general bases
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11—20 of 165 matching pages
11: Bibliography B
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W. Barrett (1981)
Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V.
Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.
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- External Links:
- ISSN 0080-4614, FullText, MathReview (J. Meixner), ZentralBlatt
- Referenced by:
- §28.8(iv).
- Encodings:
- BibTeX
12: 24.16 Generalizations
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24.16.1
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : factorial (as in ), : generalized Bernoulli polynomials, : integer, : integer, : real or complex and : real or complex
- Permalink:
- http://dlmf.nist.gov/24.16.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24
24.16.2
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : factorial (as in ), : generalized Euler polynomials, : integer, : integer, : real or complex and : real or complex
- Permalink:
- http://dlmf.nist.gov/24.16.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24
24.16.9
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : factorial (as in ), : generalized Bernoulli polynomials, : integer, : real or complex and : real or complex
- Permalink:
- http://dlmf.nist.gov/24.16.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24
24.16.10
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- Symbols:
- : base of natural logarithm, : factorial (as in ), : integer, : real or complex, : Dirichlet character, : conductor and : Generalized Bernoulli numbers
- Permalink:
- http://dlmf.nist.gov/24.16.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.16(ii), §24.16 and Ch.24
13: Philip J. Davis
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►The surface color map can be changed from height-based to phase-based for complex valued functions, and density plots can be generated through strategic scaling.
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14: 7 Error Functions, Dawson’s and Fresnel Integrals
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15: 14.24 Analytic Continuation
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14.24.1
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- Symbols:
- : gamma function, : associated Legendre function of the first kind, : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : cosine function, : base of natural logarithm, : imaginary unit, : sine function, : complex variable, : general order, : general degree and : arbitrary integer
- Referenced by:
- §14.24, §14.24
- Permalink:
- http://dlmf.nist.gov/14.24.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.24 and Ch.14
14.24.2
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- Symbols:
- : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : complex variable, : general order, : general degree and : arbitrary integer
- Referenced by:
- §14.23, §14.24
- Permalink:
- http://dlmf.nist.gov/14.24.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.24 and Ch.14
14.24.3
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- Symbols:
- : associated Legendre function of the first kind, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : complex variable, : general order, : general degree and : arbitrary integer
- Permalink:
- http://dlmf.nist.gov/14.24.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.24 and Ch.14
14.24.4
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- Symbols:
- : gamma function, : associated Legendre function of the first kind, : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : sine function, : complex variable, : general order, : general degree and : arbitrary integer
- Referenced by:
- §14.24
- Permalink:
- http://dlmf.nist.gov/14.24.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.24 and Ch.14
16: 14.23 Values on the Cut
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14.23.1
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- Symbols:
- : Ferrers function of the first kind, : associated Legendre function of the first kind, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : real variable, : general order and : general degree
- Referenced by:
- §14.23
- Permalink:
- http://dlmf.nist.gov/14.23.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.23 and Ch.14
14.23.2
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- Symbols:
- : gamma function, : Ferrers function of the first kind, : Ferrers function of the second kind, : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : real variable, : general order and : general degree
- Referenced by:
- §14.23
- Permalink:
- http://dlmf.nist.gov/14.23.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.23 and Ch.14
14.23.4
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- Symbols:
- : Ferrers function of the first kind, : associated Legendre function of the first kind, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : real variable, : general order and : general degree
- A&S Ref:
- 8.3.2 (corrected)
- Referenced by:
- §14.23, §14.23, §14.3(iv)
- Permalink:
- http://dlmf.nist.gov/14.23.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.23 and Ch.14
14.23.5
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- Symbols:
- : gamma function, : Ferrers function of the second kind, : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : real variable, : general order and : general degree
- A&S Ref:
- 8.3.4 (modified)
- Referenced by:
- §14.23
- Permalink:
- http://dlmf.nist.gov/14.23.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.23 and Ch.14
14.23.6
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- Symbols:
- : gamma function, : Ferrers function of the second kind, : associated Legendre function of the first kind, : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : real variable, : general order and : general degree
- Referenced by:
- §14.23, §14.23
- Permalink:
- http://dlmf.nist.gov/14.23.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.23 and Ch.14
17: 14.15 Uniform Asymptotic Approximations
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14.15.4
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- Symbols:
- : order not exceeding, : gamma function, : Ferrers function of the first kind, : base of natural logarithm, : real variable, : general order, : general degree, , and
- Referenced by:
- §14.15(ii)
- Permalink:
- http://dlmf.nist.gov/14.15.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.15(ii), §14.15 and Ch.14
14.15.9
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- Symbols:
- : order not exceeding, : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : modified Bessel function of the first kind, : real variable, : general order, : general degree, and
- Permalink:
- http://dlmf.nist.gov/14.15.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.15(ii), §14.15 and Ch.14
14.15.20
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- Symbols:
- : base of natural logarithm, : general order, : general degree and
- Permalink:
- http://dlmf.nist.gov/14.15.E20
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.15(iv), §14.15 and Ch.14
18: 5.9 Integral Representations
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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):
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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
19: 2.11 Remainder Terms; Stokes Phenomenon
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2.11.5
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- Symbols:
- : gamma function, : differential of , : base of natural logarithm, : generalized exponential integral and : integral
- Referenced by:
- §2.11(iii)
- Permalink:
- http://dlmf.nist.gov/2.11.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §2.11(ii), §2.11 and Ch.2
2.11.6
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- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : Poincaré asymptotic expansion, : base of natural logarithm and : generalized exponential integral
- Referenced by:
- §2.11(ii), §2.11(iii), §2.11(iii), §2.11(iv), §2.11(iv)
- Permalink:
- http://dlmf.nist.gov/2.11.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §2.11(ii), §2.11 and Ch.2
2.11.7
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- Symbols:
- : gamma function, : Pochhammer’s symbol (or shifted factorial), : Poincaré asymptotic expansion, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : generalized exponential integral and : imaginary unit
- Referenced by:
- §2.11(ii), §2.11(iv), §2.11(iv), §2.11(iv)
- Permalink:
- http://dlmf.nist.gov/2.11.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §2.11(ii), §2.11 and Ch.2
2.11.10
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- Symbols:
- : gamma function, : Pochhammer’s symbol (or shifted factorial), : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : generalized exponential integral and : terminant function
- Referenced by:
- §2.11(iii), §2.11(iv)
- Permalink:
- http://dlmf.nist.gov/2.11.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §2.11(iii), §2.11 and Ch.2
2.11.11
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- Defines:
- : terminant function
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : generalized exponential integral and : integral
- Permalink:
- http://dlmf.nist.gov/2.11.E11
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §2.11(iii), §2.11 and Ch.2
20: 19 Elliptic Integrals
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