Olver’s
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11—20 of 146 matching pages
11: 13.4 Integral Representations
…
βΊ
13.4.1
,
βΊ
β
- 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
,
βΊ
β
- 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
.
…
βΊ
β
- 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
13.4.9
, .
…
βΊ
β
- 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, : imaginary unit, : integral, : real part and : complex variable
- Permalink:
- http://dlmf.nist.gov/13.4.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.4(ii), §13.4 and Ch.13
13.4.12
, .
…
β
- 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
12: 14.19 Toroidal (or Ring) Functions
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βΊ
14.19.3
…
βΊ
β
- Symbols:
- : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : hyperbolic cosine function, or : Olver’s hypergeometric function, : general order, : general degree and : variable
- Permalink:
- http://dlmf.nist.gov/14.19.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.19(ii), §14.19 and Ch.14
14.19.5
.
…
βΊ
β
- Symbols:
- : gamma function, : Olver’s associated Legendre function, : differential of , : hyperbolic cosine function, : hyperbolic sine function, : integral, : nonnegative integer, : nonnegative integer and : variable
- A&S Ref:
- 8.11.4 (modified and corrected)
- Permalink:
- http://dlmf.nist.gov/14.19.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.19(iii), §14.19 and Ch.14
14.19.6
.
…
βΊ
β
- Symbols:
- : gamma function, : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : cosine function, : hyperbolic cosine function, : hyperbolic sine function, : real part, : nonnegative integer, : general order and : variable
- Permalink:
- http://dlmf.nist.gov/14.19.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.19(iv), §14.19 and Ch.14
14.19.7
βΊ
β
- Symbols:
- : gamma function, : associated Legendre function of the first kind, : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : hyperbolic cosine function, : hyperbolic cotangent function, : hyperbolic sine function, : nonnegative integer, : nonnegative integer and : variable
- Permalink:
- http://dlmf.nist.gov/14.19.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.19(v), §14.19 and Ch.14
14.19.8
β
- Symbols:
- : gamma function, : associated Legendre function of the first kind, : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : hyperbolic cosine function, : hyperbolic cotangent function, : hyperbolic sine function, : nonnegative integer, : nonnegative integer and : variable
- Permalink:
- http://dlmf.nist.gov/14.19.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.19(v), §14.19 and Ch.14
13: 14.24 Analytic Continuation
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βΊ
14.24.1
βΊ
β
- Symbols:
- : gamma function, : associated Legendre function of the first kind, : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : cosine function, : base of natural logarithm, : imaginary unit, : sine function, : complex variable, : general order, : general degree and : arbitrary integer
- Referenced by:
- §14.24, §14.24
- Permalink:
- http://dlmf.nist.gov/14.24.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.24 and Ch.14
14.24.2
…
βΊ
β
- Symbols:
- : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : complex variable, : general order, : general degree and : arbitrary integer
- Referenced by:
- §14.23, §14.24
- Permalink:
- http://dlmf.nist.gov/14.24.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.24 and Ch.14
14.24.4
…
β
- Symbols:
- : gamma function, : associated Legendre function of the first kind, : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : sine function, : complex variable, : general order, : general degree and : arbitrary integer
- Referenced by:
- §14.24
- Permalink:
- http://dlmf.nist.gov/14.24.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.24 and Ch.14
14: 15.14 Integrals
…
βΊ
15.14.1
.
…
β
- 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
15: 14.4 Graphics
…
16: 15.2 Definitions and Analytical Properties
…
βΊ
15.2.2
,
…
βΊ
β
- 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
.
…
βΊ
β
- 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
.
…
β
- 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
17: 14.12 Integral Representations
…
βΊ
14.12.9
…
βΊ
β
- Symbols:
- : Olver’s associated Legendre function, : differential of , : factorial (as in ), : hyperbolic cosine function, : integral, : real variable, : nonnegative integer, : nonnegative integer and
- Permalink:
- http://dlmf.nist.gov/14.12.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.12(ii), §14.12 and Ch.14
14.12.11
βΊ
β
- Symbols:
- : Olver’s associated Legendre function, : differential of , : factorial (as in ), : integral, : real variable, : nonnegative integer and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/14.12.E11
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.12(ii), §14.12 and Ch.14
14.12.12
.
…
βΊ
β
- Symbols:
- : associated Legendre function of the first kind, : Olver’s associated Legendre function, : differential of , : factorial (as in ), : integral, : real variable, : nonnegative integer and : nonnegative integer
- Referenced by:
- §14.17(i)
- Permalink:
- http://dlmf.nist.gov/14.12.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.12(ii), §14.12 and Ch.14
14.12.13
…
βΊ
β
- Symbols:
- : Legendre polynomial, : differential of , : factorial (as in ), : integral, : Olver’s associated Legendre function, : real variable and : nonnegative integer
- A&S Ref:
- 8.8.3 (modified)
- Permalink:
- http://dlmf.nist.gov/14.12.E13
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.12(ii), §14.12(ii), §14.12 and Ch.14
14.12.14
…
β
- Symbols:
- : differential of , : factorial (as in ), : hyperbolic cosine function, : integral, : Olver’s associated Legendre function, : real variable and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/14.12.E14
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.12(ii), §14.12(ii), §14.12 and Ch.14
18: 15.6 Integral Representations
…
βΊ
15.6.1
; .
βΊ
β
- 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 Olver hypergeometric function , 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
; , .
βΊ
β
- 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 Olver hypergeometric function , 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_5
; .
βΊ
β
- 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:
- §15.6, Erratum (V1.1.5) for Additions
- Permalink:
- http://dlmf.nist.gov/15.6.E2_5
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.1.5):
- This equation was added.
- See also:
- Annotations for §15.6 and Ch.15
15.6.3
; , .
…
βΊ
β
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : 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, §15.6
- Permalink:
- http://dlmf.nist.gov/15.6.E3
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.23):
- The Olver hypergeometric function , 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
; .
…
β
- 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 Olver hypergeometric function , 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
19: 14.23 Values on the Cut
…
βΊ
14.23.2
…
βΊ
β
- Symbols:
- : gamma function, : Ferrers function of the first kind, : Ferrers function of the second kind, : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : real variable, : general order and : general degree
- Referenced by:
- §14.23
- Permalink:
- http://dlmf.nist.gov/14.23.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.23 and Ch.14
14.23.3
…
βΊ
β
- Symbols:
- : gamma function, : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, or : Olver’s hypergeometric function, : real variable, : general order and : general degree
- Referenced by:
- §14.23
- Permalink:
- http://dlmf.nist.gov/14.23.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.23 and Ch.14
14.23.5
…
βΊ
β
- Symbols:
- : gamma function, : Ferrers function of the second kind, : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : real variable, : general order and : general degree
- A&S Ref:
- 8.3.4 (modified)
- Referenced by:
- §14.23
- Permalink:
- http://dlmf.nist.gov/14.23.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.23 and Ch.14
14.23.6
…
β
- Symbols:
- : gamma function, : Ferrers function of the second kind, : associated Legendre function of the first kind, : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : real variable, : general order and : general degree
- Referenced by:
- §14.23, §14.23
- Permalink:
- http://dlmf.nist.gov/14.23.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.23 and Ch.14
20: 13.2 Definitions and Basic Properties
…
βΊ
13.2.3
…
βΊ
β
- Defines:
- : Olver’s confluent hypergeometric function
- Symbols:
- : gamma function, : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : nonnegative integer and : complex variable
- Referenced by:
- §13.2(i)
- Permalink:
- http://dlmf.nist.gov/13.2.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13
13.2.4
…
βΊ
β
- Symbols:
- : gamma function, : Kummer confluent hypergeometric function, : Olver’s confluent hypergeometric function and : complex variable
- Referenced by:
- §13.2(ii)
- Permalink:
- http://dlmf.nist.gov/13.2.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13
13.2.5
…
βΊ
β
- Symbols:
- : gamma function, : Kummer confluent hypergeometric function, : Olver’s confluent hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : nonnegative integer and : complex variable
- Permalink:
- http://dlmf.nist.gov/13.2.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13
13.2.23
,
…
βΊ
β
- Symbols:
- : gamma function, : Olver’s confluent hypergeometric function, : asymptotic equality, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : phase, : complex variable and : small positive constant
- Permalink:
- http://dlmf.nist.gov/13.2.E23
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.2(iv), §13.2 and Ch.13
13.2.33
…
β
- Symbols:
- : Olver’s confluent hypergeometric function, : Wronskian, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : sine function and : complex variable
- A&S Ref:
- 13.1.20
- Permalink:
- http://dlmf.nist.gov/13.2.E33
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.2(vi), §13.2 and Ch.13