real part
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1—10 of 249 matching pages
1: 27.4 Euler Products and Dirichlet Series
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27.4.3
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- Symbols:
- : Riemann zeta function, : real part, : positive integer and : prime numbers
- A&S Ref:
- 23.2.1 and 23.2.2
- Permalink:
- http://dlmf.nist.gov/27.4.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §27.4 and Ch.27
27.4.5
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- Symbols:
- : Möbius function, : Riemann zeta function, : real part and : positive integer
- A&S Ref:
- 24.3.1 I.B
- Permalink:
- http://dlmf.nist.gov/27.4.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §27.4 and Ch.27
27.4.11
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ⓘ
- Symbols:
- : Riemann zeta function, : sum of powers of divisors of a number, : real part and : positive integer
- A&S Ref:
- 24.3.3 I.B
- Permalink:
- http://dlmf.nist.gov/27.4.E11
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §27.4 and Ch.27
27.4.12
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ⓘ
- Symbols:
- : Mangoldt’s function, : Riemann zeta function, : real part and : positive integer
- Referenced by:
- §27.4
- Permalink:
- http://dlmf.nist.gov/27.4.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §27.4 and Ch.27
27.4.13
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- Symbols:
- : Riemann zeta function, : principal branch of logarithm function, : real part and : positive integer
- Referenced by:
- §27.4
- Permalink:
- http://dlmf.nist.gov/27.4.E13
- Encodings:
- pMML, png, TeX
- Change of Notation (effective with 1.0.10):
- The notation for logarithm has been changed to from .
- See also:
- Annotations for §27.4 and Ch.27
2: 8.14 Integrals
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8.14.1
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ⓘ
- Symbols:
- : gamma function, : differential of , : base of natural logarithm, : incomplete gamma function, : integral, : real part, : real variable and : parameter
- Referenced by:
- §8.14, §8.14
- Permalink:
- http://dlmf.nist.gov/8.14.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.14 and Ch.8
8.14.2
, .
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ⓘ
- Symbols:
- : gamma function, : differential of , : base of natural logarithm, : incomplete gamma function, : integral, : real part, : real variable and : parameter
- A&S Ref:
- 6.5.36
- Referenced by:
- §8.14, §8.14
- Permalink:
- http://dlmf.nist.gov/8.14.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.14 and Ch.8
8.14.3
, ,
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ⓘ
- Symbols:
- : gamma function, : differential of , : incomplete gamma function, : integral, : real part, : real variable and : parameter
- Referenced by:
- §8.14
- Permalink:
- http://dlmf.nist.gov/8.14.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.14 and Ch.8
8.14.4
, ,
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ⓘ
- Symbols:
- : gamma function, : differential of , : incomplete gamma function, : integral, : real part, : real variable and : parameter
- A&S Ref:
- 6.5.37
- Permalink:
- http://dlmf.nist.gov/8.14.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.14 and Ch.8
8.14.6
, , .
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ⓘ
- Symbols:
- : gamma function, or : Gauss’ hypergeometric function, : differential of , : base of natural logarithm, : incomplete gamma function, : integral, : real part, : real variable and : parameter
- Referenced by:
- §8.14
- Permalink:
- http://dlmf.nist.gov/8.14.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.14 and Ch.8
3: 13.10 Integrals
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13.10.3
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- Symbols:
- : gamma function, : Olver’s confluent hypergeometric function, : differential of , : base of natural logarithm, or : scaled (or Olver’s) generalized hypergeometric function, : integral, : real part and : complex variable
- Referenced by:
- §8.14
- Permalink:
- http://dlmf.nist.gov/13.10.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.10(ii), §13.10 and Ch.13
13.10.5
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ⓘ
- Symbols:
- : gamma function, : Olver’s confluent hypergeometric function, : differential of , : base of natural logarithm, : integral and : real part
- Permalink:
- http://dlmf.nist.gov/13.10.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.10(ii), §13.10 and Ch.13
13.10.10
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ⓘ
- Symbols:
- : gamma function, : Olver’s confluent hypergeometric function, : differential of , : integral and : real part
- Keywords:
- Mellin transform
- Referenced by:
- §8.14
- Permalink:
- http://dlmf.nist.gov/13.10.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.10(iii), §13.10 and Ch.13
13.10.11
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ⓘ
- Symbols:
- : gamma function, : Kummer confluent hypergeometric function, : differential of , : integral and : real part
- Keywords:
- Mellin transform
- Referenced by:
- §8.14
- Permalink:
- http://dlmf.nist.gov/13.10.E11
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.10(iii), §13.10 and Ch.13
13.10.13
, ,
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- Symbols:
- : Bessel function of the first kind, : Olver’s confluent hypergeometric function, : differential of , : base of natural logarithm, : integral, : real part and : real variable
- Permalink:
- http://dlmf.nist.gov/13.10.E13
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §13.10(v), §13.10 and Ch.13
4: 25.13 Periodic Zeta Function
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►where if is an integer, otherwise.
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25.13.2
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- Symbols:
- : gamma function, : Hurwitz zeta function, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : periodic zeta function, : real part, : real variable and : complex variable
- Keywords:
- connection formula
- Source:
- Apostol (1976, Exercise 2, p. 273)
- Permalink:
- http://dlmf.nist.gov/25.13.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §25.13 and Ch.25
25.13.3
if ; if .
ⓘ
- Symbols:
- : gamma function, : Hurwitz zeta function, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : periodic zeta function, : real part, : real variable and : complex variable
- Keywords:
- connection formula
- Source:
- Apostol (1976, (10), p. 257)
- Notes:
- This formula is equivalent to (25.11.9).
- Referenced by:
- Erratum (V1.1.4) for Equation (25.13.3)
- Permalink:
- http://dlmf.nist.gov/25.13.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §25.13 and Ch.25
5: 7.9 Continued Fractions
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7.9.1
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : complementary error function, : base of natural logarithm, : real part and : complex variable
- A&S Ref:
- 7.1.14 (in different form)
- Referenced by:
- §7.9
- Permalink:
- http://dlmf.nist.gov/7.9.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.9 and Ch.7
7.9.2
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : complementary error function, : base of natural logarithm, : real part and : complex variable
- Referenced by:
- §7.9
- Permalink:
- http://dlmf.nist.gov/7.9.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.9 and Ch.7
6: 7.14 Integrals
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7.14.2
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : complementary error function, : differential of , : error function, : base of natural logarithm, : integral, : phase and : real part
- Keywords:
- Laplace transform
- A&S Ref:
- 7.4.17
- Referenced by:
- §7.14(i)
- Permalink:
- http://dlmf.nist.gov/7.14.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7
7.14.3
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ⓘ
- Symbols:
- : differential of , : error function, : base of natural logarithm, : integral and : real part
- Keywords:
- Laplace transform
- A&S Ref:
- 7.4.19
- Referenced by:
- §7.14(i)
- Permalink:
- http://dlmf.nist.gov/7.14.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7
7.14.4
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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, : integral, : phase and : real part
- Keywords:
- Laplace transform
- A&S Ref:
- 7.4.21
- Referenced by:
- §3.5(ix), §7.14(i)
- Permalink:
- http://dlmf.nist.gov/7.14.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7
7.14.5
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ⓘ
- Symbols:
- : auxiliary function for Fresnel integrals, : Fresnel integral, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : integral and : real part
- Keywords:
- Laplace transform
- A&S Ref:
- 7.4.27 (in different notation)
- Referenced by:
- §7.14(ii)
- Permalink:
- http://dlmf.nist.gov/7.14.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7
7.14.6
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- Symbols:
- : auxiliary function for Fresnel integrals, : Fresnel integral, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : integral and : real part
- Keywords:
- Laplace transform
- A&S Ref:
- 7.4.28 (in different notation)
- Referenced by:
- §7.14(ii)
- Permalink:
- http://dlmf.nist.gov/7.14.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7
7: 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
8: 15.14 Integrals
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15.14.1
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- Symbols:
- : gamma function, : differential of , : integral, : real part, or : Olver’s hypergeometric function, : real variable, : nonnegative integer, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Keywords:
- Mellin transform
- Referenced by:
- §15.14
- Permalink:
- http://dlmf.nist.gov/15.14.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.14 and Ch.15
9: 20.10 Integrals
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20.10.1
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ⓘ
- Symbols:
- : gamma function, : theta function, : Riemann zeta function, : the ratio of the circumference of a circle to its diameter, : differential of , : imaginary unit, : integral and : real part
- Keywords:
- Mellin transform
- Referenced by:
- §20.10(i), Erratum (V1.1.5) for §20.10(i)
- Permalink:
- http://dlmf.nist.gov/20.10.E1
- Encodings:
- pMML, png, TeX
- Modification (effective with 1.1.5):
- Originally the constraint was . This can be relaxed to because as , which follows from (20.7.31) and (20.4.5).
- See also:
- Annotations for §20.10(i), §20.10 and Ch.20
20.10.2
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ⓘ
- Symbols:
- : gamma function, : theta function, : Riemann zeta function, : the ratio of the circumference of a circle to its diameter, : differential of , : imaginary unit, : integral and : real part
- Keywords:
- Mellin transform
- Referenced by:
- §20.10(i), Erratum (V1.1.5) for §20.10(i)
- Permalink:
- http://dlmf.nist.gov/20.10.E2
- Encodings:
- pMML, png, TeX
- Modification (effective with 1.1.5):
- Originally the constraint was . This can be relaxed to because as , which follows from (20.7.32) and (20.4.4).
- See also:
- Annotations for §20.10(i), §20.10 and Ch.20
20.10.3
.
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►Let , , and be constants such that , , and .
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ⓘ
- Symbols:
- : gamma function, : theta function, : Riemann zeta function, : the ratio of the circumference of a circle to its diameter, : differential of , : imaginary unit, : integral and : real part
- Keywords:
- Mellin transform
- Referenced by:
- §20.10(i), Erratum (V1.1.5) for §20.10(i)
- Permalink:
- http://dlmf.nist.gov/20.10.E3
- Encodings:
- pMML, png, TeX
- Modification (effective with 1.1.5):
- Originally the constraint was . This can be relaxed to because as , which follows from (20.7.33) and (20.4.3).
- See also:
- Annotations for §20.10(i), §20.10 and Ch.20
10: 28.25 Asymptotic Expansions for Large
§28.25 Asymptotic Expansions for Large
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28.25.4
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : imaginary part, : phase, : real part, : parameter, : complex variable and : arbitrary small positive constant
- Permalink:
- http://dlmf.nist.gov/28.25.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.25 and Ch.28
28.25.5
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : imaginary part, : phase, : real part, : parameter, : complex variable and : arbitrary small positive constant
- Permalink:
- http://dlmf.nist.gov/28.25.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.25 and Ch.28