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1: 14.26 Uniform Asymptotic Expansions

… ►See also Frenzen (1990), Gil et al. (2000), Shivakumar and Wong (1988), Ursell (1984), and Wong (1989) for uniform asymptotic approximations obtained from integral representations.

2: 9.13 Generalized Airy Functions

… ►

§9.13(ii) Generalizations from Integral Representations

… ►Each of the functions $A_k(z,p)$ and $B_k(z,p)$ satisfies the differential equation …and the difference equation … ►Connection formulas for the solutions of (9.13.31) include … ►

3: 22.10 Maclaurin Series

… ►Further terms may be derived from the differential equations (22.13.13), (22.13.14), (22.13.15), or from the integral representations of the inverse functions in §22.15(ii). …

4: 19.16 Definitions

… ►

19.16.2_5 $$R_G(x,y,z)=\frac{1}{4}{\int }_0^{\mathrm{\infty }}\frac{1}{s(t)}\left(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)tdt.$$

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Defines:

$R_G(x,y,z)$: symmetric elliptic integral of second kind

Symbols:

$dx$: differential of $x$ , $\int$: integral and $s(t)$: scaling factor

Source:

Carlson (1977b, (9.1-9))

Referenced by:

§19.16(i), §19.16(i), §19.21(ii), (19.23.7), (19.23.7), §19.23, Erratum (V1.1.0) for Rearrangement

Permalink:

http://dlmf.nist.gov/19.16.E2_5

Encodings:

pMML, png, TeX

Rearrangement (effective with 1.1.0):

This integralrepresentation was moved from (19.23.7) and is taken as the definition of $R_G(x,y,z)$. We have also employed the notation $s(t)$ for the factor of radicals.

See also:

Annotations for §19.16(i), §19.16 and Ch.19

…

5: 18.10 Integral Representations

§18.10 Integral Representations

… ►

Ultraspherical

… ►

Legendre

… ►

Jacobi

… ►

Ultraspherical

…

6: 25.11 Hurwitz Zeta Function

… ►

25.11.27 $$\zeta (s,a)=\frac{1}{2}{a}^{-s}+\frac{a^{1-s}}{s-1}+\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\left(\frac{1}{{\mathrm{e}}^x-1}-\frac{1}{x}+\frac{1}{2}\right)x^{s-1}{\mathrm{e}}^{-ax}dx,$$ $\mathrm{\Re }s>-1$, $s\ne 1$, $\mathrm{\Re }a>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

improper integral, integralrepresentation

Proof sketch:

Derivable from (25.11.25) by first rewriting $(1-{\mathrm{e}}^{-x})={\mathrm{e}}^{-x}({\mathrm{e}}^x-1)$, then replacing $x^{s-1}\mapsto x^{s-1}(1+{\mathrm{e}}^{-x}-{\mathrm{e}}^{-x})$, using the identity ${\mathrm{e}}^{-x}=(1-{\mathrm{e}}^{-x})/({\mathrm{e}}^x-1)$, and replacing ${({\mathrm{e}}^x-1)}^{-1}\mapsto ({({\mathrm{e}}^x-1)}^{-1}-1/x+1/2-1/2+1/x)$ in the resulting integrand with (5.2.1), (5.5.1).

Referenced by:

(25.11.28)

Permalink:

http://dlmf.nist.gov/25.11.E27

Encodings:

pMML, png, TeX

Update (effective with 1.1.9):

The fraction in the integrand $\frac{x^{s-1}}{{\mathrm{e}}^{ax}}$ was rewritten to read $x^{s-1}{\mathrm{e}}^{-ax}$.

See also:

Annotations for §25.11(vii), §25.11 and Ch.25

►

25.11.28 $$\zeta (s,a)=\frac{1}{2}{a}^{-s}+\frac{a^{1-s}}{s-1}+\sum _{k=1}^n\frac{B_{2k}}{(2k)!}{\left(s\right)}_{2k-1}{a}^{1-s-2k}+\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\left(\frac{1}{{\mathrm{e}}^x-1}-\frac{1}{x}+\frac{1}{2}-\sum _{k=1}^n\frac{B_{2k}}{(2k)!}{x}^{2k-1}\right)x^{s-1}{\mathrm{e}}^{-ax}dx,$$ $\mathrm{\Re }s>-(2n+1)$, $s\ne 1$, $\mathrm{\Re }a>0$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

improper integral, integralrepresentation

Proof sketch:

Derivable from (25.11.27) by adding and subtracting ${\sum }_{k=1}^n{B}_{2k}{x}^{2k-1}/(2k)!$ in the integrand, using (5.2.1), (5.2.5), and recognizing that ${({\mathrm{e}}^x-1)}^{-1}-x^{-1}+1/2-{\sum }_{k=1}^n{B}_{2k}{x}^{2k-1}/(2k)!=O\left(x^{2n+1}\right)$ as $x\to 0$, demonstrating the region of convergence.

Referenced by:

Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/25.11.E28

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

We have rewritten the original summation ${\sum }_{k=1}^n\frac{\mathrm{\Gamma }\left(s+2k-1\right)}{\mathrm{\Gamma }\left(s\right)}\frac{B_{2k}}{(2k)!}{a}^{-2k-s+1}$ more concisely as ${\sum }_{k=1}^n\frac{B_{2k}}{(2k)!}{\left(s\right)}_{2k-1}{a}^{1-s-2k}$ using the Pochhammer symbol.

See also:

Annotations for §25.11(vii), §25.11 and Ch.25

… ►

25.11.35 $$\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{{(n+a)}^s}=\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}{\mathrm{e}}^{-ax}}{1+{\mathrm{e}}^{-x}}dx=2^{-s}\left(\zeta (s,\frac{1}{2}a)-\zeta (s,\frac{1}{2}(1+a))\right),$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }s>0$; or $\mathrm{\Re }a=0$, $\mathrm{\Im }a\ne 0$, .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\int$: integral, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

improper integral, infinite series, integralrepresentation, series representation

Proof sketch:

The infinite series connection to the difference of Hurwitz zeta functions is a restatement of (25.11.8). The integral representation is derivable from the difference of Hurwitz zeta functions by substituting (25.11.25) in each, making a change of variables $x=2t$, factoring, and expressing $1-{\mathrm{e}}^{-2t}=(1+{\mathrm{e}}^{-t})(1-{\mathrm{e}}^{-t})$.

Referenced by:

(25.11.31), §25.11(x), §25.11(x)

Permalink:

http://dlmf.nist.gov/25.11.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(x), §25.11 and Ch.25

…

7: 14.20 Conical (or Mehler) Functions

… ►

§14.20(iv) Integral Representation

… ►From (14.20.9) or (14.20.10) it is evident that ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }\left(\cos \theta \right)$ is positive for real $\theta$. … ►

14.20.12 $$g(x)={\int }_0^{\mathrm{\infty }}{P}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)f(\tau )d\tau .$$

ⓘ

Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $x$: real variable, $\tau$: real variable and $\mu$: general order

Permalink:

http://dlmf.nist.gov/14.20.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(vi), §14.20 and Ch.14

… ►

§14.20(x) Zeros and Integrals

… ►For integrals with respect to $\tau$ involving ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$, see Prudnikov et al. (1990, pp. 218–228).

8: 25.5 Integral Representations

… ►

25.5.2 $$\zeta \left(s\right)=\frac{1}{\mathrm{\Gamma }\left(s+1\right)}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^x{x}^s}{{({\mathrm{e}}^x-1)}^2}dx,$$ $\mathrm{\Re }s>1$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integralrepresentation

Proof sketch:

Derivable from (25.5.1) by integration by parts.

Permalink:

http://dlmf.nist.gov/25.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(i), §25.5 and Ch.25

… ►

25.5.4 $$\zeta \left(s\right)=\frac{1}{(1-2^{1-s})\mathrm{\Gamma }\left(s+1\right)}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^x{x}^s}{{({\mathrm{e}}^x+1)}^2}dx,$$ $\mathrm{\Re }s>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integralrepresentation

Proof sketch:

Derivable from (25.5.3) by integration by parts.

Permalink:

http://dlmf.nist.gov/25.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(i), §25.5 and Ch.25

… ►

25.5.6 $$\zeta \left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\left(\frac{1}{{\mathrm{e}}^x-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{{\mathrm{e}}^x}dx,$$ $\mathrm{\Re }s>-1$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integralrepresentation

Proof sketch:

Derivable from (25.5.1) by first replacing $x^{s-1}\mapsto x^{s-1}(1+{\mathrm{e}}^{-x}-{\mathrm{e}}^{-x})$, using the identity ${\mathrm{e}}^{-x}=(1-{\mathrm{e}}^{-x})/({\mathrm{e}}^x-1)$, and replacing ${({\mathrm{e}}^x-1)}^{-1}\mapsto ({({\mathrm{e}}^x-1)}^{-1}-1/x+1/2-1/2+1/x)$ in the resulting integrand with (5.2.1), (5.5.1).

Referenced by:

(25.5.7)

Permalink:

http://dlmf.nist.gov/25.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(i), §25.5 and Ch.25

►

25.5.7 $$\zeta \left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\sum _{m=1}^n\frac{B_{2m}}{(2m)!}{\left(s\right)}_{2m-1}+\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\left(\frac{1}{{\mathrm{e}}^x-1}-\frac{1}{x}+\frac{1}{2}-\sum _{m=1}^n\frac{B_{2m}}{(2m)!}{x}^{2m-1}\right)\frac{x^{s-1}}{{\mathrm{e}}^x}dx,$$ $\mathrm{\Re }s>-(2n+1)$, $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\mathrm{\Re }$: real part, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integralrepresentation

Proof sketch:

Derivable from (25.5.6) by adding and subtracting ${\sum }_{m=1}^n{B}_{2m}{x}^{2m-1}/(2m)!$ in the integrand, using (5.2.1), (5.2.5), and recognizing that ${({\mathrm{e}}^x-1)}^{-1}-x^{-1}+1/2-{\sum }_{m=1}^n{B}_{2m}{x}^{2m-1}/(2m)!=O\left(x^{2n+1}\right)$ as $x\to 0$, demonstrating the region of convergence.

Referenced by:

Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/25.5.E7

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

We have rewritten the original summation ${\sum }_{m=1}^n\frac{B_{2m}}{(2m)!}\frac{\mathrm{\Gamma }\left(s+2m-1\right)}{\mathrm{\Gamma }\left(s\right)}$ more concisely as ${\sum }_{m=1}^n\frac{B_{2m}}{(2m)!}{\left(s\right)}_{2m-1}$ using the Pochhammer symbol.

See also:

Annotations for §25.5(i), §25.5 and Ch.25

…

9: 6.7 Integral Representations

§6.7 Integral Representations

… ► ►

§6.7(ii) Sine and Cosine Integrals

… ►

§6.7(iii) Auxiliary Functions

… ►For collections of integral representations see Bierens de Haan (1939, pp. 56–59, 72–73, 82–84, 121, 133–136, 155, 179–181, 223, 225–227, 230, 259–260, 374, 377, 397–398, 408, 416, 424, 431, 438–439, 442–444, 488, 496–500, 567–571, 585, 602, 638, 675–677), Corrington (1961), Erdélyi et al. (1954a, vol. 1, pp. 267–270), Geller and Ng (1969), Nielsen (1906b), Oberhettinger (1974, pp. 244–246), Oberhettinger and Badii (1973, pp. 364–371), and Watrasiewicz (1967).

10: 12.14 The Function $W(a,x)$

… ►Other expansions, involving $\cos \left(\frac{1}{4}{x}^2\right)$ and $\sin \left(\frac{1}{4}{x}^2\right)$, can be obtained from (12.4.3) to (12.4.6) by replacing $a$ by $-\mathrm{i}a$ and $z$ by $x{\mathrm{e}}^{\pi \mathrm{i}/4}$; see Miller (1955, p. 80), and also (12.14.15) and (12.14.16). ►

§12.14(vi) Integral Representations

►These follow from the contour integrals of §12.5(ii), which are valid for general complex values of the argument $z$ and parameter $a$. … ►follows from (12.2.3), and has solutions $W(\frac{1}{2}{\mu }^2,\pm \mu t\sqrt{2})$. …In the following expansions, obtained from Olver (1959), $\mu$ is large and positive, and $\delta$ is again an arbitrary small positive constant. …