Euler integral
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1—10 of 206 matching pages
1: 17.13 Integrals
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17.13.2
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- Symbols:
- : integral, : -gamma function, : -Pochhammer symbol (or -shifted factorial), : -differential, : complex base and : real variable
- Permalink:
- http://dlmf.nist.gov/17.13.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.13 and Ch.17
17.13.3
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- Symbols:
- : gamma function, : differential of , : integral, : -gamma function, : -Pochhammer symbol (or -shifted factorial) and : complex base
- Referenced by:
- Erratum (V1.0.1) for Equation (17.13.3)
- Permalink:
- http://dlmf.nist.gov/17.13.E3
- Encodings:
- pMML, png, TeX
- Errata (effective with 1.0.1):
- Originally the differential was identified incorrectly as a -differential; the correct differential is .
- See also:
- Annotations for §17.13, §17.13 and Ch.17
17.13.4
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- Symbols:
- : integral, : -gamma function, : -Pochhammer symbol (or -shifted factorial), : -differential and : complex base
- Permalink:
- http://dlmf.nist.gov/17.13.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.13, §17.13 and Ch.17
2: 5.2 Definitions
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Euler’s Integral
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5.2.1
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- Defines:
- : gamma function
- Symbols:
- : differential of , : base of natural logarithm, : integral, : real part and : complex variable
- A&S Ref:
- 6.1.1
- Referenced by:
- §10.43(iii), (25.11.27), (25.11.28), (25.5.6), (25.5.7), §5.9(i), §5.9(ii), §8.21(ii), (9.12.17)
- Permalink:
- http://dlmf.nist.gov/5.2.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §5.2(i), §5.2(i), §5.2 and Ch.5
3: 6.11 Relations to Other Functions
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6.11.1
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- Symbols:
- : exponential integral, : incomplete gamma function and : complex variable
- A&S Ref:
- 5.1.45 (special case of)
- Referenced by:
- §6.11
- Permalink:
- http://dlmf.nist.gov/6.11.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §6.11, §6.11 and Ch.6
4: 35.3 Multivariate Gamma and Beta Functions
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35.3.1
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- Symbols:
- : exponential of trace, : integral, : multivariate gamma function, : real part, : complex variable, : real symmetric matrix, : determinant of , : differential of matrix, : space of positive-definite real symmetric matrices and : positive integer
- Permalink:
- http://dlmf.nist.gov/35.3.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §35.3(i), §35.3 and Ch.35
35.3.2
, ,
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- Symbols:
- : complex plane, : element of, : exponential of trace, : integral, : multivariate gamma function, : real part, : real symmetric matrix, : determinant of , : th principal minor of , : differential of matrix, : space of positive-definite real symmetric matrices, : nonnegative integer and : positive integer
- Permalink:
- http://dlmf.nist.gov/35.3.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §35.3(i), §35.3 and Ch.35
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
5: 35.6 Confluent Hypergeometric Functions of Matrix Argument
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35.6.2
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- Defines:
- : confluent hypergeometric function of matrix argument (second kind)
- Symbols:
- : element of, : exponential of trace, : integral, : multivariate gamma function, : real part, : complex variable, : identity matrix, : real symmetric 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/35.6.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §35.6(i), §35.6 and Ch.35
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
35.6.5
, .
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ⓘ
- Symbols:
- or : generalized hypergeometric function of matrix argument, : exponential of trace, : integral, : multivariate gamma function, : real part, : complex variable, : identity matrix, : real symmetric matrix, : real symmetric matrix, : real symmetric matrix, : determinant of , : differential of matrix, : space of positive-definite real symmetric matrices, : complex variable, : greater-than for matrices and : positive integer
- Referenced by:
- §35.6(ii)
- Permalink:
- http://dlmf.nist.gov/35.6.E5
- 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
35.6.8
,
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- Symbols:
- : confluent hypergeometric function of matrix argument (second kind), : integral, : multivariate gamma function, : real part, : complex variable, : real symmetric matrix, : determinant of , : differential of matrix, : space of positive-definite real symmetric matrices, : complex variable, : complex variable and : positive integer
- Permalink:
- http://dlmf.nist.gov/35.6.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §35.6(ii), §35.6 and Ch.35
6: 24.13 Integrals
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§24.13(ii) Euler Polynomials
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24.13.7
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- Symbols:
- : Euler polynomials, : differential of , : integral, : integer and : real or complex
- A&S Ref:
- 23.1.11
- Referenced by:
- §24.13(ii)
- Permalink:
- http://dlmf.nist.gov/24.13.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.13(ii), §24.13 and Ch.24
24.13.8
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- Symbols:
- : Bernoulli numbers, : Euler polynomials, : differential of , : integral, : integer and : real or complex
- Permalink:
- http://dlmf.nist.gov/24.13.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.13(ii), §24.13 and Ch.24
24.13.10
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- Symbols:
- : Euler numbers, : Euler polynomials, : differential of , : integral, : integer and : real or complex
- Referenced by:
- §24.13(ii)
- Permalink:
- http://dlmf.nist.gov/24.13.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.13(ii), §24.13 and Ch.24
§24.13(iii) Compendia
…7: 8.22 Mathematical Applications
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8.22.1
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- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : generalized exponential integral, : incomplete gamma function, : terminant function, : complex variable and : parameter
- Permalink:
- http://dlmf.nist.gov/8.22.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.22(i), §8.22 and Ch.8
8.22.2
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- Defines:
- : incomplete Riemann zeta function (locally)
- Symbols:
- : gamma function, : differential of , : base of natural logarithm, : integral, : real part and : real variable
- Permalink:
- http://dlmf.nist.gov/8.22.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.22(ii), §8.22 and Ch.8
8: 24.7 Integral Representations
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§24.7(i) Bernoulli and Euler Numbers
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24.7.5
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- Symbols:
- : Bernoulli numbers, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : integral, : principal branch of logarithm function, : integer and : real or complex
- Referenced by:
- §24.7(i)
- Permalink:
- http://dlmf.nist.gov/24.7.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.7(i), §24.7 and Ch.24
24.7.6
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- Symbols:
- : Euler numbers, : the ratio of the circumference of a circle to its diameter, : differential of , : hyperbolic secant function, : integral, : integer and : real or complex
- Referenced by:
- §24.7(i)
- Permalink:
- http://dlmf.nist.gov/24.7.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.7(i), §24.7 and Ch.24
§24.7(ii) Bernoulli and Euler Polynomials
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24.7.9
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- Symbols:
- : Euler polynomials, : the ratio of the circumference of a circle to its diameter, : cosine function, : differential of , : hyperbolic cosine function, : integral, : sine function, : integer, : real or complex and : real or complex
- Permalink:
- http://dlmf.nist.gov/24.7.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.7(ii), §24.7 and Ch.24
9: 6.6 Power Series
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6.6.1
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- Symbols:
- : Euler’s constant, : exponential integral, : factorial (as in ), : principal branch of logarithm function, : real variable and : nonnegative integer
- A&S Ref:
- 5.1.10
- Referenced by:
- §6.5
- Permalink:
- http://dlmf.nist.gov/6.6.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §6.6 and Ch.6
6.6.2
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- Symbols:
- : Euler’s constant, : exponential integral, : factorial (as in ), : principal branch of logarithm function, : complex variable and : nonnegative integer
- A&S Ref:
- 5.1.11
- Referenced by:
- §6.4, §6.5
- Permalink:
- http://dlmf.nist.gov/6.6.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §6.6 and Ch.6
6.6.6
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- Symbols:
- : Euler’s constant, : cosine integral, : factorial (as in ), : principal branch of logarithm function, : complex variable and : nonnegative integer
- A&S Ref:
- 5.2.16
- Referenced by:
- §6.5
- Permalink:
- http://dlmf.nist.gov/6.6.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §6.6 and Ch.6
10: 8.4 Special Values
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8.4.1
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : differential of , : error function, : base of natural logarithm, : incomplete gamma function, : integral and : complex variable
- A&S Ref:
- 6.5.16
- Permalink:
- http://dlmf.nist.gov/8.4.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.4 and Ch.8
8.4.4
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- Symbols:
- : differential of , : base of natural logarithm, : exponential integral, : incomplete gamma function, : integral and : complex variable
- A&S Ref:
- 6.5.15
- Permalink:
- http://dlmf.nist.gov/8.4.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.4 and Ch.8
8.4.6
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : complementary error function, : differential of , : base of natural logarithm, : incomplete gamma function, : integral and : complex variable
- A&S Ref:
- 6.5.17
- Referenced by:
- §8.4
- Permalink:
- http://dlmf.nist.gov/8.4.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.4 and Ch.8
8.4.13
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- Symbols:
- : generalized exponential integral, : incomplete gamma function, : complex variable and : nonnegative integer
- Referenced by:
- §8.19(iv)
- Permalink:
- http://dlmf.nist.gov/8.4.E13
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.4 and Ch.8
8.4.15
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- Symbols:
- : psi (or digamma) function, : base of natural logarithm, : exponential integral, : factorial (as in ), : incomplete gamma function, : principal branch of logarithm function, : complex variable, : nonnegative integer and : nonnegative integer
- A&S Ref:
- 6.5.19
- Referenced by:
- §8.19(iii), §8.19(iv), §8.4
- Permalink:
- http://dlmf.nist.gov/8.4.E15
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.4 and Ch.8