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11—20 of 296 matching pages
11: 15.1 Special Notation
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real variable. | |
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real or complex parameters. | |
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15.1.1
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- Symbols:
- or : Gauss’ hypergeometric function, : notation for Gauss’ hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/15.1.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.1 and Ch.15
15.1.2
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- 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
12: 31.9 Orthogonality
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31.9.1
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- Symbols:
- : Heun functions, : complex variable, : real or complex parameter, : real or complex parameter, : nonnegative integer, : complex parameter, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/31.9.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.9(i), §31.9 and Ch.31
31.9.2
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- Symbols:
- : Kronecker delta, : differential of , : integral, : real or complex parameter, : real or complex parameter, : real or complex parameter, : nonnegative integer, : complex parameter, : point and : normalization constant
- Permalink:
- http://dlmf.nist.gov/31.9.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.9(i), §31.9 and Ch.31
31.9.3
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- Defines:
- : normalization constant (locally)
- Symbols:
- : Wronskian, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : partial derivative of with respect to , : partial differential of , : real or complex parameter, : real or complex parameter, : real or complex parameter, : nonnegative integer, : complex parameter, : real or complex parameter, : point and , : coefficients
- Permalink:
- http://dlmf.nist.gov/31.9.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.9(i), §31.9 and Ch.31
31.9.6
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- Defines:
- : weight function (locally)
- Symbols:
- : real or complex parameter, : real or complex parameter, : real or complex parameter and : complex parameter
- Permalink:
- http://dlmf.nist.gov/31.9.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.9(ii), §31.9 and Ch.31
13: 28.14 Fourier Series
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28.14.1
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- Symbols:
- : Mathieu function, : base of natural logarithm, : imaginary unit, : integer, : parameter, : complex variable, : complex parameter and : coefficients
- A&S Ref:
- 20.3.8 (in slightly different form)
- Referenced by:
- §28.22(ii)
- Permalink:
- http://dlmf.nist.gov/28.14.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.14 and Ch.28
28.14.2
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- Symbols:
- : Mathieu function, : cosine function, : integer, : parameter, : complex variable, : complex parameter and : coefficients
- Permalink:
- http://dlmf.nist.gov/28.14.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.14 and Ch.28
28.14.5
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- Symbols:
- : integer, : parameter, : complex parameter and : coefficients
- Referenced by:
- item (d)
- Permalink:
- http://dlmf.nist.gov/28.14.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.14 and Ch.28
28.14.7
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- Symbols:
- : integer, : parameter, : complex parameter and : coefficients
- Permalink:
- http://dlmf.nist.gov/28.14.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.14 and Ch.28
28.14.8
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- Symbols:
- : integer, : parameter, : complex parameter and : coefficients
- Permalink:
- http://dlmf.nist.gov/28.14.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.14 and Ch.28
14: 31.11 Expansions in Series of Hypergeometric Functions
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31.11.3
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- Defines:
- (locally) and (locally)
- Symbols:
- : real or complex parameter, : real or complex parameter, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/31.11.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.11(ii), §31.11 and Ch.31
31.11.6
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- Symbols:
- : real or complex parameter, : nonnegative integer, : real or complex parameter, : real or complex parameter, , and : coefficients
- Referenced by:
- Erratum (V1.1.7) for Equation (31.11.6)
- Permalink:
- http://dlmf.nist.gov/31.11.E6
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.1.7):
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The sign has been corrected and the final term in the numerator
has been corrected to be .
Suggested 2022-09-14 by Hans Volkmer
- See also:
- Annotations for §31.11(ii), §31.11 and Ch.31
31.11.7
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- Symbols:
- : real or complex parameter, : nonnegative integer, : complex parameter, : real or complex parameter, : real or complex parameter, : real or complex parameter, , and : coefficients
- Permalink:
- http://dlmf.nist.gov/31.11.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.11(ii), §31.11 and Ch.31
31.11.8
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- Symbols:
- : real or complex parameter, : nonnegative integer, : real or complex parameter, : real or complex parameter, , and : coefficients
- Referenced by:
- Erratum (V1.1.7) for Equation (31.11.8)
- Permalink:
- http://dlmf.nist.gov/31.11.E8
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.1.7):
-
The sign has been corrected and the final term in the numerator
has been corrected to be .
Suggested 2022-09-14 by Hans Volkmer
- See also:
- Annotations for §31.11(ii), §31.11 and Ch.31
31.11.12
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- Symbols:
- : notation for Gauss’ hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : complex variable, : real or complex parameter, : real or complex parameter, : nonnegative integer, : real or complex parameter, : real or complex parameter and : first Kummer solution at of generalized hypergeometric differential equation
- Referenced by:
- §31.11(iii), §31.11(iii), §31.11(iii), Erratum (V1.1.7) for Subsection 31.11(iii)
- Permalink:
- http://dlmf.nist.gov/31.11.E12
- Encodings:
- pMML, png, TeX
- Notational Change (effective with 1.1.7):
- We have rewritten the gamma functions in the prefactor more concisely using Pochhammer symbols.
- See also:
- Annotations for §31.11(iii), §31.11 and Ch.31
15: 31.16 Mathematical Applications
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31.16.1
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- Symbols:
- : Heun polynomials, : Jacobi polynomial, : cosine function, : sine function, : real or complex parameter, : real or complex parameter, : real or complex parameter, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex parameter, : real or complex parameter, : real or complex parameter, : angle, : angle, : variable, : variable and : coefficients
- Referenced by:
- §31.16(ii), §31.16(ii)
- Permalink:
- http://dlmf.nist.gov/31.16.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.16(ii), §31.16 and Ch.31
31.16.3
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- Symbols:
- : Heun polynomials, : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : real or complex parameter, : real or complex parameter, : nonnegative integer, : nonnegative integer, : complex parameter, : real or complex parameter, : real or complex parameter, : coefficients, and
- Referenced by:
- §31.16(ii), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3)
- Permalink:
- http://dlmf.nist.gov/31.16.E3
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.0.21):
- The initial data , previously missing, has now been included.
- See also:
- Annotations for §31.16(ii), §31.16 and Ch.31
31.16.5
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- Defines:
- (locally)
- Symbols:
- : real or complex parameter, : real or complex parameter, : real or complex parameter, : nonnegative integer, : nonnegative integer and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/31.16.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.16(ii), §31.16 and Ch.31
31.16.6
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- Defines:
- (locally)
- Symbols:
- : real or complex parameter, : real or complex parameter, : nonnegative integer, : nonnegative integer, : complex parameter, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/31.16.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.16(ii), §31.16 and Ch.31
31.16.7
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- Defines:
- (locally)
- Symbols:
- : real or complex parameter, : real or complex parameter, : nonnegative integer and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/31.16.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.16(ii), §31.16 and Ch.31
16: 31.10 Integral Equations and Representations
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31.10.2
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- Defines:
- : weight function (locally)
- Symbols:
- : real or complex parameter, : real or complex parameter, : real or complex parameter and : complex parameter
- Permalink:
- http://dlmf.nist.gov/31.10.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.10(i), §31.10 and Ch.31
31.10.4
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- Defines:
- : Heun’s operator (locally)
- Symbols:
- : partial derivative of with respect to , : partial differential of , : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter, : complex parameter, : real or complex parameter and : real or complex parameter
- Referenced by:
- §31.10(ii)
- Permalink:
- http://dlmf.nist.gov/31.10.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.10(i), §31.10 and Ch.31
31.10.6
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- Symbols:
- : real or complex parameter, : real or complex parameter, : real or complex parameter and : complex parameter
- Referenced by:
- §31.10(ii)
- Permalink:
- http://dlmf.nist.gov/31.10.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.10(i), §31.10 and Ch.31
31.10.10
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- Symbols:
- : notation for Gauss’ hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter, : complex parameter, : real or complex parameter, : real or complex parameter, : kernel and : separation constant
- Permalink:
- http://dlmf.nist.gov/31.10.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.10(i), §31.10(i), §31.10 and Ch.31
31.10.13
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- Defines:
- : weight function (locally)
- Symbols:
- : real or complex parameter, : real or complex parameter, : real or complex parameter and : complex parameter
- Permalink:
- http://dlmf.nist.gov/31.10.E13
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.10(ii), §31.10 and Ch.31
17: 31.2 Differential Equations
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31.2.1
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- Symbols:
- : derivative of with respect to , : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter, : complex parameter, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Referenced by:
- §29.2(ii), §31.11(iii), §31.12, §31.13, §31.14(i), §31.17(i), §31.18, §31.2(v), §31.2(v), §31.2(v), §31.2(i), §31.3(i), §31.3(i), §31.3(ii), §31.3(ii), §31.3(ii), §31.3(iii), §31.3(iii), §31.4, §31.5, §31.6, §31.7(ii), §31.8, §31.8, §31.8
- Permalink:
- http://dlmf.nist.gov/31.2.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.2(i), §31.2 and Ch.31
31.2.2
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- Symbols:
- : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter, : complex parameter and : solution
- Permalink:
- http://dlmf.nist.gov/31.2.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.2(ii), §31.2 and Ch.31
31.2.6
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- Symbols:
- : cotangent function, : derivative of with respect to , : sine function, : tangent function, : real or complex parameter, : real or complex parameter, : real or complex parameter, : complex parameter, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/31.2.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.2(iii), §31.2 and Ch.31
31.2.8
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- Symbols:
- : Jacobian elliptic function, : Jacobian elliptic function, : Jacobian elliptic function, : derivative of with respect to , : real or complex parameter, : real or complex parameter, : real or complex parameter, : real or complex parameter, : real or complex parameter, : real or complex parameter, : change of variable and : modulus
- Referenced by:
- §31.7(ii)
- Permalink:
- http://dlmf.nist.gov/31.2.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31
31.2.10
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- Symbols:
- (= = ): Weierstrass -function, : Weierstrass lattice roots, : real or complex parameter, : real or complex parameter, : real or complex parameter, : lattice and : solution
- Permalink:
- http://dlmf.nist.gov/31.2.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31
18: 15.11 Riemann’s Differential Equation
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15.11.2
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- Symbols:
- : real or complex parameter, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/15.11.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.11(i), §15.11 and Ch.15
15.11.3
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- Defines:
- : Riemann’s -symbol for solutions of the generalized hypergeometric differential equation
- Symbols:
- : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter, : point, : point, : point and : solution
- Permalink:
- http://dlmf.nist.gov/15.11.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.11(i), §15.11 and Ch.15
15.11.4
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- Symbols:
- : Riemann’s -symbol for solutions of the generalized hypergeometric differential equation, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter and : solution
- Permalink:
- http://dlmf.nist.gov/15.11.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.11(i), §15.11 and Ch.15
15.11.6
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- Symbols:
- : Riemann’s -symbol for solutions of the generalized hypergeometric differential equation, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter, : point, : point, : point and : mapping
- Permalink:
- http://dlmf.nist.gov/15.11.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.11(ii), §15.11 and Ch.15
15.11.8
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- Symbols:
- : Riemann’s -symbol for solutions of the generalized hypergeometric differential equation, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter, : constant and : constant
- Permalink:
- http://dlmf.nist.gov/15.11.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.11(ii), §15.11 and Ch.15
19: 31.7 Relations to Other Functions
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31.7.1
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- Symbols:
- : notation for Gauss’ hypergeometric function, : Heun functions, : complex variable, : real or complex parameter, : real or complex parameter, : complex parameter, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/31.7.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.7(i), §31.7 and Ch.31
31.7.2
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- Symbols:
- : notation for Gauss’ hypergeometric function, : Heun functions, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/31.7.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.7(i), §31.7 and Ch.31
31.7.3
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- Symbols:
- : notation for Gauss’ hypergeometric function, : Heun functions, : complex variable, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/31.7.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.7(i), §31.7 and Ch.31
31.7.4
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- Symbols:
- : notation for Gauss’ hypergeometric function, : Heun functions, : imaginary unit, : complex variable, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/31.7.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.7(i), §31.7 and Ch.31