Mellin transforms
(0.002 seconds)
11—20 of 36 matching pages
11: 15.6 Integral Representations
…
►
15.6.1
; .
►
ⓘ
- 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:
- Mellintransform
- 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
; , .
►
ⓘ
- 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:
- Mellintransform
- 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
; .
►
ⓘ
- 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:
- Mellintransform
- 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
15.6.3
; , .
…
►
ⓘ
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : 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:
- Mellintransform
- Referenced by:
- §15.6, §15.6
- Permalink:
- http://dlmf.nist.gov/15.6.E3
- 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
; .
…
ⓘ
- 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:
- Mellintransform
- 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
12: 14.17 Integrals
…
►
§14.17(vi) Mellin Transforms
►For Mellin transforms involving associated Legendre functions see Oberhettinger (1974, pp. 69–82) and Marichev (1983, pp. 247–283), and for inverse transforms see Oberhettinger (1974, pp. 205–215).13: 16.5 Integral Representations and Integrals
…
►
16.5.2
,
►
ⓘ
- Symbols:
-
: gamma function,
or : alternatively or
generalized hypergeometric function, : differential of , : integral, : real part, : nonnegative integer, : nonnegative integer, : complex variable, : real or complex parameters and : real or complex parameters - Keywords:
- Mellintransform
- Referenced by:
- §16.5, §16.5
- Permalink:
- http://dlmf.nist.gov/16.5.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §16.5 and Ch.16
16.5.3
, ,
►
ⓘ
- Symbols:
-
: gamma function,
or : alternatively or
generalized hypergeometric function, : differential of , : base of natural logarithm, : integral, : real part, : nonnegative integer, : nonnegative integer, : complex variable, : real or complex parameters and : real or complex parameters - Keywords:
- Laplace transform, Mellintransform
- Referenced by:
- §16.5
- Permalink:
- http://dlmf.nist.gov/16.5.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §16.5 and Ch.16
16.5.4
, .
…
ⓘ
- Symbols:
-
: gamma function,
or : alternatively or
generalized hypergeometric function, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : imaginary unit, : integral, : real part, : nonnegative integer, : nonnegative integer, : complex variable, : real or complex parameters and : real or complex parameters - Keywords:
- Mellintransform
- Referenced by:
- §16.5, §16.5
- Permalink:
- http://dlmf.nist.gov/16.5.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §16.5 and Ch.16
14: 9.10 Integrals
…
►
§9.10(vi) Mellin Transform
►
9.10.17
.
…
ⓘ
- Defines:
- : parameter (locally)
- Symbols:
- : Airy function, : gamma function, : differential of , : integral and : real part
- Keywords:
- Mellintransform
- Source:
- Olver (1997b, (6.10), p. 338)
- Permalink:
- http://dlmf.nist.gov/9.10.E17
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §9.10(vi), §9.10 and Ch.9
15: 8.19 Generalized Exponential Integral
…
►
8.19.2
…
►
ⓘ
- Symbols:
- : differential of , : base of natural logarithm, : generalized exponential integral, : integral, : complex variable and : parameter
- Keywords:
- Laplace transform, Mellintransform
- Referenced by:
- §8.19(i), §8.19(viii)
- Permalink:
- http://dlmf.nist.gov/8.19.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.19(i), §8.19 and Ch.8
8.19.3
,
►
ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : generalized exponential integral, : integral, : phase, : complex variable and : parameter
- Keywords:
- Laplace transform, Mellintransform
- A&S Ref:
- 5.1.4
- Referenced by:
- §8.19(iii)
- Permalink:
- http://dlmf.nist.gov/8.19.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.19(i), §8.19(i), §8.19 and Ch.8
8.19.4
, .
…
ⓘ
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : generalized exponential integral, : integral, : phase, : real part, : complex variable and : parameter
- Keywords:
- Laplace transform, Mellintransform
- Permalink:
- http://dlmf.nist.gov/8.19.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.19(i), §8.19(i), §8.19 and Ch.8
16: 7.7 Integral Representations
…
►
7.7.13
►
ⓘ
- Symbols:
- : gamma function, : auxiliary function for Fresnel integrals, : the ratio of the circumference of a circle to its diameter, : differential of , : imaginary unit, : integral, : complex variable, : change of variable and : constant
- Keywords:
- Mellintransform
- Referenced by:
- §7.7(ii), §7.7(ii)
- Permalink:
- http://dlmf.nist.gov/7.7.E13
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7
7.7.14
…
ⓘ
- Symbols:
- : gamma function, : auxiliary function for Fresnel integrals, : the ratio of the circumference of a circle to its diameter, : differential of , : imaginary unit, : integral, : complex variable, : change of variable and : constant
- Keywords:
- Mellintransform
- Referenced by:
- §7.7(ii), §7.7(ii)
- Permalink:
- http://dlmf.nist.gov/7.7.E14
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7
17: 8.6 Integral Representations
…
►
8.6.10
, ,
►
ⓘ
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : imaginary unit, : incomplete gamma function, : integral, : phase, : complex variable, : parameter and : real constant
- Keywords:
- Mellintransform
- Referenced by:
- §8.6(ii), §8.6(ii), §8.6(ii), Erratum (V1.0.17) for Paragraph Mellin–Barnes Integrals (in §8.6(ii))
- Permalink:
- http://dlmf.nist.gov/8.6.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.6(ii), §8.6(ii), §8.6 and Ch.8
8.6.11
,
►
ⓘ
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : imaginary unit, : incomplete gamma function, : integral, : phase, : complex variable, : parameter and : real constant
- Keywords:
- Mellintransform
- Referenced by:
- §8.19(i), §8.6(ii), §8.6(ii), Erratum (V1.0.17) for Paragraph Mellin–Barnes Integrals (in §8.6(ii))
- Permalink:
- http://dlmf.nist.gov/8.6.E11
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.6(ii), §8.6(ii), §8.6 and Ch.8
8.6.12
, .
…
ⓘ
- 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, : phase, : sine function, : complex variable, : parameter and : real constant
- Keywords:
- Mellintransform
- Referenced by:
- §8.19(i), §8.6(ii), §8.6(ii), §8.6(ii), Erratum (V1.0.17) for Paragraph Mellin–Barnes Integrals (in §8.6(ii))
- Permalink:
- http://dlmf.nist.gov/8.6.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.6(ii), §8.6(ii), §8.6 and Ch.8
18: 11.7 Integrals and Sums
…
►
11.7.10
,
►
ⓘ
- Symbols:
- : gamma function, : Struve function, : the ratio of the circumference of a circle to its diameter, : differential of , : integral, : real part and : real or complex order
- Keywords:
- Mellintransform
- Permalink:
- http://dlmf.nist.gov/11.7.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §11.7(ii), §11.7 and Ch.11
11.7.11
,
,
►
ⓘ
- Symbols:
- : gamma function, : Struve function, : the ratio of the circumference of a circle to its diameter, : differential of , : integral, : real part, : tangent function and : real or complex order
- Keywords:
- Mellintransform
- A&S Ref:
- 12.1.27
- Permalink:
- http://dlmf.nist.gov/11.7.E11
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §11.7(ii), §11.7 and Ch.11
11.7.12
.
…
ⓘ
- Symbols:
- : gamma function, : Struve function, : the ratio of the circumference of a circle to its diameter, : differential of , : integral, : real part and : real or complex order
- Keywords:
- Mellintransform
- Permalink:
- http://dlmf.nist.gov/11.7.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §11.7(ii), §11.7 and Ch.11
19: 10.43 Integrals
…
►
10.43.19
.
…
►
ⓘ
- Symbols:
- : gamma function, : differential of , : integral, : modified Bessel function of the second kind, : real part and : complex parameter
- Keywords:
- Mellintransform
- A&S Ref:
- 11.4.22, 11.4.23
- Permalink:
- http://dlmf.nist.gov/10.43.E19
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.43(iv), §10.43 and Ch.10
10.43.22
…
►
ⓘ
- Symbols:
- : gamma function, : Ferrers function of the first kind, : associated Legendre function of the first kind, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : integral, : modified Bessel function of the second kind, : real part and : complex parameter
- Keywords:
- Laplace transform, Mellintransform
- Referenced by:
- §10.43(iv)
- Permalink:
- http://dlmf.nist.gov/10.43.E22
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.43(iv), §10.43 and Ch.10
10.43.23
,
…
►
ⓘ
- Symbols:
- : differential of , : exponential function, : integral, : modified Bessel function of the first kind, : real part and : complex parameter
- Keywords:
- Mellintransform
- Referenced by:
- §10.43(iv)
- Permalink:
- http://dlmf.nist.gov/10.43.E23
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.43(iv), §10.43 and Ch.10
10.43.26
.
…
►
ⓘ
- Symbols:
- : Bessel function of the first kind, : gamma function, : differential of , : imaginary part, : integral, : modified Bessel function of the second kind, : real part, or : Olver’s hypergeometric function and : complex parameter
- Keywords:
- Mellintransform
- Permalink:
- http://dlmf.nist.gov/10.43.E26
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.43(iv), §10.43 and Ch.10
10.43.27
.
…
ⓘ
- Symbols:
- : Bessel function of the first kind, : gamma function, : differential of , : imaginary part, : integral, : modified Bessel function of the second kind, : real part and : complex parameter
- Keywords:
- Mellintransform
- Permalink:
- http://dlmf.nist.gov/10.43.E27
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.43(iv), §10.43 and Ch.10
20: 13.16 Integral Representations
…
►
13.16.10
,
…
►
ⓘ
- Symbols:
- : gamma function, : Whittaker confluent hypergeometric function, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : imaginary unit, : integral, : phase and : complex variable
- Keywords:
- Mellintransform
- Permalink:
- http://dlmf.nist.gov/13.16.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.16(iii), §13.16 and Ch.13
13.16.11
,
…
►
ⓘ
- Symbols:
- : gamma function, : Whittaker confluent hypergeometric function, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : imaginary unit, : integral, : phase and : complex variable
- Keywords:
- Mellintransform
- Permalink:
- http://dlmf.nist.gov/13.16.E11
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.16(iii), §13.16 and Ch.13
13.16.12
,
…
ⓘ
- Symbols:
- : gamma function, : Whittaker confluent hypergeometric function, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : imaginary unit, : integral, : phase and : complex variable
- Keywords:
- Mellintransform
- Permalink:
- http://dlmf.nist.gov/13.16.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.16(iii), §13.16 and Ch.13