罗伯特哥顿大学会计文凭证书【购证 微kaa77788】iii

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11: 5.2 Definitions

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5.2.3 $$\gamma =\underset{n\to \mathrm{\infty }}{lim}\left(1+\frac{1}{2}+\frac{1}{3}+\mathrm{\cdots }+\frac{1}{n}-\ln n\right)=\mathrm{0.57721\hspace{0.33em}56649\hspace{0.33em}01532\hspace{0.33em}86060}\mathrm{\dots }.$$

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Defines:

$\gamma$: Euler’s constant

Symbols:

$\ln z$: principal branch of logarithm function and $n$: nonnegative integer

A&S Ref:

6.1.3 (where the 10D value is given, and Table 1.1 where the 24D value is given.)

Notes:

For more digits see OEIS Sequence A001620; see also Sloane (2003).

Referenced by:

§4.4(iii)

Permalink:

http://dlmf.nist.gov/5.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.2(ii), §5.2 and Ch.5

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§5.2(iii) Pochhammer’s Symbol

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5.2.5 $${\left(a\right)}_n=\mathrm{\Gamma }\left(a+n\right)/\mathrm{\Gamma }\left(a\right),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $a$: real or complex variable

A&S Ref:

6.1.22

Referenced by:

(25.11.28), (25.5.7), (25.8.2)

Permalink:

http://dlmf.nist.gov/5.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.2(iii), §5.2 and Ch.5

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5.2.6 $${\left(-a\right)}_n={(-1)}^n{\left(a-n+1\right)}_n,$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $a$: real or complex variable

Referenced by:

§5.2(iii), Erratum (V1.0.17) for Subsection 5.2(iii)

Permalink:

http://dlmf.nist.gov/5.2.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation was added.

See also:

Annotations for §5.2(iii), §5.2 and Ch.5

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5.2.7

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.2.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation was added.

See also:

Annotations for §5.2(iii), §5.2 and Ch.5

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12: 5.5 Functional Relations

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§5.5(iii) Multiplication

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5.5.5 $$\mathrm{\Gamma }\left(2z\right)={\pi }^{-1/2}{2}^{2z-1}\mathrm{\Gamma }\left(z\right)\mathrm{\Gamma }\left(z+\frac{1}{2}\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

A&S Ref:

6.1.18

Referenced by:

§14.19(ii), §15.4(i), §15.4(iii), (25.9.2), §5.5(iii)

Permalink:

http://dlmf.nist.gov/5.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

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5.5.6 $$\mathrm{\Gamma }\left(nz\right)={(2\pi )}^{(1-n)/2}{n}^{nz-(1/2)}\prod _{k=0}^{n-1}\mathrm{\Gamma }\left(z+\frac{k}{n}\right).$$

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Symbols:

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

A&S Ref:

6.1.20

Referenced by:

§15.4(iii), §5.5(iii)

Permalink:

http://dlmf.nist.gov/5.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

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5.5.8 $$\psi \left(2z\right)=\frac{1}{2}\left(\psi \left(z\right)+\psi \left(z+\frac{1}{2}\right)\right)+\ln 2,$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

6.3.8

Referenced by:

§5.5(iii)

Permalink:

http://dlmf.nist.gov/5.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

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5.5.9 $$\psi \left(nz\right)=\frac{1}{n}\sum _{k=0}^{n-1}\psi \left(z+\frac{k}{n}\right)+\ln n.$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer, $k$: nonnegative integer and $z$: complex variable

Referenced by:

§5.5(iii)

Permalink:

http://dlmf.nist.gov/5.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

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13: 7.22 Methods of Computation

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§7.22(iii) Repeated Integrals of the Complementary Error Function

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14: 34 3j, 6j, 9j Symbols

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15: 9.15 Mathematical Applications

… ►For descriptions of, and references to, the underlying theory see §§2.4(v) and 2.8(iii).

16: 4.47 Approximations

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§4.47(iii) Padé Approximations

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17: 4.48 Software

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§4.48(iii) General Precision

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18: 8.24 Physical Applications

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§8.24(iii) Generalized Exponential Integral

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19: 11.16 Software

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§11.16(iii) Integrals of Struve Functions

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20: 12.21 Software

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§12.21(iii) Complex Arguments and Parameters

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