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11: 10.64 Integral Representations
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10.64.1
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- Symbols:
- : Kelvin function, : the ratio of the circumference of a circle to its diameter, : cosine function, : differential of , : hyperbolic cosine function, : integral, : sine function, : integer and : real variable
- Permalink:
- http://dlmf.nist.gov/10.64.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.64, §10.64 and Ch.10
10.64.2
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- Symbols:
- : Kelvin function, : the ratio of the circumference of a circle to its diameter, : differential of , : hyperbolic sine function, : integral, : sine function, : integer and : real variable
- Permalink:
- http://dlmf.nist.gov/10.64.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.64, §10.64 and Ch.10
12: 18.33 Polynomials Orthogonal on the Unit Circle
§18.33 Polynomials Orthogonal on the Unit Circle
►§18.33(i) Definition
… ►§18.33(ii) Recurrence Relations
… ►§18.33(v) Biorthogonal Polynomials on the Unit Circle
… ►Recurrence Relations
…13: 4.4 Special Values and Limits
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4.4.2
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : imaginary unit and : principal branch of logarithm function
- A&S Ref:
- 4.1.14 (modified)
- Permalink:
- http://dlmf.nist.gov/4.4.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.4(i), §4.4 and Ch.4
4.4.3
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : imaginary unit and : principal branch of logarithm function
- A&S Ref:
- 4.1.15
- Permalink:
- http://dlmf.nist.gov/4.4.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.4(i), §4.4 and Ch.4
4.4.5
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : base of natural logarithm and : imaginary unit
- A&S Ref:
- 4.2.26
- Permalink:
- http://dlmf.nist.gov/4.4.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.4(ii), §4.4 and Ch.4
4.4.6
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : base of natural logarithm and : imaginary unit
- A&S Ref:
- 4.2.27
- Permalink:
- http://dlmf.nist.gov/4.4.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.4(ii), §4.4 and Ch.4
4.4.7
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : element of, : base of natural logarithm, : imaginary unit, : set of all integers and : integer
- A&S Ref:
- 4.2.28
- Permalink:
- http://dlmf.nist.gov/4.4.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.4(ii), §4.4 and Ch.4
14: 24.7 Integral Representations
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24.7.1
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- Symbols:
- : Bernoulli numbers, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : hyperbolic secant function, : integral, : integer and : real or complex
- Permalink:
- http://dlmf.nist.gov/24.7.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.7(i), §24.7 and Ch.24
24.7.2
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- Symbols:
- : Bernoulli numbers, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : hyperbolic cosecant function, : integral, : integer and : real or complex
- Permalink:
- http://dlmf.nist.gov/24.7.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.7(i), §24.7 and Ch.24
24.7.3
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- Symbols:
- : Bernoulli numbers, : the ratio of the circumference of a circle to its diameter, : differential of , : hyperbolic secant function, : integral, : integer and : real or complex
- Permalink:
- http://dlmf.nist.gov/24.7.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.7(i), §24.7 and Ch.24
24.7.4
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- Symbols:
- : Bernoulli numbers, : the ratio of the circumference of a circle to its diameter, : differential of , : hyperbolic cosecant function, : integral, : integer and : real or complex
- Permalink:
- http://dlmf.nist.gov/24.7.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.7(i), §24.7 and Ch.24
24.7.6
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- Symbols:
- : Euler numbers, : the ratio of the circumference of a circle to its diameter, : differential of , : hyperbolic secant function, : integral, : integer and : real or complex
- Referenced by:
- §24.7(i)
- Permalink:
- http://dlmf.nist.gov/24.7.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.7(i), §24.7 and Ch.24
15: 6.9 Continued Fraction
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6.9.1
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : exponential integral, : phase and : complex variable
- A&S Ref:
- 5.1.22 (special case of)
- Referenced by:
- §6.18(i)
- Permalink:
- http://dlmf.nist.gov/6.9.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §6.9 and Ch.6
16: 7.7 Integral Representations
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7.7.1
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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 : complex variable
- A&S Ref:
- 7.4.11 (in different form)
- Referenced by:
- §7.11, §7.7(i)
- Permalink:
- http://dlmf.nist.gov/7.7.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.7(i), §7.7 and Ch.7
7.7.2
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : differential of , : Faddeeva (or Faddeyeva) function, : base of natural logarithm, : imaginary part, : imaginary unit, : integral and : complex variable
- A&S Ref:
- 7.1.4 (in different form)
- Referenced by:
- §7.19(i), §7.22(i), §7.7(i)
- Permalink:
- http://dlmf.nist.gov/7.7.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.7(i), §7.7 and Ch.7
7.7.3
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ⓘ
- Symbols:
- : Dawson’s integral, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : imaginary unit, : integral, : real part and : complex variable
- A&S Ref:
- 7.4.6 7.4.7 (in different notation)
- Referenced by:
- §7.14(i), §7.7(i)
- Permalink:
- http://dlmf.nist.gov/7.7.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.7(i), §7.7 and Ch.7
7.7.4
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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, : real part and : complex variable
- Keywords:
- Laplace transform
- A&S Ref:
- 7.4.8
- Referenced by:
- §7.7(i)
- Permalink:
- http://dlmf.nist.gov/7.7.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.7(i), §7.7 and Ch.7
7.7.9
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : differential of , : error function, : base of natural logarithm, : integral and : real variable
- A&S Ref:
- 7.4.35 (in different form)
- Referenced by:
- §7.7(i)
- Permalink:
- http://dlmf.nist.gov/7.7.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.7(i), §7.7 and Ch.7
17: 17.18 Methods of Computation
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►Method (1) is applicable within the circles of convergence of the defining series, although it is often cumbersome owing to slowness of convergence and/or severe cancellation.
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18: 4.19 Maclaurin Series and Laurent Series
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4.19.3
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- Symbols:
- : Bernoulli numbers, : the ratio of the circumference of a circle to its diameter, : factorial (as in ), : tangent function, : integer and : complex variable
- A&S Ref:
- 4.3.67
- Referenced by:
- §4.19
- Permalink:
- http://dlmf.nist.gov/4.19.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.19 and Ch.4
4.19.4
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- Symbols:
- : Bernoulli numbers, : the ratio of the circumference of a circle to its diameter, : cosecant function, : factorial (as in ), : integer and : complex variable
- A&S Ref:
- 4.3.68
- Referenced by:
- §4.33
- Permalink:
- http://dlmf.nist.gov/4.19.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.19 and Ch.4
4.19.5
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- Symbols:
- : Euler numbers, : the ratio of the circumference of a circle to its diameter, : factorial (as in ), : secant function, : integer and : complex variable
- A&S Ref:
- 4.3.69
- Referenced by:
- §24.1, §24.19(i), §24.2(ii)
- Permalink:
- http://dlmf.nist.gov/4.19.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.19 and Ch.4
4.19.6
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- Symbols:
- : Bernoulli numbers, : the ratio of the circumference of a circle to its diameter, : cotangent function, : factorial (as in ), : integer and : complex variable
- A&S Ref:
- 4.3.70
- Permalink:
- http://dlmf.nist.gov/4.19.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.19 and Ch.4
4.19.7
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- Symbols:
- : Bernoulli numbers, : the ratio of the circumference of a circle to its diameter, : factorial (as in ), : principal branch of logarithm function, : sine function, : integer and : complex variable
- A&S Ref:
- 4.3.71
- Permalink:
- http://dlmf.nist.gov/4.19.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.19 and Ch.4
19: 24.9 Inequalities
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24.9.4
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- Symbols:
- : Bernoulli polynomials, : the ratio of the circumference of a circle to its diameter, : factorial (as in ), : integer and : real or complex
- Referenced by:
- §24.9
- Permalink:
- http://dlmf.nist.gov/24.9.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.9 and Ch.24
24.9.5
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ⓘ
- Symbols:
- : Euler polynomials, : the ratio of the circumference of a circle to its diameter, : factorial (as in ), : integer and : real or complex
- A&S Ref:
- 23.1.14
- Referenced by:
- §24.9, §24.9
- Permalink:
- http://dlmf.nist.gov/24.9.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.9 and Ch.24
24.9.6
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- Symbols:
- : Bernoulli numbers, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm and : integer
- Referenced by:
- §24.9, §24.9
- Permalink:
- http://dlmf.nist.gov/24.9.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.9 and Ch.24
24.9.8
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- Symbols:
- : Bernoulli numbers, : the ratio of the circumference of a circle to its diameter, : factorial (as in ), : integer and : quantity
- Referenced by:
- §24.9
- Permalink:
- http://dlmf.nist.gov/24.9.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.9 and Ch.24
24.9.9
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : principal branch of logarithm function and : quantity
- Permalink:
- http://dlmf.nist.gov/24.9.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.9 and Ch.24
20: 4.8 Identities
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4.8.2
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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:
- 4.1.7
- Permalink:
- http://dlmf.nist.gov/4.8.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.8(i), §4.8 and Ch.4
4.8.7
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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
- Referenced by:
- §4.2(i), §4.8(i)
- Permalink:
- http://dlmf.nist.gov/4.8.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.8(i), §4.8 and Ch.4
4.8.9
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : exponential function, : imaginary part, : principal branch of logarithm function and : complex variable
- A&S Ref:
- 4.2.3
- Permalink:
- http://dlmf.nist.gov/4.8.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.8(i), §4.8 and Ch.4
4.8.12
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : imaginary unit, : principal branch of logarithm function, : integer, : real or complex constant and : complex variable
- Permalink:
- http://dlmf.nist.gov/4.8.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.8(i), §4.8 and Ch.4
4.8.15
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : phase, : real or complex constant and : complex variable
- A&S Ref:
- 4.2.16
- Permalink:
- http://dlmf.nist.gov/4.8.E15
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.8(ii), §4.8 and Ch.4