Euler%E2%80%93Maclaurin%20formula

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11: 5.8 Infinite Products

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5.8.1 $$\mathrm{\Gamma }\left(z\right)=\underset{k\to \mathrm{\infty }}{lim}\frac{k!k^z}{z(z+1)\mathrm{\cdots }(z+k)},$$ $z\ne 0,-1,-2,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.2

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E1

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Annotations for §5.8 and Ch.5

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5.8.2 $$\frac{1}{\mathrm{\Gamma }\left(z\right)}=z{\mathrm{e}}^{\gamma z}\prod _{k=1}^{\mathrm{\infty }}\left(1+\frac{z}{k}\right){\mathrm{e}}^{-z/k},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.3

Permalink:

http://dlmf.nist.gov/5.8.E2

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Annotations for §5.8 and Ch.5

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5.8.3 $${\left|\frac{\mathrm{\Gamma }\left(x\right)}{\mathrm{\Gamma }\left(x+\mathrm{i}y\right)}\right|}^2=\prod _{k=0}^{\mathrm{\infty }}\left(1+\frac{y^2}{{(x+k)}^2}\right),$$ $x\ne 0,-1,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable and $y$: real variable

A&S Ref:

6.1.25 (where the formula for the reciprocal is given.)

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E3

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Annotations for §5.8 and Ch.5

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5.8.5 $$\prod _{k=0}^{\mathrm{\infty }}\frac{(a_1+k)(a_2+k)\mathrm{\cdots }(a_m+k)}{(b_1+k)(b_2+k)\mathrm{\cdots }(b_m+k)}=\frac{\mathrm{\Gamma }\left(b_1\right)\mathrm{\Gamma }\left(b_2\right)\mathrm{\cdots }\mathrm{\Gamma }\left(b_m\right)}{\mathrm{\Gamma }\left(a_1\right)\mathrm{\Gamma }\left(a_2\right)\mathrm{\cdots }\mathrm{\Gamma }\left(a_m\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $m$: nonnegative integer, $k$: nonnegative integer, $a_k$: coefficient and $b_k$: coefficient

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E5

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Annotations for §5.8 and Ch.5

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12: 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

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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

13: 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

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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:

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See also:

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

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14: 24.6 Explicit Formulas

§24.6 Explicit Formulas

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24.6.4 $$E_{2n}=\sum _{k=1}^n\frac{1}{2^{k-1}}\sum _{j=1}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{2k}{k-j}\right)j^{2n},$$

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Symbols:

$E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E4

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Annotations for §24.6 and Ch.24

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24.6.6 $$E_{2n}=\sum _{k=1}^{2n}\frac{{(-1)}^k}{2^{k-1}}\left(\genfrac{}{}{0pt}{}{2n+1}{k+1}\right)\sum _{j=0}^{\lfloor \frac{1}{2}k-\frac{1}{2}\rfloor }\left(\genfrac{}{}{0pt}{}{k}{j}\right){(k-2j)}^{2n}.$$

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Symbols:

$E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\lfloor x\rfloor$: floor of $x$ , $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E6

Encodings:

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See also:

Annotations for §24.6 and Ch.24

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24.6.7 $$B_n\left(x\right)=\sum _{k=0}^n\frac{1}{k+1}\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{k}{j}\right){(x+j)}^n,$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer, $n$: integer and $x$: real or complex

Referenced by:

§24.6, §24.6

Permalink:

http://dlmf.nist.gov/24.6.E7

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Annotations for §24.6 and Ch.24

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24.6.12 $$E_{2n}=\sum _{k=0}^{2n}\frac{1}{2^k}\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{k}{j}\right){(1+2j)}^{2n}.$$

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Symbols:

$E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6, §24.6

Permalink:

http://dlmf.nist.gov/24.6.E12

Encodings:

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See also:

Annotations for §24.6 and Ch.24

15: Bibliography D

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On the real roots of Euler polynomials. Monatsh. Math. 106 (2), pp. 115–138.

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External Links:

ISSN 0026-9255, MathReview (H. M. Srivastava), ZentralBlatt

Referenced by:

§24.12(ii).

Encodings:

BibTeX

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Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials. J. Approx. Theory 49 (4), pp. 321–330.

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External Links:

ISSN 0021-9045, Document, MathReview (Guillermo López Lagomasino), ZentralBlatt

Referenced by:

§24.11.

Encodings:

BibTeX

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Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters. J. Number Theory 25 (1), pp. 72–80.

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External Links:

ISSN 0022-314X, Document, MathReview (T. Metsänkylä)

Referenced by:

§24.12(iii).

Encodings:

BibTeX

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Zeros of Bernoulli, generalized Bernoulli and Euler polynomials. Mem. Amer. Math. Soc. 73 (386), pp. iv+94.

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External Links:

ISSN 0065-9266, MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i), §24.12(iii), §24.16(ii).

Encodings:

BibTeX

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Error analysis in a uniform asymptotic expansion for the generalised exponential integral. J. Comput. Appl. Math. 80 (1), pp. 127–161.

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External Links:

ISSN 0377-0427, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§2.11(iii), §8.20(ii).

Encodings:

BibTeX

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16: 25.16 Mathematical Applications

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§25.16(ii) Euler Sums

►Euler sums have the form … ► $H\left(s\right)$ is the special case $H(s,1)$ of the function …which satisfies the reciprocity law …when both $H(s,z)$ and $H(z,s)$ are finite. …

17: 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 …

18: 24.5 Recurrence Relations

§24.5 Recurrence Relations

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24.5.2 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)E_k\left(x\right)+E_n\left(x\right)=2x^n,$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(i), §24.5 and Ch.24

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24.5.4 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{2n}{2k}\right)E_{2k}=0,$$ $n=1,2,\mathrm{\dots }$,

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Symbols:

$E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer

Referenced by:

§24.19(i)

Permalink:

http://dlmf.nist.gov/24.5.E4

Encodings:

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See also:

Annotations for §24.5(i), §24.5 and Ch.24

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24.5.5 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)2^k{E}_{n-k}+E_n=2.$$

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Symbols:

$E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/24.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(i), §24.5 and Ch.24

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§24.5(iii) Inversion Formulas

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19: 30.11 Radial Spheroidal Wave Functions

… ►Then solutions of (30.2.1) with $\mu =m$ and $\lambda ={\lambda }_n^m\left({\gamma }^2\right)$ are given by …Here $a_{n,k}^{-m}({\gamma }^2)$ is defined by (30.8.2) and (30.8.6), and … ►

Connection Formulas

… ►For fixed $\gamma$, as $z\to \mathrm{\infty }$ in the sector $|\mathrm{ph}z|\le \pi -\delta$ (), … ►where …

20: 5.4 Special Values and Extrema

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$\mathrm{\Gamma }\left(1\right)=1,$

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$n!=\mathrm{\Gamma }\left(n+1\right).$

… ►(The second line of Formula (5.4.2) also applies when $n=-1$.) … ►

5.4.11 $${\mathrm{\Gamma }}^{\prime }\left(1\right)=-\gamma .$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $\gamma$: Euler’s constant

A&S Ref:

6.3.2

Referenced by:

§5.4(i)

Permalink:

http://dlmf.nist.gov/5.4.E11

Encodings:

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See also:

Annotations for §5.4(i), §5.4 and Ch.5

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$\psi \left(1\right)=-\gamma ,$

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