§14.5 Special Values
Contents
§14.5(i)
- Notes:
- These results may be derived from (14.3.11)–(14.3.14) and also (14.10.4), with replaced by .
- Permalink:
- http://dlmf.nist.gov/14.5.i
14.5.1
- Symbols:
- : gamma function, : Ferrers function of the first kind, : general order and : general degree
- A&S Ref:
- 8.6.1 (modified)
- Referenced by:
- §10.59, §14.30(ii)
- Permalink:
- http://dlmf.nist.gov/14.5.E1
- Encodings:
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14.5.2
- Symbols:
- : gamma function, : Ferrers function of the first kind, : derivative of with respect to , : real variable, : general order and : general degree
- A&S Ref:
- 8.6.3 (modified)
- Permalink:
- http://dlmf.nist.gov/14.5.E2
- Encodings:
- TeX, pMML, png
14.5.3,
- Symbols:
- : gamma function, : Ferrers function of the second kind, : sine function, : general order and : general degree
- A&S Ref:
- 8.6.2
- Permalink:
- http://dlmf.nist.gov/14.5.E3
- Encodings:
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14.5.4.
- Symbols:
- : gamma function, : Ferrers function of the second kind, : cosine function, : derivative of with respect to , : real variable, : general order and : general degree
- A&S Ref:
- 8.6.4
- Permalink:
- http://dlmf.nist.gov/14.5.E4
- Encodings:
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§14.5(ii) ,
14.5.5
- Symbols:
- : Ferrers function of the first kind, : associated Legendre function of the first kind and : real variable
- A&S Ref:
- 8.4.1
- Permalink:
- http://dlmf.nist.gov/14.5.E5
- Encodings:
- TeX, pMML, png
14.5.6
- Symbols:
- : Ferrers function of the first kind, : associated Legendre function of the first kind and : real variable
- A&S Ref:
- 8.4.3
- Permalink:
- http://dlmf.nist.gov/14.5.E6
- Encodings:
- TeX, pMML, png
14.5.7
- Symbols:
- : Ferrers function of the second kind, : principal branch of logarithm function and : real variable
- A&S Ref:
- 8.4.2
- Permalink:
- http://dlmf.nist.gov/14.5.E7
- Encodings:
- TeX, pMML, png
14.5.8
- Symbols:
- : Ferrers function of the second kind, : principal branch of logarithm function and : real variable
- A&S Ref:
- 8.4.4
- Permalink:
- http://dlmf.nist.gov/14.5.E8
- Encodings:
- TeX, pMML, png
14.5.9
- Symbols:
- : Olver’s associated Legendre function, : principal branch of logarithm function and : real variable
- A&S Ref:
- 8.4.2 (modified)
- Permalink:
- http://dlmf.nist.gov/14.5.E9
- Encodings:
- TeX, pMML, png
14.5.10
- Symbols:
- : Olver’s associated Legendre function, : principal branch of logarithm function and : real variable
- A&S Ref:
- 8.4.4 (modified)
- Permalink:
- http://dlmf.nist.gov/14.5.E10
- Encodings:
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§14.5(iii)
- Notes:
- See Erdélyi et al. (1953a, p. 150).
- Keywords:
- associated Legendre functions
- Permalink:
- http://dlmf.nist.gov/14.5.iii
In this subsection and the next two, and .
14.5.11
- Symbols:
- : Ferrers function of the first kind, : cosine function, : sine function, : general degree and : variable
- A&S Ref:
- 8.6.12
- Permalink:
- http://dlmf.nist.gov/14.5.E11
- Encodings:
- TeX, pMML, png
14.5.12
- Symbols:
- : Ferrers function of the first kind, : cosine function, : sine function, : general degree and : variable
- A&S Ref:
- 8.6.14
- Permalink:
- http://dlmf.nist.gov/14.5.E12
- Encodings:
- TeX, pMML, png
14.5.13
- Symbols:
- : Ferrers function of the second kind, : cosine function, : sine function, : general degree and : variable
- A&S Ref:
- 8.6.13
- Permalink:
- http://dlmf.nist.gov/14.5.E13
- Encodings:
- TeX, pMML, png
14.5.14
- Symbols:
- : Ferrers function of the second kind, : cosine function, : sine function, : general degree and : variable
- A&S Ref:
- 8.6.15 (corrected)
- Permalink:
- http://dlmf.nist.gov/14.5.E14
- Encodings:
- TeX, pMML, png
14.5.15
- Symbols:
- : associated Legendre function of the first kind, : hyperbolic cosine function, : hyperbolic sine function, : general degree and : variable
- A&S Ref:
- 8.6.8 (modified)
- Permalink:
- http://dlmf.nist.gov/14.5.E15
- Encodings:
- TeX, pMML, png
14.5.16
- Symbols:
- : associated Legendre function of the first kind, : hyperbolic cosine function, : hyperbolic sine function, : general degree and : variable
- A&S Ref:
- 8.6.9 (modified)
- Permalink:
- http://dlmf.nist.gov/14.5.E16
- Encodings:
- TeX, pMML, png
14.5.17
- Symbols:
- : gamma function, : Olver’s associated Legendre function, : exponential function, : hyperbolic cosine function, : hyperbolic sine function, : general degree and : variable
- A&S Ref:
- 8.6.10 (corrected) 8.6.11 (corrected)
- Permalink:
- http://dlmf.nist.gov/14.5.E17
- Encodings:
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§14.5(iv)
14.5.18
- Symbols:
- : gamma function, : Ferrers function of the first kind, : cosine function, : sine function, : general degree and : variable
- A&S Ref:
- 8.6.17
- Permalink:
- http://dlmf.nist.gov/14.5.E18
- Encodings:
- TeX, pMML, png
14.5.19
- Symbols:
- : gamma function, : associated Legendre function of the first kind, : hyperbolic cosine function, : hyperbolic sine function, : general degree and : variable
- A&S Ref:
- 8.6.16 (modified)
- Permalink:
- http://dlmf.nist.gov/14.5.E19
- Encodings:
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§14.5(v)
- Notes:
- See Magnus et al. (1966, p. 173). These results may be verified by comparing the hypergeometric representations of the Legendre functions and elliptic integrals (§§14.3 and 19.5).
- Keywords:
- Ferrers functions, associated Legendre functions, elliptic integrals
- Referenced by:
- §14.19(i)
- Permalink:
- http://dlmf.nist.gov/14.5.v
In this subsection and denote the complete elliptic integrals of the first and second kinds; see §19.2(ii).
14.5.20
- Symbols:
- : Ferrers function of the first kind, : Legendre’s complete elliptic integral of the first kind, : Legendre’s complete elliptic integral of the second kind, : cosine function, : sine function and : variable
- A&S Ref:
- 8.13.11 (in slightly different form)
- Permalink:
- http://dlmf.nist.gov/14.5.E20
- Encodings:
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14.5.21
- Symbols:
- : Ferrers function of the first kind, : Legendre’s complete elliptic integral of the first kind, : cosine function, : sine function and : variable
- A&S Ref:
- 8.13.9
- Permalink:
- http://dlmf.nist.gov/14.5.E21
- Encodings:
- TeX, pMML, png
14.5.22
- Symbols:
- : Ferrers function of the second kind, : Legendre’s complete elliptic integral of the first kind, : Legendre’s complete elliptic integral of the second kind, : cosine function and : variable
- A&S Ref:
- 8.13.12 (in slightly different form)
- Permalink:
- http://dlmf.nist.gov/14.5.E22
- Encodings:
- TeX, pMML, png
14.5.23
- Symbols:
- : Ferrers function of the second kind, : Legendre’s complete elliptic integral of the first kind, : cosine function and : variable
- A&S Ref:
- 8.13.10 (in slightly different form)
- Permalink:
- http://dlmf.nist.gov/14.5.E23
- Encodings:
- TeX, pMML, png
14.5.24
- Symbols:
- : associated Legendre function of the first kind, : Legendre’s complete elliptic integral of the second kind, : base of exponential function, : hyperbolic cosine function and : variable
- A&S Ref:
- 8.13.6
- Permalink:
- http://dlmf.nist.gov/14.5.E24
- Encodings:
- TeX, pMML, png
14.5.25
- Symbols:
- : associated Legendre function of the first kind, : Legendre’s complete elliptic integral of the first kind, : hyperbolic cosine function, : hyperbolic tangent function and : variable
- A&S Ref:
- 8.13.2
- Permalink:
- http://dlmf.nist.gov/14.5.E25
- Encodings:
- TeX, pMML, png
14.5.26
- Symbols:
- : Olver’s associated Legendre function, : Legendre’s complete elliptic integral of the first kind, : Legendre’s complete elliptic integral of the second kind, : hyperbolic cosine function, : hyperbolic secant function and : variable
- A&S Ref:
- 8.13.7 (in slightly different form)
- Permalink:
- http://dlmf.nist.gov/14.5.E26
- Encodings:
- TeX, pMML, png
14.5.27
- Symbols:
- : Olver’s associated Legendre function, : Legendre’s complete elliptic integral of the first kind, : base of exponential function, : hyperbolic cosine function and : variable
- A&S Ref:
- 8.13.4 (modified)
- Permalink:
- http://dlmf.nist.gov/14.5.E27
- Encodings:
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