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
27.4.10
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ⓘ
- Symbols:
- : Riemann zeta function, : divisor function, : real part, : positive integer and : positive integer
- Referenced by:
- §27.4
- Permalink:
- http://dlmf.nist.gov/27.4.E10
- 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
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: 25.14 Lerch’s Transcendent
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25.14.1
; .
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ⓘ
- Defines:
- : Lerch’s transcendent
- Symbols:
- : equals by definition, : real part, : nonnegative integer, : real or complex parameter, : complex variable and : complex variable
- Keywords:
- definition, infinite series, series representation
- Source:
- Erdélyi et al. (1953a, (1.11.1), p. 27)
- Referenced by:
- (25.14.2), (25.14.3), §25.14(ii), §25.14, Erratum (V1.0.21) for Equation (25.14.1)
- Permalink:
- http://dlmf.nist.gov/25.14.E1
- Encodings:
- pMML, png, TeX
- Clarification (effective with 1.0.21):
- The previous constraint was removed. A clarification regarding the correct constraints for Lerch’s transcendent has been added in the text immediately below.
- See also:
- Annotations for §25.14(i), §25.14 and Ch.25
25.14.2
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ⓘ
- Symbols:
- : Hurwitz zeta function, : Lerch’s transcendent, : real part, : real or complex parameter and : complex variable
- Keywords:
- specialization
- Proof sketch:
- Derivable from (25.11.1), (25.14.1).
- Permalink:
- http://dlmf.nist.gov/25.14.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §25.14(i), §25.14 and Ch.25
25.14.3
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ⓘ
- Symbols:
- : Lerch’s transcendent, : polylogarithm, : real part, : complex variable and : complex variable
- Keywords:
- specialization
- Proof sketch:
- Derivable from (25.12.10) and (25.14.1).
- Referenced by:
- (25.12.12)
- Permalink:
- http://dlmf.nist.gov/25.14.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §25.14(i), §25.14 and Ch.25
25.14.5
, if ;
, if .
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ⓘ
- Symbols:
- : gamma function, : Lerch’s transcendent, : half-closed interval, : complex plane, : differential of , : element of, : base of natural logarithm, : integral, : real part, : set subtraction, : real variable, : real or complex parameter, : complex variable and : complex variable
- Keywords:
- improper integral, integral representation
- Source:
- Erdélyi et al. (1953a, (1.11.3), p. 27)
- Referenced by:
- Erratum (V1.1.4) for Equation (25.14.5)
- Permalink:
- http://dlmf.nist.gov/25.14.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §25.14(ii), §25.14 and Ch.25
25.14.6
if ;
, if .
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ⓘ
- Symbols:
- : Lerch’s transcendent, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : integral, : arctangent function, : principal branch of logarithm function, : real part, : sine function, : real variable, : real or complex parameter, : complex variable and : complex variable
- Keywords:
- improper integral, integral representation
- Source:
- Erdélyi et al. (1953a, (1.11.4), p. 28)
- Notes:
- For the case see Erdélyi et al. (1953a, (1.11.4), p. 28). In the case one checks that the first integral converges absolutely iff .
- Referenced by:
- Erratum (V1.1.4) for Equation (25.14.6)
- Permalink:
- http://dlmf.nist.gov/25.14.E6
- Encodings:
- pMML, png, TeX
- Clarification (effective with 1.1.4):
-
The constraint which originally read
“ if ; if ” has been improved to be
“ if ;
, if ”.
Suggested 2021-08-23 by Gergő Nemes
- See also:
- Annotations for §25.14(ii), §25.14 and Ch.25
5: 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
5.13.3
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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
6: 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
7: 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
8: 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
9: 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
10: 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