1}^{\mathrm{\infty }}\frac{x^n}{n!n},$$ $x>0$.

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ei}\left(x\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $x$: real variable and $n$: nonnegative integer

A&S Ref:

5.1.10

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.6.E1

Encodings:

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Annotations for §6.6 and Ch.6

►

6.6.2 $$E_1\left(z\right)=-\gamma -\ln z-\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^n}{n!n}.$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

5.1.11

Referenced by:

§6.4, §6.5

Permalink:

http://dlmf.nist.gov/6.6.E2

Encodings:

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Annotations for §6.6 and Ch.6

►

6.6.3 $$E_1\left(z\right)=-\ln z+{\mathrm{e}}^{-z}\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{n!}\psi \left(n+1\right),$$

ⓘ

Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§2.5(iii), §6.6, §8.19(iv)

Permalink:

http://dlmf.nist.gov/6.6.E3

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Annotations for §6.6 and Ch.6

… ►

6.6.4 $$\mathrm{Ein}\left(z\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n-1}{z}^n}{n!n},$$

ⓘ

Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.6.E4

Encodings:

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Annotations for §6.6 and Ch.6

►

6.6.5 $$\mathrm{Si}\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^{2n+1}}{(2n+1)!(2n+1)},$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\mathrm{Si}\left(z\right)$: sine integral, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

5.2.14

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.6.E5

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Annotations for §6.6 and Ch.6

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