Euler integral
(0.011 seconds)
31—40 of 206 matching pages
31: 25.19 Tables
32: 8.6 Integral Representations
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8.6.3
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- Symbols:
- : differential of , : exponential function, : base of natural logarithm, : incomplete gamma function, : integral, : real part, : complex variable and : parameter
- Referenced by:
- §8.6(i)
- Permalink:
- http://dlmf.nist.gov/8.6.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.6(i), §8.6 and Ch.8
8.6.4
, ,
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- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : incomplete gamma function, : integral, : phase, : real part, : complex variable and : parameter
- Referenced by:
- §8.25(ii), §8.6(i)
- Permalink:
- http://dlmf.nist.gov/8.6.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.6(i), §8.6 and Ch.8
8.6.6
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- Symbols:
- : gamma function, : differential of , : base of natural logarithm, : incomplete gamma function, : integral, : modified Bessel function of the second kind, : real part, : complex variable and : parameter
- Referenced by:
- §8.6(i)
- Permalink:
- http://dlmf.nist.gov/8.6.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.6(i), §8.6 and Ch.8
8.6.9
, ,
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►For collections of integral representations of and see Erdélyi et al. (1953b, §9.3), Oberhettinger (1972, pp. 68–69), Oberhettinger and Badii (1973, pp. 309–312), Prudnikov et al. (1992b, §3.10), and Temme (1996b, pp. 282–283).
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- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : imaginary unit, : incomplete gamma function, : integral, : real part, : complex variable and : parameter
- Referenced by:
- §8.6(ii)
- Permalink:
- http://dlmf.nist.gov/8.6.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.6(ii), §8.6 and Ch.8
33: 25.20 Approximations
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•
34: 14.12 Integral Representations
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14.12.2
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- Symbols:
- : gamma function, : Ferrers function of the first kind, : differential of , : integral, : real part, : Ferrers function of the first kind, : real variable, : general order and : general degree
- Referenced by:
- §14.17(i)
- Permalink:
- http://dlmf.nist.gov/14.12.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.12(i), §14.12(i), §14.12 and Ch.14
14.12.3
, ,
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- Symbols:
- : gamma function, : Ferrers function of the second kind, : the ratio of the circumference of a circle to its diameter, : cosine function, : differential of , : hyperbolic cosine function, : hyperbolic sine function, : imaginary unit, : integral, : real part, : sine function, : general order and : general degree
- Permalink:
- http://dlmf.nist.gov/14.12.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.12(i), §14.12(i), §14.12 and Ch.14
14.12.4
, .
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- Symbols:
- : gamma function, : associated Legendre function of the first kind, : the ratio of the circumference of a circle to its diameter, : differential of , : hyperbolic cosine function, : integral, : real part, : real variable, : general order and : general degree
- Permalink:
- http://dlmf.nist.gov/14.12.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.12(ii), §14.12 and Ch.14
14.12.5
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- Symbols:
- : gamma function, : associated Legendre function of the first kind, : Legendre polynomial, : differential of , : integral, : real part, : real variable, : general order and : general degree
- Referenced by:
- §14.17(i)
- Permalink:
- http://dlmf.nist.gov/14.12.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.12(ii), §14.12 and Ch.14
14.12.6
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- Symbols:
- : gamma function, : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : differential of , : hyperbolic cosine function, : hyperbolic sine function, : integral, : real part, : real variable, : general order and : general degree
- A&S Ref:
- 8.8.2 (modified)
- Permalink:
- http://dlmf.nist.gov/14.12.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.12(ii), §14.12 and Ch.14
35: 17.6 Function
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17.6.28
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- Symbols:
- : integral, : -gamma function, : -Pochhammer symbol (or -shifted factorial), : -differential, or : basic hypergeometric (or -hypergeometric) function, : complex base, : complex variable and : real variable
- Permalink:
- http://dlmf.nist.gov/17.6.E28
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.6(v), §17.6 and Ch.17
36: 1.15 Summability Methods
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1.15.47
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- Defines:
- : fractional integral (locally)
- Symbols:
- : gamma function, : differential of and : integral
- Referenced by:
- §1.15(vi), §1.15(vi), Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii)
- Permalink:
- http://dlmf.nist.gov/1.15.E47
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §1.15(vi), §1.15 and Ch.1
1.15.50
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- Symbols:
- : gamma function, : factorial (as in ), : integer and : fractional integral
- Referenced by:
- Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii)
- Permalink:
- http://dlmf.nist.gov/1.15.E50
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §1.15(vi), §1.15 and Ch.1
37: 15.6 Integral Representations
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15.6.1
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- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : integral, : phase, : real part, or : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Keywords:
- Mellin transform
- Referenced by:
- §15.19(iii), (15.4.34), §15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)
- Permalink:
- http://dlmf.nist.gov/15.6.E1
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.23):
- The Olver hypergeometric function , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.
- See also:
- Annotations for §15.6 and Ch.15
15.6.2
; , .
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- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : imaginary unit, : integral, : phase, : real part, or : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Keywords:
- Mellin transform
- Referenced by:
- §15.6
- Permalink:
- http://dlmf.nist.gov/15.6.E2
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.23):
- The Olver hypergeometric function , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.
- See also:
- Annotations for §15.6 and Ch.15
15.6.2_5
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- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : integral, : phase, : real part, or : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Keywords:
- Mellin transform
- Referenced by:
- §15.6, Erratum (V1.1.5) for Additions
- Permalink:
- http://dlmf.nist.gov/15.6.E2_5
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.1.5):
- This equation was added.
- See also:
- Annotations for §15.6 and Ch.15
15.6.6
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ⓘ
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : imaginary unit, : integral, : phase, or : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- A&S Ref:
- 15.3.2 (modified)
- Referenced by:
- §15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)
- Permalink:
- http://dlmf.nist.gov/15.6.E6
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.23):
- The Olver hypergeometric function , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.
- See also:
- Annotations for §15.6 and Ch.15
15.6.8
; .
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ⓘ
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : integral, : phase, : real part, or : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Keywords:
- Mellin transform
- Proof sketch:
- Follows from (15.6.9) with .
- Referenced by:
- (15.6.9), §15.6, §15.6, §15.6, §18.17(iv), Erratum (V1.0.14) for Equation (15.6.8), Erratum (V1.0.14) for Equation (15.6.8)
- Permalink:
- http://dlmf.nist.gov/15.6.E8
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.23):
- The Olver hypergeometric function , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.
- See also:
- Annotations for §15.6 and Ch.15
38: 6.15 Sums
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6.15.1
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- Symbols:
- : Euler’s constant, : the ratio of the circumference of a circle to its diameter, : cosine integral, : principal branch of logarithm function and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/6.15.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §6.15 and Ch.6
6.15.3
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- Symbols:
- : Euler’s constant, : the ratio of the circumference of a circle to its diameter, : cosine integral, : principal branch of logarithm function and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/6.15.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §6.15 and Ch.6
39: 5.13 Integrals
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5.13.1
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- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : imaginary unit, : integral, : phase, : real part, : real or complex variable, : complex variable, : real or complex variable and : real or complex variable
- Referenced by:
- §5.13, §5.13, §5.19(ii)
- Permalink:
- http://dlmf.nist.gov/5.13.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §5.13 and Ch.5
5.13.2
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ⓘ
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : imaginary unit, : integral, : sine function, : real or complex variable and : real or complex variable
- Referenced by:
- §5.13
- Permalink:
- http://dlmf.nist.gov/5.13.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §5.13 and Ch.5
5.13.3
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- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : imaginary unit, : integral, : real part, : real or complex variable and : real or complex variable
- Referenced by:
- §5.13
- Permalink:
- http://dlmf.nist.gov/5.13.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §5.13, §5.13 and Ch.5
5.13.4
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- Symbols:
- : gamma function, : differential of , : integral, : real part, : real or complex variable and : real or complex variable
- Referenced by:
- §5.13
- Permalink:
- http://dlmf.nist.gov/5.13.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §5.13, §5.13 and Ch.5
5.13.5
, .
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- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : imaginary unit, : integral, : real part, : nonnegative integer, : nonnegative integer and : real or complex variable
- Referenced by:
- §5.13
- Permalink:
- http://dlmf.nist.gov/5.13.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §5.13, §5.13 and Ch.5
40: 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