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21: 3.5 Quadrature

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3.5.38 $$G(p)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}g(t)dt,$$

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Defines:

$G(p)$: Laplace transform of $g(t)$ (locally)

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $g(t)$: function

A&S Ref:

29.2.1 (in different form)

Permalink:

http://dlmf.nist.gov/3.5.E38

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(viii), §3.5(viii), §3.5 and Ch.3

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3.5.39 $$g(t)=\frac{1}{2\pi \mathrm{i}}{\int }_{\sigma -\mathrm{i}\mathrm{\infty }}^{\sigma +\mathrm{i}\mathrm{\infty }}{\mathrm{e}}^{tp}G(p)dp,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $g(t)$: function, $G(p)$: Laplace transform of $g(t)$ and $\sigma$: parameter

A&S Ref:

29.2.1 (in different form)

Referenced by:

§3.5(viii)

Permalink:

http://dlmf.nist.gov/3.5.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(viii), §3.5(viii), §3.5 and Ch.3

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3.5.42 $$\mathrm{erfc}\lambda =\frac{1}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}{\mathrm{e}}^{\zeta -2\lambda \sqrt{\zeta }}\frac{d\zeta }{\zeta },$$ $c>0$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $\lambda$: parameter

A&S Ref:

29.3.83 (in different form)

Referenced by:

§3.5(ix)

Permalink:

http://dlmf.nist.gov/3.5.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(ix), §3.5(ix), §3.5 and Ch.3

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22: 7.18 Repeated Integrals of the Complementary Error Function

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7.18.2 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)={\int }_z^{\mathrm{\infty }}{\mathrm{i}}^{n-1}\mathrm{erfc}\left(t\right)dt=\frac{2}{\sqrt{\pi }}{\int }_z^{\mathrm{\infty }}\frac{{(t-z)}^n}{n!}{\mathrm{e}}^{-t^2}dt.$$

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Defines:

${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function

Symbols:

$\pi$: 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, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.1 (in modified form) 7.2.3

Permalink:

http://dlmf.nist.gov/7.18.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(i), §7.18 and Ch.7

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23: 2.4 Contour Integrals

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2.4.2 $$Q(z)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}q(t)dt$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $q(t)$: analytic function and $Q(z)$: Laplace transform of $q(t)$

Permalink:

http://dlmf.nist.gov/2.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(ii), §2.4 and Ch.2

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2.4.8 $$q(t)=f(t)+o\left({\mathrm{e}}^{ct}\right),$$ $t\to +\mathrm{\infty }$.

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Symbols:

$\mathrm{e}$: base of natural logarithm, $o\left(x\right)$: order less than, $q(t)$: analytic function, $c$: real constant and $f(x)$: inverse transform of comparison function

Permalink:

http://dlmf.nist.gov/2.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(ii), §2.4 and Ch.2

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2.4.9 $$q(t)=f(t)+o\left(t^{-m}{\mathrm{e}}^{ct}\right),$$ $t\to +\mathrm{\infty }$.

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Symbols:

$\mathrm{e}$: base of natural logarithm, $o\left(x\right)$: order less than, $q(t)$: analytic function, $c$: real constant and $f(x)$: inverse transform of comparison function

Permalink:

http://dlmf.nist.gov/2.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(ii), §2.4 and Ch.2

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2.4.10 $$I(z)={\int }_a^b{\mathrm{e}}^{-zp(t)}q(t)dt,$$

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Defines:

$I(z)$: contour integral (locally)

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $a$: left endpoint, $b$: right endpoint, $p(t)$: analytic function and $q(t)$: analytic function

Referenced by:

§2.4(iv)

Permalink:

http://dlmf.nist.gov/2.4.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(iii), §2.4 and Ch.2

… ►The final expansion then has the form …

24: 28.12 Definitions and Basic Properties

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28.12.6 $${\mathrm{me}}_{\nu }(z+\pi ,q)={\mathrm{e}}^{\pi \mathrm{i}\nu }{\mathrm{me}}_{\nu }(z,q),$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

20.3.4 (in different form)

Permalink:

http://dlmf.nist.gov/28.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.12(ii), §28.12 and Ch.28

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25: 7.5 Interrelations

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7.5.10 $$\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right)=\frac{1}{2}(1\pm \mathrm{i}){\mathrm{e}}^{{\zeta }^2}\mathrm{erfc}\zeta .$$

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

A&S Ref:

7.3.22 (in different form)

Referenced by:

§7.12(ii), §7.13(iv), §7.5, §7.5

Permalink:

http://dlmf.nist.gov/7.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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26: 25.12 Polylogarithms

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25.12.2 $${\mathrm{Li}}_2\left(z\right)=-{\int }_0^z{t}^{-1}\ln \left(1-t\right)dt,$$ $z\in ℂ\setminus (1,\mathrm{\infty })$.

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Symbols:

$ℂ$: complex plane, $dx$: differential of $x$ , ${\mathrm{Li}}_2\left(z\right)$: dilogarithm, $\in$: element of, $\int$: integral, $\ln z$: principal branch of logarithm function, $(a,b)$: open interval, $\setminus$: set subtraction, $x$: real variable and $z$: complex variable

Keywords:

definite integral, integral representation

Source:

Maximon (2003, (2.1), p. 2807)

A&S Ref:

27.7.1 (is a modified form with $z=1-x$)

Referenced by:

§25.12(i)

Permalink:

http://dlmf.nist.gov/25.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

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25.12.3 $${\mathrm{Li}}_2\left(z\right)+{\mathrm{Li}}_2\left(\frac{z}{z-1}\right)=-\frac{1}{2}{(\ln \left(1-z\right))}^2,$$ $z\in ℂ\setminus [1,\mathrm{\infty })$.

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Symbols:

$[a,b)$: half-closed interval, $ℂ$: complex plane, ${\mathrm{Li}}_2\left(z\right)$: dilogarithm, $\in$: element of, $\ln z$: principal branch of logarithm function, $\setminus$: set subtraction, $x$: real variable and $z$: complex variable

Keywords:

connection formula

Source:

Maximon (2003, (3.4), p. 2808)

A&S Ref:

27.7.5 (is a modified form with $z=1-x$)

Permalink:

http://dlmf.nist.gov/25.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

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27: 4.13 Lambert $W$-Function

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4.13.3_1

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\mathrm{e}$: base of natural logarithm and $x$: real variable

Proof sketch:

Take $x=W_0\left(t\right)>-1$, with $t>-{\mathrm{e}}^{-1}$. Then $W_0\left(x{\mathrm{e}}^x\right)=W_0\left(W_0\left(t\right){\mathrm{e}}^{W_0\left(t\right)}\right)=W_0\left(t\right)=x$.

Referenced by:

§4.13, Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E3_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

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4.13.3_2

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $x$: real variable

Proof sketch:

Take , with . Then $W_1\left(x{\mathrm{e}}^x-0\mathrm{i}\right)=W_1\left(W_1\left(t-0\mathrm{i}\right){\mathrm{e}}^{W_1\left(t-0\mathrm{i}\right)}-0\mathrm{i}\right)=W_1\left(t-0\mathrm{i}\right)=x$.

Referenced by:

§4.13, Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E3_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

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28: 27.12 Asymptotic Formulas: Primes

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27.12.2 $$p_n>n\ln n,$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$\ln z$: principal branch of logarithm function, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

Permalink:

http://dlmf.nist.gov/27.12.E2

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.12 and Ch.27

… ►For the logarithmic integral $\mathrm{li}\left(x\right)$ see (6.2.8). … ► $\pi \left(x\right)-\mathrm{li}\left(x\right)$ changes sign infinitely often as $x\to \mathrm{\infty }$; see Littlewood (1914), Bays and Hudson (2000). … ►

27.12.7

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number and $x$: real number

Referenced by:

§27.12

Permalink:

http://dlmf.nist.gov/27.12.E7

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.12, §27.12 and Ch.27

… ►A Mersenne prime is a prime of the form $2^p-1$. …

29: 1.8 Fourier Series

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Alternative Form

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1.8.3 $$f(x)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_n{\mathrm{e}}^{\mathrm{i}nx},$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $n$: nonnegative integer

Referenced by:

§1.17(iii), §1.18(v)

Permalink:

http://dlmf.nist.gov/1.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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1.8.4 $$c_n=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(x){\mathrm{e}}^{-\mathrm{i}nx}dx.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $n$: nonnegative integer

Referenced by:

§1.17(iii), §1.18(v), §1.8(i)

Permalink:

http://dlmf.nist.gov/1.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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1.8.9 $$L_n\sim (4/{\pi }^2)\ln n;$$

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $L_n$: Lebesgue constants

Permalink:

http://dlmf.nist.gov/1.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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1.8.10 $${\int }_a^{b}f(x){\mathrm{e}}^{\mathrm{i}\lambda x}dx\to 0,$$ as $\lambda \to \mathrm{\infty }$.

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $\lambda$: real

Referenced by:

§1.8(i)

Permalink:

http://dlmf.nist.gov/1.8.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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30: 7.12 Asymptotic Expansions

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7.12.6 $$R_n^{(\mathrm{f})}(z)=\frac{{(-1)}^n}{\pi \sqrt{2}}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-\pi z^{2}t/2}{t}^{2n-(1/2)}}{t^2+1}dt,$$

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Defines:

$R_n^{(\mathrm{f})}(z)$: remainder term (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.3.29 (in different form)

Referenced by:

§7.12(ii)

Permalink:

http://dlmf.nist.gov/7.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

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7.12.7 $$R_n^{(\mathrm{g})}(z)=\frac{{(-1)}^n}{\pi \sqrt{2}}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-\pi z^{2}t/2}{t}^{2n+(1/2)}}{t^2+1}dt.$$

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Defines:

$R_n^{(\mathrm{g})}(z)$: remainder term (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.3.30 (in different form)

Referenced by:

§7.12(ii)

Permalink:

http://dlmf.nist.gov/7.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

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