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11—20 of 151 matching pages
11: 17.2 Calculus
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17.2.1
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : complex base and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/17.2.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.2(i), §17.2 and Ch.17
17.2.3
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : complex base and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/17.2.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.2(i), §17.2 and Ch.17
17.2.4
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : complex base and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/17.2.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.2(i), §17.2 and Ch.17
17.2.7
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- Symbols:
- : binomial coefficient, : -Pochhammer symbol (or -shifted factorial), : complex base and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/17.2.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.2(i), §17.2 and Ch.17
12: 33.2 Definitions and Basic Properties
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33.2.9
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- Defines:
- : phase of Coulomb functions
- Symbols:
- : Coulomb phase shift, : the ratio of the circumference of a circle to its diameter, : principal branch of logarithm function, : nonnegative integer, : nonnegative real variable and : real parameter
- A&S Ref:
- 14.5.5
- Referenced by:
- §33.10(i), §33.11
- Permalink:
- http://dlmf.nist.gov/33.2.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.2(iii), §33.2 and Ch.33
33.2.10
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is the Coulomb phase shift.
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âșAlso, are analytic functions of when .
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- Defines:
- : Coulomb phase shift
- Symbols:
- : gamma function, : imaginary unit, : phase, : nonnegative integer and : real parameter
- A&S Ref:
- 14.5.6
- Referenced by:
- §33.10(ii), §33.10(iii), §33.13, §33.2(iii), §33.25, §5.20
- Permalink:
- http://dlmf.nist.gov/33.2.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.2(iii), §33.2 and Ch.33
13: 17.5 Functions
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17.5.1
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- Symbols:
- : binomial coefficient, : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : complex base, : nonnegative integer and : complex variable
- Referenced by:
- §17.2(iii), Erratum (V1.1.11) for Equation (17.5.1)
- Permalink:
- http://dlmf.nist.gov/17.5.E1
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.1.11):
- The constraint is not necessary and was removed.
- See also:
- Annotations for §17.5, §17.5 and Ch.17
17.5.2
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : complex base and : complex variable
- Permalink:
- http://dlmf.nist.gov/17.5.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.5, §17.5 and Ch.17
17.5.3
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : complex base, : nonnegative integer and : complex variable
- Permalink:
- http://dlmf.nist.gov/17.5.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.5, §17.5 and Ch.17
17.5.4
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â
- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : complex base, : nonnegative integer and : complex variable
- Referenced by:
- §17.2(iii)
- Permalink:
- http://dlmf.nist.gov/17.5.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.5, §17.5 and Ch.17
17.5.5
â
- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function and : complex base
- Referenced by:
- Erratum (V1.1.10) for Equation (17.5.5)
- Permalink:
- http://dlmf.nist.gov/17.5.E5
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.1.10):
- The constraint is not necessary and was removed.
- See also:
- Annotations for §17.5, §17.5 and Ch.17
14: 17.3 -Elementary and -Special Functions
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17.3.1
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- Defines:
- : -exponential function
- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer and : real variable
- Referenced by:
- §17.5
- Permalink:
- http://dlmf.nist.gov/17.3.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17
17.3.2
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- Defines:
- : -exponential function
- Symbols:
- : binomial coefficient, : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer and : real variable
- Referenced by:
- §17.5
- Permalink:
- http://dlmf.nist.gov/17.3.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17
17.3.3
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â
- Defines:
- : -sine function
- Symbols:
- : imaginary unit, : -Pochhammer symbol (or -shifted factorial), : -exponential function, : complex base, : nonnegative integer and : real variable
- Permalink:
- http://dlmf.nist.gov/17.3.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17
17.3.4
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â
- Defines:
- : -sine function
- Symbols:
- : imaginary unit, : -exponential function, : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer and : real variable
- Permalink:
- http://dlmf.nist.gov/17.3.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17
17.3.5
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- Defines:
- : -cosine function
- Symbols:
- : imaginary unit, : -Pochhammer symbol (or -shifted factorial), : -exponential function, : complex base, : nonnegative integer and : real variable
- Permalink:
- http://dlmf.nist.gov/17.3.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17
15: 18.7 Interrelations and Limit Relations
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18.7.1
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â
- Symbols:
- : Jacobi polynomial, : Pochhammer’s symbol (or shifted factorial), : ultraspherical (or Gegenbauer) polynomial, : nonnegative integer and : real variable
- A&S Ref:
- 22.5.27
- Referenced by:
- §18.15(ii), §18.7(i), §18.7(ii), §18.7(iii)
- Permalink:
- http://dlmf.nist.gov/18.7.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18
18.7.2
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- Symbols:
- : Jacobi polynomial, : Pochhammer’s symbol (or shifted factorial), : ultraspherical (or Gegenbauer) polynomial, : nonnegative integer and : real variable
- A&S Ref:
- 22.5.20
- Referenced by:
- §18.6(i), §18.7(i)
- Permalink:
- http://dlmf.nist.gov/18.7.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18
18.7.7
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- Symbols:
- : Chebyshev polynomial of the first kind, : shifted Chebyshev polynomial of the first kind, : nonnegative integer and : real variable
- A&S Ref:
- 22.5.14
- Referenced by:
- §18.7(i)
- Permalink:
- http://dlmf.nist.gov/18.7.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18
18.7.8
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- Symbols:
- : Chebyshev polynomial of the second kind, : shifted Chebyshev polynomial of the second kind, : nonnegative integer and : real variable
- A&S Ref:
- 22.5.15
- Permalink:
- http://dlmf.nist.gov/18.7.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18
18.7.10
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- Symbols:
- : Legendre polynomial, : shifted Legendre polynomial, : nonnegative integer and : real variable
- Permalink:
- http://dlmf.nist.gov/18.7.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18
16: 12.9 Asymptotic Expansions for Large Variable
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12.9.1
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- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : Poincaré asymptotic expansion, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : factorial (as in ), : parabolic cylinder function, : phase, : complex variable, : nonnegative integer, : real or complex parameter and : arbitrary small positive constant
- A&S Ref:
- 19.8.1 (modification of)
- Referenced by:
- §12.9(i), §12.9(ii)
- Permalink:
- http://dlmf.nist.gov/12.9.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §12.9(i), §12.9 and Ch.12
12.9.2
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â
- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : Poincaré asymptotic expansion, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : factorial (as in ), : parabolic cylinder function, : phase, : complex variable, : nonnegative integer, : real or complex parameter and : arbitrary small positive constant
- A&S Ref:
- 19.8.2 (modification of)
- Referenced by:
- §12.9(i), §12.9(ii)
- Permalink:
- http://dlmf.nist.gov/12.9.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §12.9(i), §12.9 and Ch.12
12.9.3
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- Symbols:
- : gamma function, : Pochhammer’s symbol (or shifted factorial), : Poincaré asymptotic expansion, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : factorial (as in ), : imaginary unit, : parabolic cylinder function, : phase, : complex variable, : nonnegative integer, : real or complex parameter and : arbitrary small positive constant
- Permalink:
- http://dlmf.nist.gov/12.9.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §12.9(i), §12.9 and Ch.12
12.9.4
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- Symbols:
- : gamma function, : Pochhammer’s symbol (or shifted factorial), : Poincaré asymptotic expansion, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : factorial (as in ), : imaginary unit, : parabolic cylinder function, : phase, : complex variable, : nonnegative integer, : real or complex parameter and : arbitrary small positive constant
- Referenced by:
- §12.9(i)
- Permalink:
- http://dlmf.nist.gov/12.9.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §12.9(i), §12.9 and Ch.12
17: 13.13 Addition and Multiplication Theorems
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13.13.1
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- Symbols:
- : Kummer confluent hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : nonnegative integer, : real variable and : real variable
- Permalink:
- http://dlmf.nist.gov/13.13.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.13(i), §13.13 and Ch.13
13.13.2
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â
- Symbols:
- : Kummer confluent hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : nonnegative integer, : real variable and : real variable
- Permalink:
- http://dlmf.nist.gov/13.13.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.13(i), §13.13 and Ch.13
13.13.3
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âș
â
- Symbols:
- : Kummer confluent hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : real part, : nonnegative integer, : real variable and : real variable
- Permalink:
- http://dlmf.nist.gov/13.13.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.13(i), §13.13 and Ch.13
13.13.4
âș
â
- Symbols:
- : Kummer confluent hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : base of natural logarithm, : factorial (as in ), : nonnegative integer, : real variable and : real variable
- Permalink:
- http://dlmf.nist.gov/13.13.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.13(i), §13.13 and Ch.13
13.13.5
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â
- Symbols:
- : Kummer confluent hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : base of natural logarithm, : factorial (as in ), : real part, : nonnegative integer, : real variable and : real variable
- Permalink:
- http://dlmf.nist.gov/13.13.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.13(i), §13.13 and Ch.13
18: 13.26 Addition and Multiplication Theorems
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13.26.1
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âș
â
- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : Whittaker confluent hypergeometric function, : base of natural logarithm, : factorial (as in ), : nonnegative integer, : real variable and : real variable
- Permalink:
- http://dlmf.nist.gov/13.26.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.26(i), §13.26 and Ch.13
13.26.2
âș
â
- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : Whittaker confluent hypergeometric function, : base of natural logarithm, : factorial (as in ), : nonnegative integer, : real variable and : real variable
- Permalink:
- http://dlmf.nist.gov/13.26.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.26(i), §13.26 and Ch.13
13.26.3
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…
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- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : Whittaker confluent hypergeometric function, : base of natural logarithm, : factorial (as in ), : real part, : nonnegative integer, : real variable and : real variable
- Referenced by:
- §13.26(i)
- Permalink:
- http://dlmf.nist.gov/13.26.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.26(i), §13.26 and Ch.13
13.26.5
âș
â
- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : Whittaker confluent hypergeometric function, : base of natural logarithm, : factorial (as in ), : nonnegative integer, : real variable and : real variable
- Permalink:
- http://dlmf.nist.gov/13.26.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.26(i), §13.26 and Ch.13
13.26.6
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- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : Whittaker confluent hypergeometric function, : base of natural logarithm, : factorial (as in ), : real part, : nonnegative integer, : real variable and : real variable
- Referenced by:
- §13.26(i)
- Permalink:
- http://dlmf.nist.gov/13.26.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.26(i), §13.26 and Ch.13
19: 15.15 Sums
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15.15.1
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- 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
20: 16.13 Appell Functions
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16.13.1
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âș
â
- Defines:
- : first Appell function
- Symbols:
- : Pochhammer’s symbol (or shifted factorial) and : factorial (as in )
- Permalink:
- http://dlmf.nist.gov/16.13.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §16.13 and Ch.16
16.13.2
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- Defines:
- : second Appell function
- Symbols:
- : Pochhammer’s symbol (or shifted factorial) and : factorial (as in )
- Permalink:
- http://dlmf.nist.gov/16.13.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §16.13 and Ch.16
16.13.3
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âș
â
- Defines:
- : third Appell function
- Symbols:
- : Pochhammer’s symbol (or shifted factorial) and : factorial (as in )
- Permalink:
- http://dlmf.nist.gov/16.13.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §16.13 and Ch.16
16.13.4
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- Defines:
- : fourth Appell function
- Symbols:
- : Pochhammer’s symbol (or shifted factorial) and : factorial (as in )
- Permalink:
- http://dlmf.nist.gov/16.13.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §16.13 and Ch.16