贝尔法斯特圣玛丽学院国际商务文凭证书【购证 微kaa77788】pi/2
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1: 4.25 Continued Fractions
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4.25.1
, , .
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : tangent function and : complex variable
- A&S Ref:
- 4.3.94
- Permalink:
- http://dlmf.nist.gov/4.25.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.25 and Ch.4
4.25.2
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : real part, : tangent function, : real or complex constant and : complex variable
- A&S Ref:
- 4.3.95
- Permalink:
- http://dlmf.nist.gov/4.25.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.25 and Ch.4
2: 19.5 Maclaurin and Related Expansions
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19.5.1
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- Symbols:
- : notation for Gauss’ hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the first kind, : factorial (as in ), : nonnegative integer and : real or complex modulus
- Source:
- Carlson (1961b, (2.4))
- Referenced by:
- §19.36(iv), §19.5
- Permalink:
- http://dlmf.nist.gov/19.5.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.5 and Ch.19
19.5.2
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ⓘ
- Symbols:
- : notation for Gauss’ hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the second kind, : factorial (as in ), : nonnegative integer and : real or complex modulus
- Source:
- Carlson (1961b, (2.5))
- Referenced by:
- §19.36(iv)
- Permalink:
- http://dlmf.nist.gov/19.5.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.5 and Ch.19
19.5.4
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ⓘ
- Symbols:
- : first Appell function, : Pochhammer’s symbol (or shifted factorial), : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the third kind, : factorial (as in ), : nonnegative integer, : nonnegative integer, : real or complex modulus and : real or complex parameter
- Source:
- Carlson (1961b, (2.10))
- Referenced by:
- §19.5
- Permalink:
- http://dlmf.nist.gov/19.5.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.5 and Ch.19
19.5.8
,
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the first kind, : nome, : nonnegative integer and : real or complex modulus
- Referenced by:
- §19.5, §19.5
- Permalink:
- http://dlmf.nist.gov/19.5.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.5 and Ch.19
19.5.10
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the first kind, : nonnegative integer and : real or complex modulus
- Referenced by:
- §19.5
- Permalink:
- http://dlmf.nist.gov/19.5.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.5 and Ch.19
3: 10.65 Power Series
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10.65.3
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- Symbols:
- : Kelvin function, : Kelvin function, : Kelvin function, : the ratio of the circumference of a circle to its diameter, : cosine function, : psi (or digamma) function, : factorial (as in ), : principal branch of logarithm function, : integer, : nonnegative integer and : real variable
- A&S Ref:
- 9.9.11
- Permalink:
- http://dlmf.nist.gov/10.65.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.65(ii), §10.65 and Ch.10
10.65.4
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ⓘ
- Symbols:
- : Kelvin function, : Kelvin function, : Kelvin function, : the ratio of the circumference of a circle to its diameter, : psi (or digamma) function, : factorial (as in ), : principal branch of logarithm function, : sine function, : integer, : nonnegative integer and : real variable
- A&S Ref:
- 9.9.11
- Permalink:
- http://dlmf.nist.gov/10.65.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.65(ii), §10.65 and Ch.10
4: 5.4 Special Values and Extrema
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5.4.16
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : psi (or digamma) function, : hyperbolic cotangent function, : imaginary part, : imaginary unit and : real variable
- A&S Ref:
- 6.3.11
- Permalink:
- http://dlmf.nist.gov/5.4.E16
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §5.4(ii), §5.4 and Ch.5
5.4.17
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : psi (or digamma) function, : hyperbolic tangent function, : imaginary part, : imaginary unit and : real variable
- A&S Ref:
- 6.3.12
- Permalink:
- http://dlmf.nist.gov/5.4.E17
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §5.4(ii), §5.4 and Ch.5
5.4.18
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : psi (or digamma) function, : hyperbolic cotangent function, : imaginary part, : imaginary unit and : real variable
- A&S Ref:
- 6.3.13
- Referenced by:
- §5.4(ii)
- Permalink:
- http://dlmf.nist.gov/5.4.E18
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §5.4(ii), §5.4 and Ch.5
5.4.19
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- Symbols:
- : Euler’s constant, : the ratio of the circumference of a circle to its diameter, : cosine function, : cotangent function, : psi (or digamma) function, : principal branch of logarithm function, : real or complex variable and : nonnegative integer
- Referenced by:
- §5.19(i), §5.4(ii)
- Permalink:
- http://dlmf.nist.gov/5.4.E19
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §5.4(ii), §5.4 and Ch.5
5: 10.15 Derivatives with Respect to Order
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10.15.3
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ⓘ
- Symbols:
- : Bessel function of the first kind, : Bessel function of the second kind, : the ratio of the circumference of a circle to its diameter, : factorial (as in ), : partial derivative of with respect to , : partial differential of , : integer, : nonnegative integer, : complex variable and : complex parameter
- A&S Ref:
- 9.1.66
- Referenced by:
- §10.15, §10.15, §10.38
- Permalink:
- http://dlmf.nist.gov/10.15.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.15, §10.15 and Ch.10
10.15.4
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ⓘ
- Symbols:
- : Bessel function of the first kind, : Bessel function of the second kind, : the ratio of the circumference of a circle to its diameter, : factorial (as in ), : partial derivative of with respect to , : partial differential of , : integer, : nonnegative integer, : complex variable and : complex parameter
- A&S Ref:
- 9.1.67
- Permalink:
- http://dlmf.nist.gov/10.15.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.15, §10.15 and Ch.10
10.15.5
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- Symbols:
- : Bessel function of the first kind, : Bessel function of the second kind, : the ratio of the circumference of a circle to its diameter, : partial derivative of with respect to , : partial differential of , : complex variable and : complex parameter
- A&S Ref:
- 9.1.68
- Referenced by:
- §10.15, §10.38
- Permalink:
- http://dlmf.nist.gov/10.15.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.15, §10.15 and Ch.10
6: 20.2 Definitions and Periodic Properties
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20.2.11
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ⓘ
- Symbols:
- : theta function, : the ratio of the circumference of a circle to its diameter, : imaginary unit, : complex, : lattice parameter and
- A&S Ref:
- 16.33.2 (in different notation)
- Referenced by:
- §20.7(iv)
- Permalink:
- http://dlmf.nist.gov/20.2.E11
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §20.2(iii), §20.2 and Ch.20
20.2.12
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ⓘ
- Symbols:
- : theta function, : the ratio of the circumference of a circle to its diameter, : complex, : lattice parameter and
- A&S Ref:
- 16.33.3 (in different notation)
- Permalink:
- http://dlmf.nist.gov/20.2.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §20.2(iii), §20.2 and Ch.20
20.2.13
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ⓘ
- Symbols:
- : theta function, : the ratio of the circumference of a circle to its diameter, : complex, : lattice parameter and
- A&S Ref:
- 16.33.4 (in different notation)
- Permalink:
- http://dlmf.nist.gov/20.2.E13
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §20.2(iii), §20.2 and Ch.20
20.2.14
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ⓘ
- Symbols:
- : theta function, : the ratio of the circumference of a circle to its diameter, : imaginary unit, : complex, : lattice parameter and
- Referenced by:
- §20.7(iv)
- Permalink:
- http://dlmf.nist.gov/20.2.E14
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §20.2(iii), §20.2 and Ch.20
7: 28.22 Connection Formulas
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28.22.5
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- Symbols:
- : Mathieu function, : the ratio of the circumference of a circle to its diameter, : integer, : parameter, , : joining factors and : Fourier coefficient
- Permalink:
- http://dlmf.nist.gov/28.22.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.22(i), §28.22 and Ch.28
28.22.6
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ⓘ
- Symbols:
- : Mathieu function, : the ratio of the circumference of a circle to its diameter, : integer, : parameter, , : joining factors and : Fourier coefficient
- Permalink:
- http://dlmf.nist.gov/28.22.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.22(i), §28.22 and Ch.28
28.22.7
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ⓘ
- Symbols:
- : Mathieu function, : the ratio of the circumference of a circle to its diameter, : integer, : parameter, , : joining factors and : Fourier coefficient
- Permalink:
- http://dlmf.nist.gov/28.22.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.22(i), §28.22 and Ch.28
28.22.8
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ⓘ
- Symbols:
- : Mathieu function, : the ratio of the circumference of a circle to its diameter, : integer, : parameter, , : joining factors and : Fourier coefficient
- Permalink:
- http://dlmf.nist.gov/28.22.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.22(i), §28.22 and Ch.28
8: 22.11 Fourier and Hyperbolic Series
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22.11.3
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ⓘ
- Symbols:
- : Jacobian elliptic function, : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the first kind, : cosine function, : nome, : complex, : modulus and : change of variable
- A&S Ref:
- 16.23.3
- Permalink:
- http://dlmf.nist.gov/22.11.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.11 and Ch.22
22.11.7
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ⓘ
- Symbols:
- : Jacobian elliptic function, : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the first kind, : cosecant function, : nome, : sine function, : complex, : modulus and : change of variable
- A&S Ref:
- 16.23.10
- Referenced by:
- §22.11
- Permalink:
- http://dlmf.nist.gov/22.11.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.11 and Ch.22
22.11.8
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ⓘ
- Symbols:
- : Jacobian elliptic function, : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the first kind, : cosecant function, : nome, : sine function, : complex, : modulus and : change of variable
- A&S Ref:
- 16.23.11
- Permalink:
- http://dlmf.nist.gov/22.11.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.11 and Ch.22
22.11.9
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ⓘ
- Symbols:
- : Jacobian elliptic function, : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the first kind, : cotangent function, : nome, : sine function, : complex, : modulus and : change of variable
- A&S Ref:
- 16.23.12
- Permalink:
- http://dlmf.nist.gov/22.11.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.11 and Ch.22
22.11.10
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ⓘ
- Symbols:
- : Jacobian elliptic function, : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the first kind, : cosine function, : nome, : secant function, : complex, : modulus and : change of variable
- A&S Ref:
- 16.23.7
- Permalink:
- http://dlmf.nist.gov/22.11.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.11 and Ch.22
9: 10.67 Asymptotic Expansions for Large Argument
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10.67.1
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- Symbols:
- : Kelvin function, : Poincaré asymptotic expansion, : the ratio of the circumference of a circle to its diameter, : cosine function, : base of natural logarithm, : nonnegative integer, : real variable, : complex parameter and : polynomial coefficient
- A&S Ref:
- 9.10.3, 9.10.5--9.10.7 (modified)
- Referenced by:
- §10.61(v), §10.67(ii), §10.68(iii)
- Permalink:
- http://dlmf.nist.gov/10.67.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.67(i), §10.67 and Ch.10
10.67.2
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ⓘ
- Symbols:
- : Kelvin function, : Poincaré asymptotic expansion, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : sine function, : nonnegative integer, : real variable, : complex parameter and : polynomial coefficient
- A&S Ref:
- 9.10.4, 9.10.5--9.10.7 (modified)
- Referenced by:
- §10.61(v)
- Permalink:
- http://dlmf.nist.gov/10.67.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.67(i), §10.67 and Ch.10
10.67.5
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ⓘ
- Symbols:
- : Kelvin function, : Poincaré asymptotic expansion, : the ratio of the circumference of a circle to its diameter, : cosine function, : base of natural logarithm, : nonnegative integer, : real variable, : complex parameter and : polynomial coefficient
- A&S Ref:
- §9.10 (modified)
- Permalink:
- http://dlmf.nist.gov/10.67.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.67(i), §10.67 and Ch.10
10.67.6
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ⓘ
- Symbols:
- : Kelvin function, : Poincaré asymptotic expansion, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : sine function, : nonnegative integer, : real variable, : complex parameter and : polynomial coefficient
- A&S Ref:
- §9.10 (modified)
- Permalink:
- http://dlmf.nist.gov/10.67.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.67(i), §10.67 and Ch.10
10.67.13
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- Symbols:
- : Kelvin function, : Kelvin function, : Poincaré asymptotic expansion, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm and : real variable
- A&S Ref:
- 9.10.31
- Permalink:
- http://dlmf.nist.gov/10.67.E13
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.67(ii), §10.67 and Ch.10
10: 8.11 Asymptotic Approximations and Expansions
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8.11.6
, .
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ⓘ
- Symbols:
- : Poincaré asymptotic expansion, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : incomplete gamma function, : phase, : real variable, : complex variable, : parameter, : nonnegative integer, : small positive constant, : parameter and : coefficients
- Notes:
- See Paris (2002b)
- Referenced by:
- §8.11(iii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)
- Permalink:
- http://dlmf.nist.gov/8.11.E6
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.14):
- Originally this equation was stated for real variables and , . It has been extended to allow for complex variables and , where and .
- See also:
- Annotations for §8.11(iii), §8.11 and Ch.8
8.11.7
, .
…
►The expansion (8.11.7) also applies when is replaced by , and with , .
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ⓘ
- Symbols:
- : Poincaré asymptotic expansion, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : incomplete gamma function, : phase, : real variable, : complex variable, : parameter, : nonnegative integer, : small positive constant, : parameter and : coefficients
- Referenced by:
- §8.11(iii), §8.11(iii), §8.11(iii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)
- Permalink:
- http://dlmf.nist.gov/8.11.E7
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.14):
- Originally this equation was stated for real variables and , . It has been extended to allow for complex variables and , where and .
- See also:
- Annotations for §8.11(iii), §8.11 and Ch.8
8.11.12
.
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ⓘ
- Symbols:
- : Poincaré asymptotic expansion, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : incomplete gamma function, : phase, : complex variable and : small positive constant
- A&S Ref:
- 6.5.35 (Modified form)
- Referenced by:
- §8.11(v), Firefox 3 users should upgrade, Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)
- Permalink:
- http://dlmf.nist.gov/8.11.E12
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.14):
- The sector of validity for the above equation was increased from to .
- See also:
- Annotations for §8.11(v), §8.11 and Ch.8