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11—20 of 135 matching pages
11: 36.2 Catastrophes and Canonical Integrals
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36.2.1
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- Defines:
- : cuspoid catastrophe of codimension
- Symbols:
- : integer, : variable, : codimension and : real parameter
- Referenced by:
- §36.10(i), §36.12(i), §36.5(ii), §36.7(ii), §36.8
- Permalink:
- http://dlmf.nist.gov/36.2.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36
36.2.2
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- Defines:
- : elliptic umbilic catastrophe
- Symbols:
- : real parameter, : real parameter, : variable, : variable and : real parameter
- Referenced by:
- §36.10(iii), §36.2(i), §36.7(iii), §36.8
- Permalink:
- http://dlmf.nist.gov/36.2.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36
36.2.3
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- Defines:
- : hyperbolic umbilic catastrophe
- Symbols:
- : real parameter, : real parameter, : variable, : variable and : real parameter
- Referenced by:
- §36.10(iii), §36.2(i), §36.8
- Permalink:
- http://dlmf.nist.gov/36.2.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36
36.2.10
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►For more extensive lists of normal forms of catastrophes (umbilic and beyond) involving two variables (“corank two”) see Arnol’d (1972, 1974, 1975).
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- Defines:
- : diffraction catastrophe
- Symbols:
- : cuspoid catastrophe of codimension , : differential of , : exponential function, : imaginary unit, : integral, : variable, : variable and : codimension
- Referenced by:
- §36.12(i)
- Permalink:
- http://dlmf.nist.gov/36.2.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36
12: 5.2 Definitions
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5.2.5
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- Symbols:
- : gamma function, : Pochhammer’s symbol (or shifted factorial), : nonnegative integer and : 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
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:
- 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
13: 1.9 Calculus of a Complex Variable
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1.9.63
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- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : variable, : nonnegative integer, : nonnegative integer, : radius of convergence and : absolute value of
- Permalink:
- http://dlmf.nist.gov/1.9.E63
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §1.9(vi), §1.9(vi), §1.9 and Ch.1
14: 18.28 Askey–Wilson Class
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18.28.4
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : multiple -Pochhammer symbol, : real variable and
- Permalink:
- http://dlmf.nist.gov/18.28.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18
18.28.17
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- Symbols:
- : Kronecker delta, : the ratio of the circumference of a circle to its diameter, : continuous -Hermite polynomial, : cosine function, : differential of , : base of natural logarithm, : imaginary unit, : integral, : -Pochhammer symbol (or -shifted factorial), : real variable, : nonnegative integer and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/18.28.E17
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §18.28(vi), §18.28 and Ch.18
18.28.25
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- Symbols:
- : open interval, : -Pochhammer symbol (or -shifted factorial), : continuous -ultraspherical polynomial, : real variable, : nonnegative integer and : real variable
- Proved:
- Gasper and Rahman (2004, (7.5.33))(proved)
- Permalink:
- http://dlmf.nist.gov/18.28.E25
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §18.28(ix), §18.28 and Ch.18
18.28.27
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- Symbols:
- : little -Jacobi polynomial, : -Pochhammer symbol (or -shifted factorial), : real variable, : nonnegative integer and : real variable
- Proved:
- Koornwinder (1993, (6.4)); The limit formula there is equivalent to the present formula(proved)
- Proof sketch:
- Proceed in a similar way as in the proof of (18.28.26). After taking the limit one now obtains a which equals (18.27.14_1) by (18.27.13).
- Referenced by:
- §18.28(x)
- Permalink:
- http://dlmf.nist.gov/18.28.E27
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18
18.28.32
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- Symbols:
- : continuous -Hermite polynomial, : -Pochhammer symbol (or -shifted factorial), : continuous -ultraspherical polynomial, : real variable, : nonnegative integer and : real variable
- Source:
- Koekoek et al. (2010, (14.10.34))
- Referenced by:
- §18.28(x)
- Permalink:
- http://dlmf.nist.gov/18.28.E32
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18
15: 15.8 Transformations of Variable
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15.8.6
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- Symbols:
- or : Gauss’ hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : complex variable, : real or complex parameter, : real or complex parameter and : integer
- Referenced by:
- §15.8(ii), §15.8(ii), §18.17(vii)
- Permalink:
- http://dlmf.nist.gov/15.8.E6
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.1.2):
- Pochhammer symbols now link to their definition.
- See also:
- Annotations for §15.8(ii), §15.8 and Ch.15
15.8.7
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- Symbols:
- or : Gauss’ hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : complex variable, : real or complex parameter, : real or complex parameter and : integer
- Referenced by:
- §15.19(iv), §15.8(ii), §15.8(ii), (18.35.4_5)
- Permalink:
- http://dlmf.nist.gov/15.8.E7
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.1.2):
- Pochhammer symbols now link to their definition.
- See also:
- Annotations for §15.8(ii), §15.8 and Ch.15
15.8.8
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- Symbols:
- : gamma function, : Pochhammer’s symbol (or shifted factorial), : the ratio of the circumference of a circle to its diameter, : psi (or digamma) function, : factorial (as in ), : principal branch of logarithm function, : phase, or : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter, : integer and : integer
- Referenced by:
- §15.12(i), §15.19(iv), §15.8(ii), §15.8(ii), §16.11(i)
- Permalink:
- http://dlmf.nist.gov/15.8.E8
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.1.2):
- Pochhammer symbols now link to their definition.
- See also:
- Annotations for §15.8(ii), §15.8 and Ch.15
15.8.9
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ⓘ
- Symbols:
- : gamma function, : Pochhammer’s symbol (or shifted factorial), : the ratio of the circumference of a circle to its diameter, : psi (or digamma) function, : factorial (as in ), : principal branch of logarithm function, : phase, or : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter, : integer and : integer
- Referenced by:
- §15.8(ii)
- Permalink:
- http://dlmf.nist.gov/15.8.E9
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.1.2):
- Pochhammer symbols now link to their definition.
- See also:
- Annotations for §15.8(ii), §15.8 and Ch.15
15.8.10
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- Symbols:
- : gamma function, : Pochhammer’s symbol (or shifted factorial), : the ratio of the circumference of a circle to its diameter, : psi (or digamma) function, : factorial (as in ), : principal branch of logarithm function, : phase, or : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter, : integer and : integer
- Referenced by:
- §15.4(ii), §15.8(ii)
- Permalink:
- http://dlmf.nist.gov/15.8.E10
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.1.2):
- Pochhammer symbols now link to their definition.
- See also:
- Annotations for §15.8(ii), §15.8 and Ch.15
16: 17.14 Constant Term Identities
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17.14.1
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : nonnegative integer, : complex base, : nonnegative integer, : nonnegative integer and : real variable
- Referenced by:
- §17.14
- Permalink:
- http://dlmf.nist.gov/17.14.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.14, §17.14 and Ch.17
17.14.2
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer, : complex variable and : LHS of (17.2.50)
- Referenced by:
- §17.14
- Permalink:
- http://dlmf.nist.gov/17.14.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.14, §17.14 and Ch.17
17.14.3
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer, : complex variable and : LHS of (17.2.49)
- Permalink:
- http://dlmf.nist.gov/17.14.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.14, §17.14 and Ch.17
17.14.4
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer, : complex variable and : LHS of (17.2.49)
- Permalink:
- http://dlmf.nist.gov/17.14.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.14, §17.14 and Ch.17
17.14.5
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer, : complex variable and : LHS of (17.2.50)
- Referenced by:
- §17.14
- Permalink:
- http://dlmf.nist.gov/17.14.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.14, §17.14 and Ch.17
17: 15.16 Products
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15.16.1
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- Symbols:
- or : Gauss’ hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : nonnegative integer, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter and : coefficients
- Permalink:
- http://dlmf.nist.gov/15.16.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.16 and Ch.15
15.16.3
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- Symbols:
- or : Gauss’ hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : nonnegative integer, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/15.16.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.16 and Ch.15
18: 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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- 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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- 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
17.3.6
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- Defines:
- : -cosine 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.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17
19: 8.21 Generalized Sine and Cosine Integrals
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8.21.16
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- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : generalized sine integral, : spherical Bessel function of the first kind, : complex variable, : parameter and : nonnegative integer
- Referenced by:
- §8.21(vi)
- Permalink:
- http://dlmf.nist.gov/8.21.E16
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8
8.21.17
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- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : generalized cosine integral, : spherical Bessel function of the first kind, : complex variable, : parameter and : nonnegative integer
- Referenced by:
- §8.21(vi)
- Permalink:
- http://dlmf.nist.gov/8.21.E17
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8
8.21.26
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- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : Poincaré asymptotic expansion, : complex variable, : parameter, : nonnegative integer and : auxiliary function
- Permalink:
- http://dlmf.nist.gov/8.21.E26
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.21(viii), §8.21 and Ch.8
8.21.27
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- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : Poincaré asymptotic expansion, : complex variable, : parameter, : nonnegative integer and : auxiliary function
- Permalink:
- http://dlmf.nist.gov/8.21.E27
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.21(viii), §8.21 and Ch.8
20: 13.29 Methods of Computation
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13.29.3
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- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : base of natural logarithm, : factorial (as in ), : nonnegative integer, : complex variable and : solution
- Permalink:
- http://dlmf.nist.gov/13.29.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13
13.29.6
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- Symbols:
- : Kummer confluent hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : nonnegative integer and : complex variable
- Permalink:
- http://dlmf.nist.gov/13.29.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13
13.29.7
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- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : nonnegative integer and : complex variable
- Permalink:
- http://dlmf.nist.gov/13.29.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13