for%20ratios
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1—10 of 33 matching pages
1: 9.9 Zeros
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9.9.6
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : th zero of Airy , : nonnegative integer and : expansion
- Source:
- Olver (1954, (A 20))
- Permalink:
- http://dlmf.nist.gov/9.9.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §9.9(iv), §9.9 and Ch.9
9.9.7
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- Symbols:
- : Airy function, : the ratio of the circumference of a circle to its diameter, : th zero of Airy , : nonnegative integer and : expansion
- Source:
- Olver (1954, (A 20))
- Permalink:
- http://dlmf.nist.gov/9.9.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §9.9(iv), §9.9 and Ch.9
9.9.8
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : th zero of Airy , : nonnegative integer and : expansion
- Source:
- Olver (1954, (A 20))
- Permalink:
- http://dlmf.nist.gov/9.9.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §9.9(iv), §9.9 and Ch.9
9.9.9
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- Symbols:
- : Airy function, : the ratio of the circumference of a circle to its diameter, : th zero of Airy , : nonnegative integer and : expansion
- Source:
- Olver (1954, (A 20))
- Permalink:
- http://dlmf.nist.gov/9.9.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §9.9(iv), §9.9 and Ch.9
2: 7.8 Inequalities
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7.8.5
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- Symbols:
- : Mills’ ratio and : real variable
- Referenced by:
- §7.8
- Permalink:
- http://dlmf.nist.gov/7.8.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.8 and Ch.7
3: 20.7 Identities
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20.7.17
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- Symbols:
- : theta function, : the ratio of the circumference of a circle to its diameter, : complex, : lattice parameter and
- Permalink:
- http://dlmf.nist.gov/20.7.E17
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §20.7(vi), §20.7 and Ch.20
20.7.18
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- Symbols:
- : theta function, : the ratio of the circumference of a circle to its diameter, : complex, : lattice parameter and
- Permalink:
- http://dlmf.nist.gov/20.7.E18
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §20.7(vi), §20.7 and Ch.20
§20.7(vii) Derivatives of Ratios of Theta Functions
… ►See Lawden (1989, pp. 19–20). … ►
20.7.34
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- Symbols:
- : theta function, : imaginary unit, : complex and : nome
- Referenced by:
- §20.7(iv), Erratum (V1.0.5) for Chapters 8, 20, 36
- Permalink:
- http://dlmf.nist.gov/20.7.E34
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.5):
- This equation has been added. See Walker (2012).
- See also:
- Annotations for §20.7(ix), §20.7 and Ch.20
4: 22.3 Graphics
5: 36.2 Catastrophes and Canonical Integrals
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36.2.28
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- Symbols:
- : Bessel function of the first kind, : the ratio of the circumference of a circle to its diameter, : complex conjugate, : elliptic umbilic canonical integral function, : exponential function, : imaginary unit and : real parameter
- Referenced by:
- Erratum (V1.0.5) for Chapters 8, 20, 36
- Permalink:
- http://dlmf.nist.gov/36.2.E28
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.5):
- This equation has been added. For the proof see Berry and Howls (2010).
- See also:
- Annotations for §36.2(iv), §36.2 and Ch.36
6: 36.4 Bifurcation Sets
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36.4.11
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : exponential function, : imaginary unit, : real parameter, : real parameter, : integer and : real parameter
- Referenced by:
- §36.7(iii)
- Permalink:
- http://dlmf.nist.gov/36.4.E11
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36
7: Bibliography G
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W. Gautschi (1964a)
Algorithm 222: Incomplete beta function ratios.
Comm. ACM 7 (3), pp. 143–144.
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W. Gautschi (1994)
Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules.
ACM Trans. Math. Software 20 (1), pp. 21–62.
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A. Gil, J. Segura, and N. M. Temme (2014)
Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992)
Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys. 217 (1), pp. 1–20.
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B. N. Gupta (1970)
On Mill’s ratio.
Proc. Cambridge Philos. Soc. 67, pp. 363–364.
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- Notes:
- Variable-precision Algol program. For a Certification see same journal 7 (1964), p. 244.
- External Links:
- ISSN 0001-0782, Document, Certification
- Referenced by:
- §8.28(iv).
- Encodings:
- BibTeX
ⓘ
- Notes:
- For remark see same journal 24(1998), p. 355.
- External Links:
- Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt
- Referenced by:
- 2nd item, §3.5(v), §3.5(v).
- Encodings:
- BibTeX
ⓘ
- Notes:
- Includes Fortran program claiming 12S accuracy
- External Links:
- ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
- Referenced by:
- 3rd item, §10.77(ix).
- Encodings:
- BibTeX
ⓘ
- External Links:
- ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt
- Referenced by:
- §18.38(iii).
- Encodings:
- BibTeX
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- External Links:
- MathReview (C. G. Esseen), ZentralBlatt
- Referenced by:
- §7.8.
- Encodings:
- BibTeX
8: 25.6 Integer Arguments
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25.6.2
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- Symbols:
- : Bernoulli numbers, : Riemann zeta function, : the ratio of the circumference of a circle to its diameter, : factorial (as in ) and : nonnegative integer
- Sources:
- Magnus et al. (1966, p. 19); Apostol (1976, (22), p. 266)
- A&S Ref:
- 23.2.16
- Referenced by:
- (25.11.21)
- Permalink:
- http://dlmf.nist.gov/25.6.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §25.6(i), §25.6 and Ch.25
25.6.3
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- Symbols:
- : Bernoulli numbers, : Riemann zeta function and : nonnegative integer
- Source:
- Apostol (1976, (20), p. 266); with
- A&S Ref:
- 23.2.15 (in slightly different form)
- Permalink:
- http://dlmf.nist.gov/25.6.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §25.6(i), §25.6 and Ch.25
25.6.6
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- Symbols:
- : Bernoulli polynomials, : Riemann zeta function, : the ratio of the circumference of a circle to its diameter, : cotangent function, : differential of , : factorial (as in ), : integral and : nonnegative integer
- Source:
- Nörlund (1924, (81*), p. 66); with
- A&S Ref:
- 23.2.17
- Permalink:
- http://dlmf.nist.gov/25.6.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §25.6(i), §25.6 and Ch.25
25.6.11
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- Symbols:
- : Riemann zeta function, : the ratio of the circumference of a circle to its diameter and : principal branch of logarithm function
- Source:
- Apostol (1985a, p. 223)
- A&S Ref:
- 23.2.13
- Permalink:
- http://dlmf.nist.gov/25.6.E11
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §25.6(ii), §25.6 and Ch.25
25.6.12
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- Symbols:
- : Euler’s constant, : Riemann zeta function, : Stieltjes constants, : the ratio of the circumference of a circle to its diameter and : principal branch of logarithm function
- Source:
- Apostol (1985a, pp. 226, 227, 229)
- Referenced by:
- Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series
- Permalink:
- http://dlmf.nist.gov/25.6.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §25.6(ii), §25.6 and Ch.25
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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- 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
20.10.4
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- Symbols:
- : theta function, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : hyperbolic secant function, : hyperbolic sine function, : imaginary unit, : integral, : parameter, : parameter and : parameter
- Permalink:
- http://dlmf.nist.gov/20.10.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §20.10(ii), §20.10 and Ch.20
20.10.5
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- Symbols:
- : theta function, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : hyperbolic cosecant function, : hyperbolic cosine function, : imaginary unit, : integral, : parameter, : parameter and : parameter
- Permalink:
- http://dlmf.nist.gov/20.10.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §20.10(ii), §20.10 and Ch.20
10: 9.7 Asymptotic Expansions
…
►Lastly, for we define
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9.7.3
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- Defines:
- : function (locally)
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter and : equals by definition
- Source:
- Olver (1997b, (13.02), p. 225)
- Referenced by:
- Erratum (V1.0.17) for Equations (9.7.3), (9.7.4)
- Permalink:
- http://dlmf.nist.gov/9.7.E3
- Encodings:
- pMML, png, TeX
- Modification (effective with 1.0.17):
- Originally the function was presented with argument given by a positive integer . It has now been clarified to be valid for argument given by a positive real number .
- See also:
- Annotations for §9.7(i), §9.7 and Ch.9
9.7.4
►Numerical values of are given in Table 9.7.1 for to 2D.
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- Symbols:
- : asymptotic equality, : the ratio of the circumference of a circle to its diameter and : function
- Source:
- Olver (1997b, p. 225)
- Referenced by:
- Erratum (V1.0.17) for Equations (9.7.3), (9.7.4)
- Permalink:
- http://dlmf.nist.gov/9.7.E4
- Encodings:
- pMML, png, TeX
- Modification (effective with 1.0.17):
- Originally the formula was presented with argument given by a positive integer . It has now been clarified to be valid for argument given by a positive real number .
- See also:
- Annotations for §9.7(i), §9.7 and Ch.9
9.7.5
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- Symbols:
- : Airy function, : Poincaré asymptotic expansion, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : phase, : nonnegative integer, : complex variable, : small positive constant, : change of variable and : expansion coefficient
- Source:
- Olver (1997b, (1.07), pp. 392 and 413)
- Referenced by:
- (9.10.4), (9.10.6), §9.7(iii), §9.7(iv), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iv)
- Permalink:
- http://dlmf.nist.gov/9.7.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §9.7(ii), §9.7 and Ch.9