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1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
►►Let T be a linear operator on V with dense domain 𝒟 ⁢ ( T ) and with range ℛ ⁢ ( T ) = { T ⁢ v ∣ v 𝒟 ⁢ ( T ) } . …
2: 2.5 Mellin Transform Methods
►The domain of analyticity of ℳ ⁡ f ⁡ ( z ) is usually an infinite strip a < ⁡ z < b parallel to the imaginary axis. … ►
Table 2.5.1: Domains of convergence for Mellin transforms.
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Transform Domain of Convergence
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►Next from Table 2.5.1 we observe that the integrals for the transform pair ℳ ⁡ f j ⁡ ( 1 z ) and ℳ ⁡ h k ⁡ ( z ) are absolutely convergent in the domain D j ⁢ k specified in Table 2.5.2, and these domains are nonempty as a consequence of (2.5.19) and (2.5.20). … ►From Table 2.5.2, we see that each G j ⁢ k ⁡ ( z ) is analytic in the domain D j ⁢ k . … ►Put x = 1 / ζ and break the integration range at t = 1 , as in (2.5.23) and (2.5.24). …
3: Bibliography G
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  • W. Gautschi (2016) Algorithm 957: evaluation of the repeated integral of the coerror function by half-range Gauss-Hermite quadrature. ACM Trans. Math. Softw. 42 (1), pp. 9:1–9:10.
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  • A. Gil, J. Segura, and N. M. Temme (2000) Computing toroidal functions for wide ranges of the parameters. J. Comput. Phys. 161 (1), pp. 204–217.
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  • S. G. Gindikin (1964) Analysis in homogeneous domains. Uspehi Mat. Nauk 19 (4 (118)), pp. 3–92 (Russian).
  • 4: Bibliography F
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  • J. Faraut (1982) Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un. J. Funct. Anal. 49 (2), pp. 230–268.
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  • T. Fort (1948) Finite Differences and Difference Equations in the Real Domain. Clarendon Press, Oxford.
  • 5: 2.8 Differential Equations with a Parameter
    ►in which ξ ranges over a bounded or unbounded interval or domain 𝚫 , and ψ ⁡ ( ξ ) is C or analytic on 𝚫 . …
    6: 2.4 Contour Integrals
    ►Assume that p ⁡ ( t ) and q ⁡ ( t ) are analytic on an open domain 𝐓 that contains 𝒫 , with the possible exceptions of t = a and t = b . … ►
  • (b)

    z ranges along a ray or over an annular sector θ 1 θ θ 2 , | z | Z , where θ = ph ⁡ z , θ 2 θ 1 < π , and Z > 0 . I ⁡ ( z ) converges at b absolutely and uniformly with respect to z .

  • 7: 18.39 Applications in the Physical Sciences
    ►If Ψ ⁢ ( x , t = 0 ) = χ ⁢ ( x ) is an arbitrary unit normalized function in the domain of ℋ then, by self-adjointness, … ►Kuijlaars and Milson (2015, §1) refer to these, in this case complex zeros, as exceptional, as opposed to regular, zeros of the EOP’s, these latter belonging to the (real) orthogonality integration range. … ►Shizgal (2015, Chapter 2), contains a broad-ranged discussion of methods and applications for these, and other, non-classical weight functions. …
    8: Errata
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  • Section 14.30

    In regard to the definition of the spherical harmonics Y l , m , the domain of the integer m originally written as 0 m l has been replaced with the more general | m | l . Because of this change, in the sentence just below (14.30.2), “tesseral for m < l and sectorial for m = l ” has been replaced with “tesseral for | m | < l and sectorial for | m | = l ”. Furthermore, in (14.30.4), m has been replaced with | m | .

    Reported by Ching-Li Chai on 2019-10-05

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  • Equation (8.18.3)
    8.18.3 I x ⁡ ( a , b ) = Γ ⁡ ( a + b ) Γ ⁡ ( a ) ⁢ ( k = 0 n 1 d k ⁢ F k + O ⁡ ( a n ) ⁢ F 0 )

    The range of x was extended to include 1 . Previously this equation appeared without the order estimate as I x ⁡ ( a , b ) Γ ⁡ ( a + b ) Γ ⁡ ( a ) ⁢ k = 0 d k ⁢ F k .

    Reported 2016-08-30 by Xinrong Ma.

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  • Paragraph Case III: V ⁡ ( x ) = 𝟏 𝟐 ⁢ x 𝟐 + 𝟏 𝟒 ⁢ β ⁢ x 𝟒 (in §22.19(ii))

    Two corrections have been made in this paragraph. First, the correct range of the initial displacement a is 1 / β | a | < 2 / β . Previously it was 1 / β | a | 2 / β . Second, the correct period of the oscillations is 2 ⁢ K ⁡ ( k ) / η . Previously it was given incorrectly as 4 ⁢ K ⁡ ( k ) / η .

    Reported 2014-05-02 by Svante Janson.