with complex parameters
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1—10 of 296 matching pages
1: 31.3 Basic Solutions
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31.3.2
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- Symbols:
- : real or complex parameter, : complex parameter, : real or complex parameter and : coefficients
- Permalink:
- http://dlmf.nist.gov/31.3.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.3(i), §31.3 and Ch.31
31.3.5
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- Symbols:
- : Heun functions, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter, : complex parameter, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Referenced by:
- §31.3(iii), §31.7(ii)
- Permalink:
- http://dlmf.nist.gov/31.3.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.3(i), §31.3 and Ch.31
31.3.6
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- Symbols:
- : Heun functions, : complex variable, : real or complex parameter, : real or complex parameter, : complex parameter, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/31.3.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.3(ii), §31.3 and Ch.31
31.3.7
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- Symbols:
- : Heun functions, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter, : complex parameter, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/31.3.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.3(ii), §31.3 and Ch.31
31.3.8
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- Symbols:
- : Heun functions, : complex variable, : real or complex parameter, : real or complex parameter, : complex parameter, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/31.3.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.3(ii), §31.3 and Ch.31
2: 31.1 Special Notation
3: 15.7 Continued Fractions
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15.7.1
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- Symbols:
- or : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter, : coefficient and : coefficient
- Permalink:
- http://dlmf.nist.gov/15.7.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.7 and Ch.15
15.7.3
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- Symbols:
- or : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter, : coefficient and : coefficient
- Permalink:
- http://dlmf.nist.gov/15.7.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.7 and Ch.15
15.7.5
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- Symbols:
- or : Olver’s hypergeometric function, : real variable, : real variable, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/15.7.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.7 and Ch.15
4: 31.12 Confluent Forms of Heun’s Equation
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31.12.1
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- Symbols:
- : derivative of with respect to , : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Referenced by:
- §31.12
- Permalink:
- http://dlmf.nist.gov/31.12.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.12, §31.12 and Ch.31
31.12.2
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- Symbols:
- : derivative of with respect to , : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/31.12.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.12, §31.12 and Ch.31
31.12.3
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- Symbols:
- : derivative of with respect to , : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Referenced by:
- Erratum (V1.0.7) for Equation (31.12.3)
- Permalink:
- http://dlmf.nist.gov/31.12.E3
- Encodings:
- pMML, png, TeX
- Errata (effective with 1.0.7):
-
Originally the sign in front of the second term in this equation was .
The correct sign is .
Reported 2013-10-31 by Henryk Witek
- See also:
- Annotations for §31.12, §31.12 and Ch.31
31.12.4
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- Symbols:
- : derivative of with respect to , : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Referenced by:
- §31.12
- Permalink:
- http://dlmf.nist.gov/31.12.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.12, §31.12 and Ch.31
5: 31.14 General Fuchsian Equation
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31.14.1
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- Symbols:
- : derivative of with respect to , : complex variable, : real or complex parameter, : nonnegative integer, : complex parameter, : real or complex parameter and : number of singularities
- Referenced by:
- §31.15(i)
- Permalink:
- http://dlmf.nist.gov/31.14.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.14(i), §31.14 and Ch.31
31.14.3
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- Symbols:
- : complex variable, : real or complex parameter, : nonnegative integer, : complex parameter, : number of singularities and : solution
- Permalink:
- http://dlmf.nist.gov/31.14.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31
31.14.4
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- Defines:
- : solution (locally)
- Symbols:
- : derivative of with respect to , : complex variable, : real or complex parameter, : nonnegative integer, : complex parameter, : real or complex parameter and : number of singularities
- Permalink:
- http://dlmf.nist.gov/31.14.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31
6: 15.5 Derivatives and Contiguous Functions
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15.5.1
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- Symbols:
- or : Gauss’ hypergeometric function, : derivative of with respect to , : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/15.5.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.5(i), §15.5 and Ch.15
15.5.2
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- Symbols:
- or : Gauss’ hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : derivative of with respect to , : integer, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/15.5.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.5(i), §15.5 and Ch.15
15.5.3
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- Symbols:
- or : Gauss’ hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : derivative of with respect to , : integer, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Referenced by:
- §18.17(iv)
- Permalink:
- http://dlmf.nist.gov/15.5.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.5(i), §15.5 and Ch.15
15.5.4
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- Symbols:
- or : Gauss’ hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : derivative of with respect to , : integer, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Referenced by:
- §18.17(iv)
- Permalink:
- http://dlmf.nist.gov/15.5.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.5(i), §15.5 and Ch.15
15.5.5
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- Symbols:
- or : Gauss’ hypergeometric function, : Pochhammer’s symbol (or shifted factorial), : derivative of with respect to , : integer, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Referenced by:
- §18.17(iv)
- Permalink:
- http://dlmf.nist.gov/15.5.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.5(i), §15.5 and Ch.15
7: 9.14 Incomplete Airy Functions
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►Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter.
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8: 31.4 Solutions Analytic at Two Singularities: Heun Functions
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31.4.1
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- Symbols:
- : Heun functions, : complex variable, : real or complex parameter, : real or complex parameter, : nonnegative integer, : complex parameter, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Referenced by:
- §31.6
- Permalink:
- http://dlmf.nist.gov/31.4.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.4 and Ch.31
31.4.2
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- Symbols:
- : real or complex parameter, : complex parameter, : real or complex parameter, : coefficient, : coefficient and : coefficient
- Referenced by:
- §31.18
- Permalink:
- http://dlmf.nist.gov/31.4.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.4 and Ch.31
31.4.3
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- Symbols:
- : Heun functions, : complex variable, : real or complex parameter, : real or complex parameter, : nonnegative integer, : complex parameter, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Referenced by:
- §31.4, §31.6
- Permalink:
- http://dlmf.nist.gov/31.4.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.4 and Ch.31
9: 31.6 Path-Multiplicative Solutions
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31.6.1
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- Symbols:
- : Heun functions, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter, : nonnegative integer, : complex parameter, : real or complex parameter, : real or complex parameter, : real or complex parameter and : parameters
- Permalink:
- http://dlmf.nist.gov/31.6.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.6 and Ch.31
10: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
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31.5.1
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- Symbols:
- : real or complex parameter, : nonnegative integer, : complex parameter, : coefficient, : coefficient and : coefficient
- Permalink:
- http://dlmf.nist.gov/31.5.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.5 and Ch.31
31.5.2
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- Defines:
- : Heun polynomials
- Symbols:
- : Heun functions, : complex variable, : real or complex parameter, : real or complex parameter, : nonnegative integer, : nonnegative integer, : complex parameter, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/31.5.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.5 and Ch.31