Euler%E2%80%93Maclaurin

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1: 24.1 Special Notation

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Euler Numbers and Polynomials

… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations $E_n$, $E_n\left(x\right)$, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

2: Bibliography H

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The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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External Links:

MathReview (L. Berg)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

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Note on Dr. Vacca’s series for $\gamma$ . Quart. J. Math. 43, pp. 215–216.

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External Links:

ZentralBlatt

Referenced by:

(25.2.4), (25.2.5), §25.2(ii).

Encodings:

BibTeX

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An Euler-Maclaurin-type formula involving conjugate Bernoulli polynomials and an application to $\zeta (2m+1)$ . Commun. Appl. Anal. 1 (1), pp. 15–32.

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External Links:

ISSN 1083-2564, MathReview (Pavel Guerzhoy), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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A Boole-type Formula involving Conjugate Euler Polynomials. In Charlemagne and his Heritage. 1200 Years of Civilization and Science in Europe, Vol. 2 (Aachen, 1995), P.L. Butzer, H. Th. Jongen, and W. Oberschelp (Eds.), pp. 361–375.

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External Links:

ISBN 2-503-50674-7, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

… ►

Roots of the Euler polynomials. Pacific J. Math. 64 (1), pp. 181–191.

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External Links:

MathReview (M. Marden), ZentralBlatt

Referenced by:

§24.12(ii).

Encodings:

BibTeX

…

3: 27.2 Functions

… ►This is the number of positive integers $\le n$ that are relatively prime to $n$; $\varphi \left(n\right)$ is Euler’s totient. ►If $\left(a,n\right)=1$, then the Euler–Fermat theorem states that …The $\varphi \left(n\right)$ numbers $a,a^2,\mathrm{\dots },a^{\varphi \left(n\right)}$ are relatively prime to $n$ and distinct (mod $n$). …Note that $J_1\left(n\right)=\varphi \left(n\right)$. … ►Table 27.2.2 tabulates the Euler totient function $\varphi \left(n\right)$, the divisor function $d\left(n\right)$ ($={\sigma }_0(n)$), and the sum of the divisors $\sigma (n)$ ($={\sigma }_1(n)$), for $n=1(1)52$. …

4: 24.17 Mathematical Applications

§24.17 Mathematical Applications

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Euler–Maclaurin Summation Formula

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Euler Splines

… ►are called Euler splines of degree $n$. … ►

5: 5.7 Series Expansions

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§5.7(i) Maclaurin and Taylor Series

… ►where $c_1=1$, $c_2=\gamma$, and … ►

5.7.3 $$\ln \mathrm{\Gamma }\left(1+z\right)=-\ln \left(1+z\right)+z(1-\gamma )+\sum _{k=2}^{\mathrm{\infty }}{(-1)}^k(\zeta \left(k\right)-1)\frac{z^k}{k},$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.33

Referenced by:

§5.21, §5.7(i)

Permalink:

http://dlmf.nist.gov/5.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.7(i), §5.7 and Ch.5

… ►For 20D numerical values of the coefficients of the Maclaurin series for $\mathrm{\Gamma }\left(z+3\right)$ see Luke (1969b, p. 299). … ►

5.7.6 $$\psi \left(z\right)=-\gamma -\frac{1}{z}+\sum _{k=1}^{\mathrm{\infty }}\frac{z}{k(k+z)}=-\gamma +\sum _{k=0}^{\mathrm{\infty }}\left(\frac{1}{k+1}-\frac{1}{k+z}\right),$$

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Symbols:

$\gamma$: Euler’s constant, $\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable

Referenced by:

§5.19(i)

Permalink:

http://dlmf.nist.gov/5.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §5.7(ii), §5.7 and Ch.5

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6: 10.31 Power Series

… ►where $\psi \left(x\right)={\mathrm{\Gamma }}^{\prime }\left(x\right)/\mathrm{\Gamma }\left(x\right)$ (§5.2(i)). … ►

10.31.2 $$K_0\left(z\right)=-\left(\ln \left(\frac{1}{2}z\right)+\gamma \right)I_0\left(z\right)+\frac{\frac{1}{4}{z}^2}{{(1!)}^2}+(1+\frac{1}{2})\frac{{(\frac{1}{4}{z}^2)}^2}{{(2!)}^2}+(1+\frac{1}{2}+\frac{1}{3})\frac{{(\frac{1}{4}{z}^2)}^3}{{(3!)}^2}+\mathrm{\cdots }.$$

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Symbols:

$\gamma$: Euler’s constant, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

9.6.13

Referenced by:

§10.45

Permalink:

http://dlmf.nist.gov/10.31.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.31 and Ch.10

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10.31.3 $$I_{\nu }\left(z\right)I_{\mu }\left(z\right)={(\frac{1}{2}z)}^{\nu +\mu }\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\nu +\mu +k+1\right)}_k{(\frac{1}{4}{z}^2)}^k}{k!\mathrm{\Gamma }\left(\nu +k+1\right)\mathrm{\Gamma }\left(\mu +k+1\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.31

Permalink:

http://dlmf.nist.gov/10.31.E3

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §10.31 and Ch.10

7: 31.3 Basic Solutions

… ► $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ denotes the solution of (31.2.1) that corresponds to the exponent $0$ at $z=0$ and assumes the value $1$ there. If the other exponent is not a positive integer, that is, if $\gamma \ne 0,-1,-2,\mathrm{\dots }$, then from §2.7(i) it follows that $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ exists, is analytic in the disk , and has the Maclaurin expansion … ►Similarly, if $\gamma \ne 1,2,3,\mathrm{\dots }$, then the solution of (31.2.1) that corresponds to the exponent $1-\gamma$ at $z=0$ is … ►When $\gamma \in \mathbb{Z}$, linearly independent solutions can be constructed as in §2.7(i). … ►For example, $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ is equal to …

8: 24.18 Physical Applications

§24.18 Physical Applications

►Bernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)). ►Euler polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (Ballentine and McRae (1998)).

9: 12.12 Integrals

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12.12.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{4}{t}^2}{t}^{\mu -1}U(a,t)dt=\frac{\sqrt{\pi }{2}^{-\frac{1}{2}(\mu +a+\frac{1}{2})}\mathrm{\Gamma }\left(\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}(\mu +a+\frac{3}{2})\right)},$$ $\mathrm{\Re }\mu >0$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.12 and Ch.12

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12.12.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{3}{4}{t}^2}{t}^{-a-\frac{3}{2}}U(a,t)dt=2^{\frac{1}{4}+\frac{1}{2}a}\mathrm{\Gamma }\left(-a-\frac{1}{2}\right)\cos \left((\frac{1}{4}a+\frac{1}{8})\pi \right),$$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.12 and Ch.12

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12.12.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{4}{t}^2}{t}^{-a-\frac{1}{2}}{(x^2+t^2)}^{-1}U(a,t)dt=\sqrt{\pi /2}\mathrm{\Gamma }\left(\frac{1}{2}-a\right)x^{-a-\frac{3}{2}}{\mathrm{e}}^{\frac{1}{4}{x}^2}U(-a,x),$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part, $x$: real variable and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.12 and Ch.12

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12.12.4 $${(U(a,z))}^2+{(\overline{U}(a,z))}^2=\frac{2^{\frac{3}{2}}}{\pi }\mathrm{\Gamma }\left(\frac{1}{2}-a\right){\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2at+\frac{1}{2}{z}^2\tanh t}}{\sqrt{\sinh \left(2t\right)}}dt,$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\overline{U}(a,x)$: parabolic cylinder function, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: real or complex parameter

Notes:

See Durand (1975).

Referenced by:

§12.12

Permalink:

http://dlmf.nist.gov/12.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.12, §12.12 and Ch.12

… ►For compendia of integrals see Erdélyi et al. (1953b, v. 2, pp. 121–122), Erdélyi et al. (1954a, b, v. 1, pp. 60–61, 115, 210–211, and 336; v. 2, pp. 76–80, 115, 151, 171, and 395–398), Gradshteyn and Ryzhik (2000, §7.7), Magnus et al. (1966, pp. 330–331), Marichev (1983, pp. 190–191), Oberhettinger (1974, pp. 144–145), Oberhettinger (1990, pp. 106–108 and 192), Oberhettinger and Badii (1973, pp. 181–185), Prudnikov et al. (1986b, pp. 36–37, 155–168, 243–246, 289–290, 327–328, 419–420, and 619), Prudnikov et al. (1992a, §3.11), and Prudnikov et al. (1992b, §3.11). …

10: 25.2 Definition and Expansions

… ►where the Stieltjes constants ${\gamma }_n$ are defined via … ►

§25.2(iii) Representations by the Euler–Maclaurin Formula

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25.2.8 $$\zeta \left(s\right)=\sum _{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-s{\int }_N^{\mathrm{\infty }}\frac{x-\lfloor x\rfloor }{x^{s+1}}dx,$$ $\mathrm{\Re }s>0$, $N=1,2,3,\mathrm{\dots }$.

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\lfloor x\rfloor$: floor of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Source:

Apostol (1976, (25), p. 269)

Proof sketch:

Derivable from Apostol (1976, Theorem 12.21, p. 269) by setting $a=1$ and $N\mapsto N-1$.

A&S Ref:

23.2.9

Referenced by:

(25.2.9)

Permalink:

http://dlmf.nist.gov/25.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iii), §25.2 and Ch.25

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25.2.10 $$\zeta \left(s\right)=\frac{1}{s-1}+\frac{1}{2}+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_1^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{x^{s+2n+1}}dx,$$ $\mathrm{\Re }s>-2n$, $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler–Maclaurin formula

Proof sketch:

Derivable from (25.2.9) with $N=1$.

A&S Ref:

23.2.3

Permalink:

http://dlmf.nist.gov/25.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iii), §25.2 and Ch.25

… ►product over zeros $\rho$ of $\zeta$ with $\mathrm{\Re }\rho >0$ (see §25.10(i)); $\gamma$ is Euler’s constant (§5.2(ii)).