Olver hypergeometric function
(0.007 seconds)
1—10 of 69 matching pages
1: 15.1 Special Notation
…
►and also
►
15.1.2
…
ⓘ
- Symbols:
- : gamma function, or : Gauss’ hypergeometric function, or : scaled (or Olver’s) generalized hypergeometric function, or : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/15.1.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.1 and Ch.15
2: 15.6 Integral Representations
…
►
15.6.1
; .
►
ⓘ
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : integral, : phase, : real part, or : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Keywords:
- Mellin transform
- Referenced by:
- §15.19(iii), (15.4.34), §15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)
- Permalink:
- http://dlmf.nist.gov/15.6.E1
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.23):
- The Olverhypergeometricfunction , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.
- See also:
- Annotations for §15.6 and Ch.15
15.6.2
; , .
…
►
ⓘ
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : imaginary unit, : integral, : phase, : real part, or : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Keywords:
- Mellin transform
- Referenced by:
- §15.6
- Permalink:
- http://dlmf.nist.gov/15.6.E2
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.23):
- The Olverhypergeometricfunction , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.
- See also:
- Annotations for §15.6 and Ch.15
15.6.6
; .
►
ⓘ
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : imaginary unit, : integral, : phase, or : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- A&S Ref:
- 15.3.2 (modified)
- Referenced by:
- §15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)
- Permalink:
- http://dlmf.nist.gov/15.6.E6
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.23):
- The Olverhypergeometricfunction , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.
- See also:
- Annotations for §15.6 and Ch.15
15.6.7
; .
►
ⓘ
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : imaginary unit, : integral, : phase, or : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Referenced by:
- §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)
- Permalink:
- http://dlmf.nist.gov/15.6.E7
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.23):
- The Olverhypergeometricfunction , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.
- See also:
- Annotations for §15.6 and Ch.15
15.6.8
; .
…
ⓘ
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : integral, : phase, : real part, or : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Keywords:
- Mellin transform
- Proof sketch:
- Follows from (15.6.9) with .
- Referenced by:
- (15.6.9), §15.6, §15.6, §15.6, §18.17(iv), Erratum (V1.0.14) for Equation (15.6.8), Erratum (V1.0.14) for Equation (15.6.8)
- Permalink:
- http://dlmf.nist.gov/15.6.E8
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.23):
- The Olverhypergeometricfunction , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.
- See also:
- Annotations for §15.6 and Ch.15
3: 13.1 Special Notation
…
►The main functions treated in this chapter are the Kummer functions
and , Olver’s function
, and the Whittaker functions
and .
…
4: 14.3 Definitions and Hypergeometric Representations
…
►
14.3.3
►is Olver’s hypergeometric function (§15.1).
…
►
ⓘ
- Symbols:
- : gamma function, or : Gauss’ hypergeometric function, or : Olver’s hypergeometric function and : real variable
- Permalink:
- http://dlmf.nist.gov/14.3.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.3(i), §14.3(i), §14.3 and Ch.14
14.3.6
…
►
ⓘ
- Symbols:
- : associated Legendre function of the first kind, or : Olver’s hypergeometric function, : general order, : general degree and : real variable
- Referenced by:
- §14.21(i), §14.21(iii), §14.3(ii), §14.3(iv), §15.9(iv)
- Permalink:
- http://dlmf.nist.gov/14.3.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14
14.3.9
…
►
ⓘ
- Symbols:
- : associated Legendre function of the first kind, or : Olver’s hypergeometric function, : general order, : general degree and : real variable
- Permalink:
- http://dlmf.nist.gov/14.3.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14
14.3.15
…
ⓘ
- Symbols:
- : associated Legendre function of the first kind, or : Olver’s hypergeometric function, : general order, : general degree and : real variable
- Referenced by:
- §14.21(iii)
- Permalink:
- http://dlmf.nist.gov/14.3.E15
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.3(iii), §14.3 and Ch.14
5: 15.7 Continued Fractions
…
►
15.7.1
…
►
ⓘ
- Symbols:
- or : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter, : coefficient and : coefficient
- Permalink:
- http://dlmf.nist.gov/15.7.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.7 and Ch.15
15.7.3
…
►
ⓘ
- Symbols:
- or : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter, : coefficient and : coefficient
- Permalink:
- http://dlmf.nist.gov/15.7.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.7 and Ch.15
15.7.5
…
ⓘ
- Symbols:
- or : Olver’s hypergeometric function, : real variable, : real variable, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/15.7.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.7 and Ch.15
6: 15.15 Sums
…
►
15.15.1
…
ⓘ
- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), or : Olver’s hypergeometric function, : nonnegative integer, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Referenced by:
- §15.19(i)
- Permalink:
- http://dlmf.nist.gov/15.15.E1
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.1.2):
- The Pochhammer symbol now links to its definition.
- See also:
- Annotations for §15.15 and Ch.15
7: 15.14 Integrals
…
►
15.14.1
.
…
ⓘ
- Symbols:
- : gamma function, : differential of , : integral, : real part, or : Olver’s hypergeometric function, : real variable, : nonnegative integer, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Keywords:
- Mellin transform
- Referenced by:
- §15.14
- Permalink:
- http://dlmf.nist.gov/15.14.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.14 and Ch.15
8: 15.2 Definitions and Analytical Properties
…
►
15.2.2
,
…
►
ⓘ
- Defines:
- or : Olver’s hypergeometric function
- Symbols:
- : gamma function, : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), or : scaled (or Olver’s) generalized hypergeometric function, : nonnegative integer, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Referenced by:
- §15.12(i), §16.2(v)
- Permalink:
- http://dlmf.nist.gov/15.2.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.2(i), §15.2 and Ch.15
15.2.3
.
…
►
ⓘ
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : imaginary unit, or : Olver’s hypergeometric function, : real variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Referenced by:
- §15.2(i)
- Permalink:
- http://dlmf.nist.gov/15.2.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.2(i), §15.2 and Ch.15
15.2.3_5
.
…
ⓘ
- Symbols:
- : gamma function, or : Gauss’ hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), or : Olver’s hypergeometric function, : integer, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Referenced by:
- §15.2(ii), Erratum (V1.0.28) for Equations (15.2.3_5), (19.11.6_5)
- Permalink:
- http://dlmf.nist.gov/15.2.E3_5
- Encodings:
- pMML, png, TeX
- Clarification (effective with 1.0.28):
- This equation was given a number.
- See also:
- Annotations for §15.2(ii), §15.2 and Ch.15
9: 13.10 Integrals
…
►
13.10.1
…
►
ⓘ
- Symbols:
- : Olver’s confluent hypergeometric function, : differential of , : integral and : complex variable
- Permalink:
- http://dlmf.nist.gov/13.10.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.10(i), §13.10 and Ch.13
13.10.3
, ,
…
►
ⓘ
- Symbols:
- : gamma function, : Olver’s confluent hypergeometric function, : differential of , : base of natural logarithm, or : scaled (or Olver’s) generalized hypergeometric function, : integral, : real part and : complex variable
- Referenced by:
- §8.14
- Permalink:
- http://dlmf.nist.gov/13.10.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.10(ii), §13.10 and Ch.13
13.10.5
,
…
►
ⓘ
- Symbols:
- : gamma function, : Olver’s confluent hypergeometric function, : differential of , : base of natural logarithm, : integral and : real part
- Permalink:
- http://dlmf.nist.gov/13.10.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.10(ii), §13.10 and Ch.13
13.10.10
,
…
►
ⓘ
- Symbols:
- : gamma function, : Olver’s confluent hypergeometric function, : differential of , : integral and : real part
- Keywords:
- Mellin transform
- Referenced by:
- §8.14
- Permalink:
- http://dlmf.nist.gov/13.10.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.10(iii), §13.10 and Ch.13
13.10.14
, ,
…
ⓘ
- Symbols:
- : Bessel function of the first kind, : gamma function, : Kummer confluent hypergeometric function, : Olver’s confluent hypergeometric function, : differential of , : base of natural logarithm, : integral, : real part and : real variable
- Referenced by:
- §13.10(v)
- Permalink:
- http://dlmf.nist.gov/13.10.E14
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.10(v), §13.10 and Ch.13
10: 13.4 Integral Representations
…
►
13.4.1
,
►
ⓘ
- Symbols:
- : gamma function, : Olver’s confluent hypergeometric function, : differential of , : base of natural logarithm, : integral, : real part and : complex variable
- A&S Ref:
- 13.2.1 (in different form)
- Referenced by:
- §13.10(v), §13.29(iii)
- Permalink:
- http://dlmf.nist.gov/13.4.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.4(i), §13.4 and Ch.13
13.4.2
,
►
ⓘ
- Symbols:
- : gamma function, : Olver’s confluent hypergeometric function, : differential of , : integral, : real part, : complex variable and : arbitrary
- Referenced by:
- §13.10(vi)
- Permalink:
- http://dlmf.nist.gov/13.4.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.4(i), §13.4 and Ch.13
13.4.3
.
…
►
ⓘ
- Symbols:
- : Bessel function of the first kind, : gamma function, : Olver’s confluent hypergeometric function, : differential of , : base of natural logarithm, : integral, : real part and : complex variable
- Referenced by:
- §13.10(v)
- Permalink:
- http://dlmf.nist.gov/13.4.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.4(i), §13.4 and Ch.13
13.4.12
, .
…
►
ⓘ
- Symbols:
- : gamma function, : Olver’s confluent hypergeometric function, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, or : scaled (or Olver’s) generalized hypergeometric function, : imaginary unit, : integral, : open interval, : phase and : complex variable
- Permalink:
- http://dlmf.nist.gov/13.4.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.4(ii), §13.4 and Ch.13
13.4.13
.
…
ⓘ
- Symbols:
- : Olver’s confluent hypergeometric function, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : imaginary unit, : integral, : open interval, : phase and : complex variable
- Permalink:
- http://dlmf.nist.gov/13.4.E13
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.4(ii), §13.4 and Ch.13