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⇱ DLMF: §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function

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5 Gamma FunctionProperties5.1 Special Notation5.3 Graphics

§5.2 Definitions

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Referenced by:
§1.15(vi), §10.23(iii), §10.43(ii), §10.44(iii), §25.5(ii)
Permalink:
http://dlmf.nist.gov/5.2
See also:
Annotations for Ch.5
Contents
  1. §5.2(i) Gamma and Psi Functions
  2. §5.2(ii) Euler’s Constant
  3. §5.2(iii) Pochhammer’s Symbol

§5.2(i) Gamma and Psi Functions

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Notes:
See Olver (1997b, pp. 31 and 39) or Temme (1996b, pp. 41 and 53).
Referenced by:
§10.1, §10.22(ii), §10.31, §10.8, §13.1, §14.1, §14.17(iv), §15.1, §19.12, §2.3(ii), §2.5(i), §25.1, §32.11(ii), §33.1, §33.6, §6.10(ii), §6.6, §8.1, §8.19(iv)
Permalink:
http://dlmf.nist.gov/5.2.i
See also:
Annotations for §5.2 and Ch.5

Euler’s Integral

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5.2.1
.
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Defines:
: gamma function
Symbols:
: differential of , : base of natural logarithm, : integral, : real part and : complex variable
A&S Ref:
6.1.1
Referenced by:
§10.43(iii), (25.11.27), (25.11.28), (25.5.6), (25.5.7), §5.9(i), §5.9(ii), §8.21(ii), (9.12.17)
Permalink:
http://dlmf.nist.gov/5.2.E1
Encodings:
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See also:
Annotations for §5.2(i), §5.2(i), §5.2 and Ch.5

When , is defined by analytic continuation. It is a meromorphic function with no zeros, and with simple poles of residue at . is entire, with simple zeros at .

5.2.2
.
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Defines:
: psi (or digamma) function
Symbols:
: gamma function and : complex variable
A&S Ref:
6.3.1
Permalink:
http://dlmf.nist.gov/5.2.E2
Encodings:
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See also:
Annotations for §5.2(i), §5.2(i), §5.2 and Ch.5

is meromorphic with simple poles of residue at .

§5.2(ii) Euler’s Constant

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Keywords:
Euler’s constant
Notes:
See Olver (1997b, p. 34) or Temme (1996b, p. 10).
Referenced by:
§10.22(iii), §10.24, §10.45, §10.8, §10.9(i), §11.6(i), §13.1, §14.1, §14.8(i), §2.10(i), §2.5(iii), §2.6(iii), §25.1, §25.2(iv), §26.8(vii), §27.11, §3.12, §33.5(iii), §5.1, §6.1, §7.1, §9.12(viii)
Permalink:
http://dlmf.nist.gov/5.2.ii
See also:
Annotations for §5.2 and Ch.5
5.2.3
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Defines:
: Euler’s constant
Symbols:
: principal branch of logarithm function and : 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:
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See also:
Annotations for §5.2(ii), §5.2 and Ch.5

§5.2(iii) Pochhammer’s Symbol

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Defines:
: Pochhammer’s symbol (or shifted factorial)
Keywords:
Lah numbers, Pochhammer’s symbol, definitions
Referenced by:
§3.9(v), Erratum (V1.0.17) for Subsection 5.2(iii)
Permalink:
http://dlmf.nist.gov/5.2.iii
Addition (effective with 1.2.0):
A paragraph under (5.2.8) about Lah numbers and (5.2.9) was added.

Suggested 2024-02-26 by Eric Shirley

Addition (effective with 1.0.17):
Equations (5.2.6)–(5.2.8) were added.
See also:
Annotations for §5.2 and Ch.5
5.2.4
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5.2.5
.
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5.2.6
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Symbols:
: Pochhammer’s symbol (or shifted factorial), : nonnegative integer and : 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:
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Addition (effective with 1.0.17):
This equation was added.
See also:
Annotations for §5.2(iii), §5.2 and Ch.5
5.2.7
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Symbols:
: Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : nonnegative integer and : nonnegative integer
Permalink:
http://dlmf.nist.gov/5.2.E7
Encodings:
TeX, pMML, png
Addition (effective with 1.0.17):
This equation was added.
See also:
Annotations for §5.2(iii), §5.2 and Ch.5
5.2.8
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Symbols:
: Pochhammer’s symbol (or shifted factorial), : nonnegative integer and : 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.E8
Encodings:
TeX, TeX, pMML, pMML, png, png
Addition (effective with 1.0.17):
This equation was added.

Suggested by Tom Koornwinder

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

Pochhammer symbols (rising factorials) and falling factorials can be expressed in terms of each other via

5.2.9
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Symbols:
: Pochhammer’s symbol (or shifted factorial), : nonnegative integer, : nonnegative integer, : real variable and : Lah number
Referenced by:
§5.2(iii), Erratum (V1.2.0) for Chapter 5 Addition
Permalink:
http://dlmf.nist.gov/5.2.E9
Encodings:
TeX, TeX, pMML, pMML, png, png
Addition (effective with 1.2.0):
This equation was added.

Suggested by Eric Shirley

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

in which is the Lah number.

5.1 Special Notation5.3 Graphics
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