About the Project
NIST

Heun operator

AdvancedHelp

(0.002 seconds)

6 matching pages

1: 31.10 Integral Equations and Representations
β–Ί
31.10.3 ( π’Ÿ z - π’Ÿ t ) ⁒ 𝒦 = 0 ,
β–Ίwhere π’Ÿ z is Heun’s operator in the variable z : β–Ί
31.10.4 π’Ÿ z = z ⁒ ( z - 1 ) ⁒ ( z - a ) ⁒ ( 2 / ⁑ z 2 ) + ( Ξ³ ⁒ ( z - 1 ) ⁒ ( z - a ) + Ξ΄ ⁒ z ⁒ ( z - a ) + Ο΅ ⁒ z ⁒ ( z - 1 ) ) ⁒ ( / ⁑ z ) + Ξ± ⁒ Ξ² ⁒ z .
β–Ί
31.10.14 ( ( t - z ) ⁒ π’Ÿ s + ( z - s ) ⁒ π’Ÿ t + ( s - t ) ⁒ π’Ÿ z ) ⁒ 𝒦 = 0 ,
2: 31.17 Physical Applications
§31.17 Physical Applications
β–Ί
§31.17(i) Addition of Three Quantum Spins
β–Ί
§31.17(ii) Other Applications
β–ΊFor applications of Heun’s equation and functions in astrophysics see Debosscher (1998) where different spectral problems for Heun’s equation are also considered. …
3: Bibliography D
β–Ί
  • A. Decarreau, M.-Cl. Dumont-Lepage, P. Maroni, A. Robert, and A. Ronveaux (1978a) Formes canoniques des équations confluentes de l’équation de Heun. Ann. Soc. Sci. Bruxelles Sér. I 92 (1-2), pp. 53–78.
  • β–Ί
  • A. Decarreau, P. Maroni, and A. Robert (1978b) Sur les équations confluentes de l’équation de Heun. Ann. Soc. Sci. Bruxelles Sér. I 92 (3), pp. 151–189.
  • β–Ί
  • B. Deconinck and J. N. Kutz (2006) Computing spectra of linear operators using the Floquet-Fourier-Hill method. J. Comput. Phys. 219 (1), pp. 296–321.
  • 4: Bibliography R
    β–Ί
  • S. Ritter (1998) On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.
  • β–Ί
  • A. Ronveaux (Ed.) (1995) Heun’s Differential Equations. The Clarendon Press Oxford University Press, New York.
  • β–Ί
  • J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.
  • 5: Bibliography L
    β–Ί
  • V. LaΔ­ (1994) The two-point connection problem for differential equations of the Heun class. Teoret. Mat. Fiz. 101 (3), pp. 360–368 (Russian).
  • β–Ί
  • L. Lapointe and L. Vinet (1996) Exact operator solution of the Calogero-Sutherland model. Comm. Math. Phys. 178 (2), pp. 425–452.
  • β–Ί
  • W. Lay, K. Bay, and S. Yu. Slavyanov (1998) Asymptotic and numeric study of eigenvalues of the double confluent Heun equation. J. Phys. A 31 (42), pp. 8521–8531.
  • β–Ί
  • W. Lay and S. Yu. Slavyanov (1998) The central two-point connection problem for the Heun class of ODEs. J. Phys. A 31 (18), pp. 4249–4261.
  • β–Ί
  • W. Lay and S. Yu. Slavyanov (1999) Heun’s equation with nearby singularities. Proc. Roy. Soc. London Ser. A 455, pp. 4347–4361.
  • 6: Errata
    β–Ί
  • Equation (3.3.34)

    In the online version, the leading divided difference operators were previously omitted from these formulas, due to programming error.

    Reported by Nico Temme on 2021-06-01

  • β–Ί
  • Equation (2.3.6)
    2.3.6 𝒱 a , b ⁑ ( f ⁒ ( t ) ) = a b | f ⁒ ( t ) | ⁒ d t

    The integrand has been corrected so that the absolute value does not include the differential.

    Reported by Juan Luis Varona on 2021-02-08

  • β–Ί
  • Subsection 1.16(vii)

    Several changes have been made to

    1. (i)

      make consistent use of the Fourier transform notations β„± ⁑ ( f ) , β„± ⁑ ( Ο• ) and β„± ⁑ ( u ) where f is a function of one real variable, Ο• is a test function of n variables associated with tempered distributions, and u is a tempered distribution (see (1.14.1), (1.16.29) and (1.16.35));

    2. (ii)

      introduce the partial differential operator D in (1.16.30);

    3. (iii)

      clarify the definition (1.16.32) of the partial differential operator P ⁒ ( D ) ; and

    4. (iv)

      clarify the use of P ⁒ ( D ) and P ⁒ ( x ) in (1.16.33), (1.16.34), (1.16.36) and (1.16.37).

  • β–Ί
  • Subsections 1.15(vi), 1.15(vii), 2.6(iii)

    A number of changes were made with regard to fractional integrals and derivatives. In §1.15(vi) a reference to Miller and Ross (1993) was added, the fractional integral operator of order Ξ± was more precisely identified as the Riemann-Liouville fractional integral operator of order Ξ± , and a paragraph was added below (1.15.50) to generalize (1.15.47). In §1.15(vii) the sentence defining the fractional derivative was clarified. In §2.6(iii) the identification of the Riemann-Liouville fractional integral operator was made consistent with §1.15(vi).

  • β–Ί
  • Equation (10.17.14)
    10.17.14 | R β„“ ± ⁒ ( Ξ½ , z ) | 2 ⁒ | a β„“ ⁒ ( Ξ½ ) | ⁒ 𝒱 z , ± i ⁒ ⁑ ( t - β„“ ) ⁒ exp ⁑ ( | Ξ½ 2 - 1 4 | ⁒ 𝒱 z , ± i ⁒ ⁑ ( t - 1 ) )

    Originally the factor 𝒱 z , ± i ⁒ ⁑ ( t - 1 ) in the argument to the exponential was written incorrectly as 𝒱 z , ± i ⁒ ⁑ ( t - β„“ ) .

    Reported 2014-09-27 by GergΕ‘ Nemes.