logarithm function
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21—30 of 399 matching pages
21: 27.11 Asymptotic Formulas: Partial Sums
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27.11.2
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- Symbols:
- : order not exceeding, : Euler’s constant, : divisor function, : principal branch of logarithm function, : positive integer and : real number
- A&S Ref:
- 24.3.3 III
- Referenced by:
- §27.11, §27.11, Erratum (V1.1.8) for Section 27.11
- Permalink:
- http://dlmf.nist.gov/27.11.E2
- 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 §27.11 and Ch.27
27.11.3
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- Symbols:
- : order not exceeding, : Euler’s constant, : divisor function, : principal branch of logarithm function, : positive integer and : real number
- Referenced by:
- §27.11
- Permalink:
- http://dlmf.nist.gov/27.11.E3
- 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 §27.11 and Ch.27
27.11.10
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- Symbols:
- : order not exceeding, : principal branch of logarithm function, : prime numbers and : real number
- Permalink:
- http://dlmf.nist.gov/27.11.E10
- 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 §27.11 and Ch.27
27.11.12
,
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ⓘ
- Symbols:
- : order not exceeding, : Möbius function, : base of natural logarithm, : principal branch of logarithm function, : positive integer, : real number and : constant
- A&S Ref:
- 24.3.1 III
- Referenced by:
- §27.11
- Permalink:
- http://dlmf.nist.gov/27.11.E12
- 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 §27.11 and Ch.27
27.11.15
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- Symbols:
- : Möbius function, : principal branch of logarithm function, : positive integer and : real number
- A&S Ref:
- 24.3.1 III
- Referenced by:
- §27.11, §27.11
- Permalink:
- http://dlmf.nist.gov/27.11.E15
- 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 §27.11 and Ch.27
22: 4.37 Inverse Hyperbolic Functions
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§4.37(iv) Logarithmic Forms
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4.37.19
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- Symbols:
- : complex plane, : element of, : inverse hyperbolic cosine function, : principal branch of logarithm function, : open interval, : set subtraction and : complex variable
- Referenced by:
- §4.37(iv), §4.37(iii), §4.37(iv)
- Permalink:
- http://dlmf.nist.gov/4.37.E19
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4
4.37.21
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- Symbols:
- : complex plane, : element of, : inverse hyperbolic cosine function, : principal branch of logarithm function, : open interval, : set subtraction and : complex variable
- Permalink:
- http://dlmf.nist.gov/4.37.E21
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4
4.37.24
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►For the corresponding results for , , and , use (4.37.7)–(4.37.9); compare §4.23(iv).
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- Symbols:
- : half-closed interval, : complex plane, : element of, : inverse hyperbolic tangent function, : principal branch of logarithm function, : half-closed interval, : set subtraction, : union and : complex variable
- Permalink:
- http://dlmf.nist.gov/4.37.E24
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4
23: 25.8 Sums
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25.8.4
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- Symbols:
- : gamma function, : Riemann zeta function, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : principal branch of logarithm function, : nonnegative integer and : nonnegative integer
- Keywords:
- infinite series
- Source:
- Adamchik and Srivastava (1998, (2.7), p. 135)
- Permalink:
- http://dlmf.nist.gov/25.8.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §25.8 and Ch.25
25.8.7
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ⓘ
- Symbols:
- : gamma function, : Euler’s constant, : Riemann zeta function, : principal branch of logarithm function, : nonnegative integer and : complex variable
- Keywords:
- infinite series
- Proof sketch:
- Derivable from (25.8.5) by dividing by and integrating.
- Permalink:
- http://dlmf.nist.gov/25.8.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §25.8 and Ch.25
25.8.8
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ⓘ
- Symbols:
- : Riemann zeta function, : the ratio of the circumference of a circle to its diameter, : principal branch of logarithm function, : sine function, : nonnegative integer and : complex variable
- Keywords:
- infinite series
- Proof sketch:
- Derivable from (25.8.6) by dividing by and integrating.
- Permalink:
- http://dlmf.nist.gov/25.8.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §25.8 and Ch.25
25.8.9
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- Symbols:
- : Riemann zeta function, : principal branch of logarithm function and : nonnegative integer
- Keywords:
- infinite series
- Source:
- Srivastava and Choi (2001, (467), p. 212)
- Permalink:
- http://dlmf.nist.gov/25.8.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §25.8 and Ch.25
24: 22.14 Integrals
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22.14.1
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- Symbols:
- : Jacobian elliptic function, : Jacobian elliptic function, : Jacobian elliptic function, : differential of , : integral, : principal branch of logarithm function, : real and : modulus
- A&S Ref:
- 16.24.1
- Permalink:
- http://dlmf.nist.gov/22.14.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.14(i), §22.14 and Ch.22
22.14.7
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- Symbols:
- : Jacobian elliptic function, : Jacobian elliptic function, : Jacobian elliptic function, : differential of , : integral, : principal branch of logarithm function, : real and : modulus
- A&S Ref:
- 16.24.7
- Permalink:
- http://dlmf.nist.gov/22.14.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.14(i), §22.14 and Ch.22
22.14.10
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- Symbols:
- : Jacobian elliptic function, : Jacobian elliptic function, : Jacobian elliptic function, : differential of , : integral, : principal branch of logarithm function, : real and : modulus
- A&S Ref:
- 16.24.10
- Permalink:
- http://dlmf.nist.gov/22.14.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.14(i), §22.14 and Ch.22
22.14.11
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- Symbols:
- : Jacobian elliptic function, : Jacobian elliptic function, : Jacobian elliptic function, : differential of , : integral, : principal branch of logarithm function, : real and : modulus
- A&S Ref:
- 16.24.11
- Permalink:
- http://dlmf.nist.gov/22.14.E11
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.14(i), §22.14 and Ch.22
22.14.12
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- Symbols:
- : Jacobian elliptic function, : Jacobian elliptic function, : Jacobian elliptic function, : differential of , : integral, : principal branch of logarithm function, : real and : modulus
- A&S Ref:
- 16.24.12
- Permalink:
- http://dlmf.nist.gov/22.14.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.14(i), §22.14 and Ch.22
25: 6.14 Integrals
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6.14.1
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ⓘ
- Symbols:
- : differential of , : base of natural logarithm, : exponential integral, : integral, : principal branch of logarithm function and : real part
- A&S Ref:
- 5.1.34 (special case of)
- Permalink:
- http://dlmf.nist.gov/6.14.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §6.14(i), §6.14 and Ch.6
6.14.2
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ⓘ
- Symbols:
- : cosine integral, : differential of , : base of natural logarithm, : integral, : principal branch of logarithm function and : real part
- A&S Ref:
- 5.2.28
- Referenced by:
- §6.14(i)
- Permalink:
- http://dlmf.nist.gov/6.14.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §6.14(i), §6.14 and Ch.6
6.14.3
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- Symbols:
- : differential of , : base of natural logarithm, : integral, : arctangent function, : real part and : sine integral
- A&S Ref:
- 5.2.29
- Permalink:
- http://dlmf.nist.gov/6.14.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §6.14(i), §6.14 and Ch.6
6.14.4
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- Symbols:
- : differential of , : exponential integral, : integral and : principal branch of logarithm function
- A&S Ref:
- 5.1.33
- Permalink:
- http://dlmf.nist.gov/6.14.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §6.14(ii), §6.14 and Ch.6
6.14.7
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- Symbols:
- : cosine integral, : differential of , : integral, : principal branch of logarithm function and : sine integral
- A&S Ref:
- 5.2.32
- Referenced by:
- §6.14(ii)
- Permalink:
- http://dlmf.nist.gov/6.14.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §6.14(ii), §6.14 and Ch.6
26: 4.9 Continued Fractions
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§4.9(i) Logarithms
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4.9.1
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : principal branch of logarithm function, : phase and : complex variable
- A&S Ref:
- 4.1.39
- Permalink:
- http://dlmf.nist.gov/4.9.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.9(i), §4.9 and Ch.4
4.9.2
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- Symbols:
- : principal branch of logarithm function and : complex variable
- A&S Ref:
- 4.1.40
- Permalink:
- http://dlmf.nist.gov/4.9.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.9(i), §4.9 and Ch.4
27: 2.2 Transcendental Equations
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2.2.3
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- Symbols:
- : principal branch of logarithm function and : root
- Permalink:
- http://dlmf.nist.gov/2.2.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §2.2, §2.2 and Ch.2
2.2.5
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- Symbols:
- : order less than, : principal branch of logarithm function and : root
- Permalink:
- http://dlmf.nist.gov/2.2.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §2.2, §2.2 and Ch.2
2.2.6
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- Symbols:
- : order less than, : principal branch of logarithm function and : root
- Permalink:
- http://dlmf.nist.gov/2.2.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §2.2, §2.2 and Ch.2
28: 4.23 Inverse Trigonometric Functions
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§4.23(iv) Logarithmic Forms
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4.23.24
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- Symbols:
- : half-closed interval, : element of, : imaginary unit, : arccosine function, : principal branch of logarithm function and : real variable
- Permalink:
- http://dlmf.nist.gov/4.23.E24
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4
4.23.25
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►Care needs to be taken on the cuts, for example, if then .
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : element of, : imaginary unit, : arccosine function, : principal branch of logarithm function, : half-closed interval and : real variable
- Permalink:
- http://dlmf.nist.gov/4.23.E25
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4
4.23.36
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- Symbols:
- : imaginary unit, : arctangent function, : principal branch of logarithm function, : real variable, : real variable and : complex variable
- A&S Ref:
- 4.4.39 (with the general value.)
- Referenced by:
- (4.23.34), (4.23.35), §4.23(vi)
- Permalink:
- http://dlmf.nist.gov/4.23.E36
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.23(vi), §4.23 and Ch.4
29: 4.45 Methods of Computation
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Logarithms
►The function can always be computed from its ascending power series after preliminary scaling. … ►
4.45.2
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- Symbols:
- : principal branch of logarithm function, : integer, : real variable and : real variable
- Permalink:
- http://dlmf.nist.gov/4.45.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.45(i), §4.45(i), §4.45 and Ch.4
4.45.3
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►The trigonometric functions may be computed from the definitions (4.14.1)–(4.14.7), and their inverses from the logarithmic forms in §4.23(iv), followed by (4.23.7)–(4.23.9).
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- Symbols:
- : principal branch of logarithm function, : integer and : real variable
- Permalink:
- http://dlmf.nist.gov/4.45.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.45(i), §4.45(i), §4.45 and Ch.4
30: 5.9 Integral Representations
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Binet’s Formula
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5.9.10
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ⓘ
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : integral, : arctangent function, : general logarithm function, : principal branch of logarithm function and : complex variable
- A&S Ref:
- 6.1.50
- Referenced by:
- (5.9.10_1), §5.9(i), §5.9(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)
- Permalink:
- http://dlmf.nist.gov/5.9.E10
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.10):
-
To increase the region of validity of this equation, the logarithm of the
gamma function that appears on its left-hand side has been changed to
, where is the general logarithm.
Originally was used, where
is the principal branch of the logarithm.
Suggested 2015-02-13 by Philippe Spindel
- See also:
- Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5
5.9.10_1
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ⓘ
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : integral, : general logarithm function, : principal branch of logarithm function and : complex variable
- Proof sketch:
- Can be obtained from (5.9.10) via integration by parts.
- Referenced by:
- §5.9(i), Erratum (V1.1.7) for Additions
- Permalink:
- http://dlmf.nist.gov/5.9.E10_1
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.1.7):
- This equation was added.
- See also:
- Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5
5.9.10_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, : integral, : general logarithm function, : principal branch of logarithm function and : complex variable
- Source:
- Whittaker and Watson (1927, §12.31)
- Referenced by:
- §5.9(i), Erratum (V1.1.7) for Additions
- Permalink:
- http://dlmf.nist.gov/5.9.E10_2
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.1.7):
- This equation was added.
- See also:
- Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5
5.9.11
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- Symbols:
- : gamma function, : Euler’s constant, : Riemann zeta function, : the ratio of the circumference of a circle to its diameter, : differential of , : imaginary unit, : integral, : general logarithm function, : principal branch of logarithm function, : sine function, : real or complex variable and : complex variable
- Referenced by:
- §5.9(i), §5.9(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)
- Permalink:
- http://dlmf.nist.gov/5.9.E11
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.10):
-
To increase the region of validity of this equation, the logarithm of the
gamma function that appears on its left-hand side has been changed to
, where is the general logarithm.
Originally was used, where
is the principal branch of the logarithm.
Suggested 2015-02-13 by Philippe Spindel
- See also:
- Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5