of a complex variable
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11—20 of 469 matching pages
11: 15.15 Sums
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15.15.1
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ⓘ
- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), or : Olver’s hypergeometric function, : nonnegative integer, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Referenced by:
- §15.19(i)
- Permalink:
- http://dlmf.nist.gov/15.15.E1
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.1.2):
- The Pochhammer symbol now links to its definition.
- See also:
- Annotations for §15.15 and Ch.15
12: 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
35.3.4
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ⓘ
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : multivariate gamma function, : complex variable, : nonnegative integer and : positive integer
- Permalink:
- http://dlmf.nist.gov/35.3.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §35.3(ii), §35.3 and Ch.35
35.3.6
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ⓘ
- Symbols:
- : multivariate gamma function, : complex variable and : positive integer
- Permalink:
- http://dlmf.nist.gov/35.3.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §35.3(ii), §35.3 and Ch.35
35.3.7
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ⓘ
- Symbols:
- : multivariate beta function, : multivariate gamma function, : complex variable, : complex variable and : positive integer
- Permalink:
- http://dlmf.nist.gov/35.3.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §35.3(ii), §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
13: 9.14 Incomplete Airy Functions
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►Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter.
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14: 8.7 Series Expansions
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8.7.1
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ⓘ
- Symbols:
- : gamma function, : base of natural logarithm, : factorial (as in ), : incomplete gamma function, : complex variable, : parameter and : nonnegative integer
- A&S Ref:
- 6.5.29
- Referenced by:
- §8.11(ii), §8.14, §8.6(i), §8.7
- Permalink:
- http://dlmf.nist.gov/8.7.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.7 and Ch.8
8.7.3
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ⓘ
- Symbols:
- : gamma function, : base of natural logarithm, : factorial (as in ), : incomplete gamma function, : complex variable, : parameter and : nonnegative integer
- Referenced by:
- §8.14, §8.19(iv), §8.4, §8.7
- Permalink:
- http://dlmf.nist.gov/8.7.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.7 and Ch.8
8.7.5
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ⓘ
- Symbols:
- : gamma function, : Pochhammer’s symbol (or shifted factorial), : base of natural logarithm, : incomplete gamma function, : modified spherical Bessel function, : complex variable, : parameter and : nonnegative integer
- Referenced by:
- §8.7
- Permalink:
- http://dlmf.nist.gov/8.7.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.7 and Ch.8
8.7.6
, .
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- Symbols:
- : Laguerre (or generalized Laguerre) polynomial, : base of natural logarithm, : incomplete gamma function, : real part, : real variable, : complex variable, : parameter and : nonnegative integer
- Proof sketch:
- Integrate (18.12.13) from to and for the integral on the left-hand side use the substitution . The resulting integral is (8.6.5). The constraint follows from (18.15.14).
- Referenced by:
- Erratum (V1.1.8) for Equation (8.7.6)
- Permalink:
- http://dlmf.nist.gov/8.7.E6
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.1.8):
-
The constraint was updated to include .
Suggested 2022-10-14 by Walter Gautschi
- See also:
- Annotations for §8.7 and Ch.8
15: 16.19 Identities
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16.19.1
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ⓘ
- Symbols:
- or : Meijer -function, : nonnegative integer, : nonnegative integer, : complex variable, : real or complex parameters and : real or complex parameters
- Referenced by:
- §16.21
- Permalink:
- http://dlmf.nist.gov/16.19.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §16.19 and Ch.16
16.19.2
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ⓘ
- Symbols:
- or : Meijer -function, : nonnegative integer, : nonnegative integer, : complex variable, : real or complex parameters and : real or complex parameters
- Permalink:
- http://dlmf.nist.gov/16.19.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §16.19 and Ch.16
16.19.3
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ⓘ
- Symbols:
- or : Meijer -function, : nonnegative integer, : nonnegative integer, : complex variable, : real or complex parameters and : real or complex parameters
- Permalink:
- http://dlmf.nist.gov/16.19.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §16.19 and Ch.16
16.19.5
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ⓘ
- Symbols:
- or : Meijer -function, : nonnegative integer, : nonnegative integer, : complex variable, : real or complex parameters, : real or complex parameters and : differential operator
- Permalink:
- http://dlmf.nist.gov/16.19.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §16.19 and Ch.16
16.19.6
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ⓘ
- Symbols:
- : gamma function, or : Meijer -function, : differential of , : integral, : nonnegative integer, : nonnegative integer, : complex variable, : real or complex parameters and : real or complex parameters
- Referenced by:
- §16.19
- Permalink:
- http://dlmf.nist.gov/16.19.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §16.19 and Ch.16
16: 16.12 Products
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16.12.1
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ⓘ
- Symbols:
-
or : alternatively or
generalized hypergeometric function, : complex variable, : real or complex parameters and : real or complex parameters - Referenced by:
- (10.8.3), §10.8, §10.8, §16.12, Erratum (V1.0.17) for Section 10.8
- Permalink:
- http://dlmf.nist.gov/16.12.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §16.12 and Ch.16
16.12.2
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ⓘ
- Symbols:
-
: notation for Gauss’ hypergeometric function,
or : alternatively or
generalized hypergeometric function, : complex variable, : real or complex parameters and : real or complex parameters - Referenced by:
- §16.12, §17.9(iii), Erratum (V1.1.12) for Subsection 17.9(iii)
- Permalink:
- http://dlmf.nist.gov/16.12.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §16.12 and Ch.16
16.12.3
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- Symbols:
-
: notation for Gauss’ hypergeometric function,
or : alternatively or
generalized hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : complex variable, : real or complex parameters and : real or complex parameters - Referenced by:
- §16.12
- Permalink:
- http://dlmf.nist.gov/16.12.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §16.12 and Ch.16
17: 8.21 Generalized Sine and Cosine Integrals
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8.21.20
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- Symbols:
- : cosine function, : generalized sine integral, : sine function, : complex variable, : parameter, : auxiliary function and : auxiliary function
- Referenced by:
- §8.21(viii)
- Permalink:
- http://dlmf.nist.gov/8.21.E20
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.21(vii), §8.21 and Ch.8
8.21.21
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ⓘ
- Symbols:
- : cosine function, : generalized cosine integral, : sine function, : complex variable, : parameter, : auxiliary function and : auxiliary function
- Referenced by:
- §8.21(viii)
- Permalink:
- http://dlmf.nist.gov/8.21.E21
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.21(vii), §8.21 and Ch.8
8.21.22
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ⓘ
- Symbols:
- : differential of , : integral, : sine function, : complex variable, : parameter and : auxiliary function
- A&S Ref:
- 5.2.12 (This generalizes the form in AMS 55.)
- Referenced by:
- §8.21(vii)
- Permalink:
- http://dlmf.nist.gov/8.21.E22
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.21(vii), §8.21 and Ch.8
8.21.23
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ⓘ
- Symbols:
- : cosine function, : differential of , : integral, : complex variable, : parameter and : auxiliary function
- A&S Ref:
- 5.2.13 (This generalizes the form in AMS 55.)
- Referenced by:
- §8.21(vii)
- Permalink:
- http://dlmf.nist.gov/8.21.E23
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.21(vii), §8.21 and Ch.8
8.21.26
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- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : Poincaré asymptotic expansion, : complex variable, : parameter, : nonnegative integer and : auxiliary function
- Permalink:
- http://dlmf.nist.gov/8.21.E26
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.21(viii), §8.21 and Ch.8
18: 8.5 Confluent Hypergeometric Representations
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8.5.1
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►
ⓘ
- Symbols:
- : Kummer confluent hypergeometric function, : base of natural logarithm, : incomplete gamma function, : complex variable and : parameter
- A&S Ref:
- 6.5.12
- Referenced by:
- §8.11(ii), §8.14
- Permalink:
- http://dlmf.nist.gov/8.5.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.5 and Ch.8
8.5.2
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ⓘ
- Symbols:
- : Olver’s confluent hypergeometric function, : base of natural logarithm, : incomplete gamma function, : complex variable and : parameter
- Referenced by:
- §8.7
- Permalink:
- http://dlmf.nist.gov/8.5.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.5 and Ch.8
8.5.3
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ⓘ
- Symbols:
- : Kummer confluent hypergeometric function, : base of natural logarithm, : incomplete gamma function, : complex variable and : parameter
- Referenced by:
- §8.14, §8.19(vi)
- Permalink:
- http://dlmf.nist.gov/8.5.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.5 and Ch.8
8.5.4
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ⓘ
- Symbols:
- : Whittaker confluent hypergeometric function, : base of natural logarithm, : incomplete gamma function, : complex variable and : parameter
- Referenced by:
- §8.5
- Permalink:
- http://dlmf.nist.gov/8.5.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.5 and Ch.8
8.5.5
ⓘ
- Symbols:
- : Whittaker confluent hypergeometric function, : base of natural logarithm, : incomplete gamma function, : complex variable and : parameter
- Permalink:
- http://dlmf.nist.gov/8.5.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.5 and Ch.8
19: 15.8 Transformations of Variable
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15.8.12
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ⓘ
- Symbols:
- or : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter and : integer
- Permalink:
- http://dlmf.nist.gov/15.8.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.8(ii), §15.8 and Ch.15
15.8.13
,
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ⓘ
- Symbols:
- or : Gauss’ hypergeometric function, : the ratio of the circumference of a circle to its diameter, : phase, : complex variable, : real or complex parameter and : real or complex parameter
- Referenced by:
- §15.8(iv)
- Permalink:
- http://dlmf.nist.gov/15.8.E13
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15
15.8.14
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ⓘ
- Symbols:
- or : Gauss’ hypergeometric function, : the ratio of the circumference of a circle to its diameter, : phase, : complex variable, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/15.8.E14
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15
15.8.15
,
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ⓘ
- Symbols:
- or : Gauss’ hypergeometric function, : complex variable, : real or complex parameter and : real or complex parameter
- Referenced by:
- §15.4(i), §15.8(iv)
- Permalink:
- http://dlmf.nist.gov/15.8.E15
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15
15.8.16
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- Symbols:
- or : Gauss’ hypergeometric function, : complex variable, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/15.8.E16
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15
20: 8.2 Definitions and Basic Properties
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8.2.5
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- Symbols:
- : normalized incomplete gamma function, : normalized incomplete gamma function, : complex variable and : parameter
- Permalink:
- http://dlmf.nist.gov/8.2.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.2(i), §8.2 and Ch.8
8.2.8
,
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : incomplete gamma function, : complex variable, : parameter and : integer
- Permalink:
- http://dlmf.nist.gov/8.2.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.2(ii), §8.2 and Ch.8
8.2.9
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ⓘ
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : incomplete gamma function, : complex variable, : parameter and : integer
- Referenced by:
- §8.2(ii)
- Permalink:
- http://dlmf.nist.gov/8.2.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.2(ii), §8.2 and Ch.8
8.2.12
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- Symbols:
- : derivative of with respect to , : complex variable, : parameter and : solution
- Permalink:
- http://dlmf.nist.gov/8.2.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.2(iii), §8.2 and Ch.8
8.2.13
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- Symbols:
- : derivative of with respect to , : complex variable, : parameter and : solution
- Permalink:
- http://dlmf.nist.gov/8.2.E13
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.2(iii), §8.2 and Ch.8