Euler beta integral
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1: 35.3 Multivariate Gamma and Beta Functions
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35.3.3
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- Defines:
- : multivariate beta function
- Symbols:
- : integral, : real part, : complex variable, : identity matrix, : real symmetric matrix, : determinant of , : differential of matrix, : complex variable, : less-than for matrices, : positive integer and : matrix of zeros
- Keywords:
- definition, multivariate beta function
- Permalink:
- http://dlmf.nist.gov/35.3.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §35.3(i), §35.3 and Ch.35
35.3.8
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- Symbols:
- : integral, : multivariate beta function, : real part, : complex variable, : identity matrix, : real symmetric matrix, : determinant of , : differential of matrix, : space of positive-definite real symmetric matrices, : complex variable and : positive integer
- Permalink:
- http://dlmf.nist.gov/35.3.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §35.3(ii), §35.3 and Ch.35
2: 35.6 Confluent Hypergeometric Functions of Matrix Argument
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35.6.4
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- Symbols:
- or : generalized hypergeometric function of matrix argument, : exponential of trace, : integral, : multivariate beta function, : real part, : complex variable, : identity matrix, : real symmetric matrix, : real symmetric matrix, : determinant of , : differential of matrix, : complex variable, : less-than for matrices, : positive integer and : matrix of zeros
- Permalink:
- http://dlmf.nist.gov/35.6.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §35.6(ii), §35.6 and Ch.35
35.6.6
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- Symbols:
- or : generalized hypergeometric function of matrix argument, : integral, : multivariate beta function, : real part, : complex variable, : real symmetric matrix, : real symmetric matrix, : determinant of , : differential of matrix, : complex variable, : less-than for matrices, : positive integer and : matrix of zeros
- Referenced by:
- §35.6(ii)
- Permalink:
- http://dlmf.nist.gov/35.6.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §35.6(ii), §35.6 and Ch.35
3: 35.4 Partitions and Zonal Polynomials
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35.4.9
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- Symbols:
- : integral, : multivariate beta function, : partitional shifted factorial, : zonal polynomial, : complex variable, : identity matrix, : real symmetric matrix, : real symmetric matrix, : determinant of , : differential of matrix, : complex variable, : less-than for matrices, : positive integer, : matrix of zeros and : vector of nonnegative integers
- Permalink:
- http://dlmf.nist.gov/35.4.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35
4: 5.12 Beta Function
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Euler’s Beta Integral
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5.12.1
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- Defines:
- : beta function
- Symbols:
- : gamma function, : differential of , : integral, : real or complex variable and : real or complex variable
- A&S Ref:
- 6.2.1 and 6.2.2
- Referenced by:
- §10.22(ii), §10.43(iii), §19.20(i), §19.20(iv), §2.6(iii), §5.12, §5.12, §7.7(ii)
- Permalink:
- http://dlmf.nist.gov/5.12.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §5.12, §5.12 and Ch.5
5.12.3
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- Symbols:
- : beta function, : differential of , : integral, : real or complex variable and : real or complex variable
- A&S Ref:
- 6.2.1 and 6.2.2
- Referenced by:
- §5.12
- Permalink:
- http://dlmf.nist.gov/5.12.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §5.12, §5.12 and Ch.5
5.12.4
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- Symbols:
- : beta function, : the ratio of the circumference of a circle to its diameter, : differential of , : integral, : phase, : complex variable, : real or complex variable and : real or complex variable
- Referenced by:
- §5.12, §5.12
- Permalink:
- http://dlmf.nist.gov/5.12.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §5.12, §5.12 and Ch.5
5.12.7
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- Symbols:
- : beta function, : differential of , : hyperbolic cosine function, : integral, : real part, : real or complex variable and : real or complex variable
- Referenced by:
- §5.12
- Permalink:
- http://dlmf.nist.gov/5.12.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §5.12, §5.12 and Ch.5
5: 35.7 Gaussian Hypergeometric Function of Matrix Argument
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35.7.5
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- Symbols:
- or : generalized hypergeometric function of matrix argument, : integral, : multivariate beta function, : real part, : complex variable, : identity matrix, : real symmetric matrix, : real symmetric matrix, : determinant of , : differential of matrix, : complex variable, : less-than for matrices, : complex variable, : positive integer and : matrix of zeros
- Referenced by:
- §35.7(iv)
- Permalink:
- http://dlmf.nist.gov/35.7.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35
6: 35.8 Generalized Hypergeometric Functions of Matrix Argument
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35.8.13
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- Symbols:
- or : generalized hypergeometric function of matrix argument, : integral, : multivariate beta function, : real part, : complex variable, : identity matrix, : real symmetric matrix, : real symmetric matrix, : determinant of , : differential of matrix, : complex variable, : less-than for matrices, : nonnegative integer, : nonnegative integer, : positive integer and : matrix of zeros
- Permalink:
- http://dlmf.nist.gov/35.8.E13
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35
7: 19.28 Integrals of Elliptic Integrals
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19.28.1
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- Symbols:
- : symmetric elliptic integral of first kind, : beta function, : differential of , : integral and : parameter
- Referenced by:
- §19.28, §19.28
- Permalink:
- http://dlmf.nist.gov/19.28.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.28 and Ch.19
19.28.2
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- Symbols:
- : symmetric elliptic integral of second kind, : beta function, : differential of , : integral and : parameter
- Permalink:
- http://dlmf.nist.gov/19.28.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.28 and Ch.19
19.28.3
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- Symbols:
- : elliptic integral symmetric in only two variables, : beta function, : differential of , : integral and : parameter
- Referenced by:
- §19.28, §19.28
- Permalink:
- http://dlmf.nist.gov/19.28.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.28 and Ch.19
8: 5.13 Integrals
§5.13 Integrals
… ►Barnes’ Beta Integral
… ►Ramanujan’s Beta Integral
… ►de Branges–Wilson Beta Integral
… ►9: 19.33 Triaxial Ellipsoids
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19.33.11
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- Symbols:
- : symmetric elliptic integral of first kind, : differential of , : integral, : positive constant, : positive constant, : positive constant, : energy and : function
- Permalink:
- http://dlmf.nist.gov/19.33.E11
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.33(iv), §19.33 and Ch.19
10: 16.15 Integral Representations and Integrals
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16.15.1
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- Symbols:
- : first Appell function, : gamma function, : differential of , : integral and : real part
- Keywords:
- Mellin transform
- Permalink:
- http://dlmf.nist.gov/16.15.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §16.15 and Ch.16
16.15.2
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- Symbols:
- : second Appell function, : gamma function, : differential of , : integral and : real part
- Keywords:
- Mellin transform
- Permalink:
- http://dlmf.nist.gov/16.15.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §16.15 and Ch.16
16.15.3
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- Symbols:
- : third Appell function, : gamma function, : differential of and : real part
- Keywords:
- Mellin transform
- Notes:
- Erdélyi et al. (1953a, 5.8.1(3)) contains a typo.
- Referenced by:
- Erratum (V1.0.14) for Equation (16.15.3), Erratum (V1.0.14) for Equation (16.15.3)
- Permalink:
- http://dlmf.nist.gov/16.15.E3
- Encodings:
- pMML, png, TeX
- Clarification (effective with 1.0.14):
- A stray “” was removed; it appeared after the comma at the end of this equation.
- See also:
- Annotations for §16.15 and Ch.16
16.15.4
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- Symbols:
- : fourth Appell function, : gamma function, : differential of , : integral and : real part
- Keywords:
- Mellin transform
- Source:
- Burchnall and Chaundy (1940, (68))
- Referenced by:
- §16.15
- Permalink:
- http://dlmf.nist.gov/16.15.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §16.15 and Ch.16