Olver’s
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21—30 of 146 matching pages
21: 14.25 Integral Representations
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14.25.2
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- Symbols:
- : gamma function, : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : differential of , : hyperbolic cosine function, : hyperbolic sine function, : integral, : real part, : complex variable, : general order and : general degree
- Referenced by:
- §14.25
- Permalink:
- http://dlmf.nist.gov/14.25.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.25 and Ch.14
22: 8.5 Confluent Hypergeometric Representations
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8.5.2
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- Symbols:
- : Olver’s confluent hypergeometric function, : base of natural logarithm, : incomplete gamma function, : complex variable and : parameter
- Referenced by:
- §8.7
- Permalink:
- http://dlmf.nist.gov/8.5.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.5 and Ch.8
23: 14.22 Graphics
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24: 14.9 Connection Formulas
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14.9.12
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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, : real variable, : general order and : general degree
- A&S Ref:
- 8.2.2 (modified)
- Referenced by:
- §14.20(i), §14.8(ii), §14.8(iii)
- Permalink:
- http://dlmf.nist.gov/14.9.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.9(iii), §14.9 and Ch.14
14.9.14
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- Symbols:
- : Olver’s associated Legendre function, : real variable, : general order and : general degree
- A&S Ref:
- 8.2.6 (modified)
- Referenced by:
- §14.23, §14.9(iv)
- Permalink:
- http://dlmf.nist.gov/14.9.E14
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.9(iii), §14.9 and Ch.14
14.9.15
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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, : sine function, : real variable, : general order and : general degree
- A&S Ref:
- 8.2.5 (modified)
- Permalink:
- http://dlmf.nist.gov/14.9.E15
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.9(iii), §14.9 and Ch.14
14.9.16
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- Symbols:
- : associated Legendre function of the first kind, : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : real variable, : general order and : general degree
- A&S Ref:
- 8.2.7 (modified)
- Referenced by:
- §14.19(v), §14.21(iii), §14.9(iv)
- Permalink:
- http://dlmf.nist.gov/14.9.E16
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.9(iv), §14.9 and Ch.14
14.9.17
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- Symbols:
- : associated Legendre function of the first kind, : Olver’s associated Legendre function, : the ratio of the circumference of a circle to its diameter, : real variable, : general order and : general degree
- Referenced by:
- §14.19(v), §14.9(iv)
- Permalink:
- http://dlmf.nist.gov/14.9.E17
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.9(iv), §14.9 and Ch.14
25: 14.8 Behavior at Singularities
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14.8.9
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- Symbols:
- : order not exceeding, : gamma function, : Euler’s constant, : psi (or digamma) function, : principal branch of logarithm function, : Olver’s associated Legendre function, : real variable and : general degree
- Referenced by:
- Erratum (V1.1.5) for Equation (14.8.9)
- Permalink:
- http://dlmf.nist.gov/14.8.E9
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.1.5):
-
The symbol has been corrected to be .
Suggested 2022-02-09 by Mark Ashbaugh
- See also:
- Annotations for §14.8(ii), §14.8 and Ch.14
14.8.10
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- Symbols:
- : factorial (as in ), : Olver’s associated Legendre function, : real variable and : nonnegative integer
- Referenced by:
- §14.8(ii)
- Permalink:
- http://dlmf.nist.gov/14.8.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.8(ii), §14.8 and Ch.14
14.8.11
, .
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- Symbols:
- : gamma function, : Olver’s associated Legendre function, : asymptotic equality, : real part, : real variable, : general order and : general degree
- Referenced by:
- §14.24
- Permalink:
- http://dlmf.nist.gov/14.8.E11
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.8(ii), §14.8 and Ch.14
14.8.15
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- Symbols:
- : gamma function, : Olver’s associated Legendre function, : asymptotic equality, : the ratio of the circumference of a circle to its diameter, : real variable, : general order and : general degree
- Referenced by:
- §14.8(iii)
- Permalink:
- http://dlmf.nist.gov/14.8.E15
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.8(iii), §14.8 and Ch.14
14.8.16
, .
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- Symbols:
- : gamma function, : Olver’s associated Legendre function, : asymptotic equality, : the ratio of the circumference of a circle to its diameter, : factorial (as in ), : real variable, : nonnegative integer and : general order
- Referenced by:
- §14.21(iii), §14.8(iii)
- Permalink:
- http://dlmf.nist.gov/14.8.E16
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.8(iii), §14.8 and Ch.14
26: 14.5 Special Values
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14.5.9
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- Symbols:
- : principal branch of logarithm function, : Olver’s associated Legendre function and : real variable
- A&S Ref:
- 8.4.2 (modified)
- Permalink:
- http://dlmf.nist.gov/14.5.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.5(ii), §14.5 and Ch.14
14.5.10
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- Symbols:
- : principal branch of logarithm function, : Olver’s associated Legendre function and : real variable
- A&S Ref:
- 8.4.4 (modified)
- Permalink:
- http://dlmf.nist.gov/14.5.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.5(ii), §14.5 and Ch.14
14.5.26
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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, : Legendre’s complete elliptic integral of the second kind, : hyperbolic cosine function, : hyperbolic secant function, : Olver’s associated Legendre function and : variable
- A&S Ref:
- 8.13.7 (in slightly different form)
- Permalink:
- http://dlmf.nist.gov/14.5.E26
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.5(v), §14.5 and Ch.14
14.5.27
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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, : base of natural logarithm, : hyperbolic cosine function, : Olver’s associated Legendre function and : variable
- A&S Ref:
- 8.13.4 (modified)
- Permalink:
- http://dlmf.nist.gov/14.5.E27
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.5(v), §14.5 and Ch.14
14.5.30
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- Symbols:
- : principal branch of logarithm function, : Olver’s associated Legendre function and : real variable
- Referenced by:
- (14.5.30)
- Permalink:
- http://dlmf.nist.gov/14.5.E30
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.7):
- (14.5.30) has been added to this section.
- See also:
- Annotations for §14.5(vi), §14.5 and Ch.14
27: 2 Asymptotic Approximations
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28: 14.6 Integer Order
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14.6.5
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- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : Olver’s associated Legendre function, : derivative of with respect to , : Olver’s associated Legendre function, : real variable, : nonnegative integer and : general degree
- Referenced by:
- §14.21(iii), §14.6(i)
- Permalink:
- http://dlmf.nist.gov/14.6.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §14.6(i), §14.6 and Ch.14
29: 15.9 Relations to Other Functions
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15.9.17
and .
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ⓘ
- Symbols:
- : associated Legendre function of the first kind, : the ratio of the circumference of a circle to its diameter, : phase, or : Olver’s hypergeometric function, : complex variable, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/15.9.E17
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.9(iv), §15.9 and Ch.15
15.9.18
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ⓘ
- Symbols:
- : associated Legendre function of the first kind, : the ratio of the circumference of a circle to its diameter, : phase, or : Olver’s hypergeometric function, : complex variable, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/15.9.E18
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.9(iv), §15.9 and Ch.15
15.9.19
and .
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ⓘ
- Symbols:
- : associated Legendre function of the first kind, : the ratio of the circumference of a circle to its diameter, : phase, or : Olver’s hypergeometric function, : complex variable, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/15.9.E19
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.9(iv), §15.9 and Ch.15
15.9.20
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- Symbols:
- : associated Legendre function of the first kind, : the ratio of the circumference of a circle to its diameter, : phase, or : Olver’s hypergeometric function, : complex variable, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/15.9.E20
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.9(iv), §15.9 and Ch.15
15.9.21
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- Symbols:
- : associated Legendre function of the first kind, : the ratio of the circumference of a circle to its diameter, : phase, or : Olver’s hypergeometric function, : complex variable, : real or complex parameter and : real or complex parameter
- Referenced by:
- §15.9(iv)
- Permalink:
- http://dlmf.nist.gov/15.9.E21
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.9(iv), §15.9 and Ch.15
30: 13.11 Series
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13.11.3
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- Symbols:
- : Bessel function of the first kind, : Olver’s confluent hypergeometric function, : base of natural logarithm, : nonnegative integer, : complex variable and : coefficients
- Source:
- Slater (1960, §3.8)
- A&S Ref:
- 13.3.7
- Referenced by:
- §13.11, Erratum (V1.1.0) for Additions
- Permalink:
- http://dlmf.nist.gov/13.11.E3
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.1.0):
- This equation was added.
- See also:
- Annotations for §13.11 and Ch.13