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21: 2.6 Distributional Methods
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2.6.46 𝐼 ΞΌ f ⁑ ( x ) = s = 0 n 1 ( 1 ) s ⁒ a s s ! ⁒ Ξ“ ⁑ ( ΞΌ + 1 ) ⁒ d s + 1 d x s + 1 ⁑ ( x ΞΌ ⁒ ( ln ⁑ x Ξ³ ψ ⁑ ( ΞΌ + 1 ) ) ) s = 1 n d s Ξ“ ⁑ ( ΞΌ s + 1 ) ⁒ x ΞΌ s + 1 x n ⁒ Ξ΄ n ⁑ ( x ) ,
22: 33.23 Methods of Computation
β–ΊThe power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii ρ and r , respectively, and may be used to compute the regular and irregular solutions. … β–Ί
§33.23(iii) Integration of Defining Differential Equations
β–ΊWhen numerical values of the Coulomb functions are available for some radii, their values for other radii may be obtained by direct numerical integration of equations (33.2.1) or (33.14.1), provided that the integration is carried out in a stable direction (§3.7). …On the other hand, the irregular solutions of §§33.2(iii) and 33.14(iii) need to be integrated in the direction of decreasing radii beginning, for example, with values obtained from asymptotic expansions (§§33.11 and 33.21). … β–ΊCurtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. …
23: 18.17 Integrals
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18.17.34_5 0 e x ⁒ z ⁒ L m ( Ξ± ) ⁑ ( x ) ⁒ L n ( Ξ± ) ⁑ ( x ) ⁒ e x ⁒ x Ξ± ⁒ d x = Ξ“ ⁑ ( Ξ± + m + 1 ) ⁒ Ξ“ ⁑ ( Ξ± + n + 1 ) Ξ“ ⁑ ( Ξ± + 1 ) ⁒ m ! ⁒ n ! ⁒ z m + n ( z + 1 ) Ξ± + m + n + 1 ⁒ F 1 2 ⁑ ( m , n Ξ± + 1 ; z 2 ) , ⁑ z > 1 .
24: Errata
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  • Paragraph Inversion Formula (in §35.2)

    The wording was changed to make the integration variable more apparent.

  • β–Ί
  • Subsection 19.25(vi)

    The Weierstrass lattice roots e j , were linked inadvertently as the base of the natural logarithm. In order to resolve this inconsistency, the lattice roots e j ⁑ , and lattice invariants g 2 ⁑ , g 3 ⁑ , now link to their respective definitions (see §§23.2(i), 23.3(i)).

    Reported by Felix Ospald.

  • β–Ί
  • Paragraph Mellin–Barnes Integrals (in §8.6(ii))

    The descriptions for the paths of integration of the Mellin-Barnes integrals (8.6.10)–(8.6.12) have been updated. The description for (8.6.11) now states that the path of integration is to the right of all poles. Previously it stated incorrectly that the path of integration had to separate the poles of the gamma function from the pole at s = 0 . The paths of integration for (8.6.10) and (8.6.12) have been clarified. In the case of (8.6.10), it separates the poles of the gamma function from the pole at s = a for Ξ³ ⁑ ( a , z ) . In the case of (8.6.12), it separates the poles of the gamma function from the poles at s = 0 , 1 , 2 , .

    Reported 2017-07-10 by Kurt Fischer.

  • β–Ί
  • Equations (10.15.1), (10.38.1)

    These equations have been generalized to include the additional cases of J ν ⁑ ( z ) / ν , I ν ⁑ ( z ) / ν , respectively.

  • β–Ί
  • Subsections 14.5(ii), 14.5(vi)

    The titles have been changed to ΞΌ = 0 , Ξ½ = 0 , 1 , and Addendum to §14.5(ii) ΞΌ = 0 , Ξ½ = 2 , respectively, in order to be more descriptive of their contents.

  • 25: 31.9 Orthogonality
    β–ΊThe integration path begins at z = ΞΆ , encircles z = 1 once in the positive sense, followed by z = 0 once in the positive sense, and so on, returning finally to z = ΞΆ . The integration path is called a Pochhammer double-loop contour (compare Figure 5.12.3). … β–Ί
    31.9.3 ΞΈ m = ( 1 e 2 ⁒ Ο€ ⁒ i ⁒ Ξ³ ) ⁒ ( 1 e 2 ⁒ Ο€ ⁒ i ⁒ Ξ΄ ) ⁒ ΞΆ Ξ³ ⁒ ( 1 ΞΆ ) Ξ΄ ⁒ ( ΞΆ a ) Ο΅ ⁒ f 0 ⁑ ( q , ΞΆ ) f 1 ⁑ ( q , ΞΆ ) ⁒ q ⁑ 𝒲 ⁑ { f 0 ⁑ ( q , ΞΆ ) , f 1 ⁑ ( q , ΞΆ ) } | q = q m ,
    β–Ί
    31.9.5 β„’ 1 β„’ 2 ρ ⁑ ( s , t ) ⁒ w 1 ⁑ ( s ) ⁒ w 1 ⁑ ( t ) ⁒ w 2 ⁑ ( s ) ⁒ w 2 ⁑ ( t ) ⁒ d s ⁒ d t = 0 , | n 1 n 2 | + | m 1 m 2 | 0 ,
    β–Ίand the integration paths β„’ 1 , β„’ 2 are Pochhammer double-loop contours encircling distinct pairs of singularities { 0 , 1 } , { 0 , a } , { 1 , a } . …
    26: 18.40 Methods of Computation
    β–Ί
    A numerical approach to the recursion coefficients and quadrature abscissas and weights
    β–ΊHaving now directly connected computation of the quadrature abscissas and weights to the moments, what follows uses these for a Stieltjes–Perron inversion to regain w ⁑ ( x ) . … β–ΊResults of low ( 2 to 3 decimal digits) precision for w ⁑ ( x ) are easily obtained for N 10 to 20 . … β–ΊInterpolation of the midpoints of the jumps followed by differentiation with respect to x yields a Stieltjes–Perron inversion to obtain w RCP ⁑ ( x ) to a precision of 4 decimal digits for N = 120 . … β–ΊFurther, exponential convergence in N , via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate w ⁑ ( x ) for these OP systems on x [ 1 , 1 ] and ( , ) respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …
    27: 8.21 Generalized Sine and Cosine Integrals
    β–ΊFurthermore, si ⁑ ( a , z ) and ci ⁑ ( a , z ) are entire functions of a , and Si ⁑ ( a , z ) and Ci ⁑ ( a , z ) are meromorphic functions of a with simple poles at a = 1 , 3 , 5 , and a = 0 , 2 , 4 , , respectively. … β–Ί(obtained from (5.2.1) by rotation of the integration path) is also needed. … β–ΊIn these representations the integration paths do not cross the negative real axis, and in the case of (8.21.4) and (8.21.5) the paths also exclude the origin. … β–Ί
    §8.21(v) Special Values
    β–ΊWhen z with | ph ⁑ z | Ο€ Ξ΄ ( < Ο€ ), …
    28: 18.39 Applications in the Physical Sciences
    β–ΊWhile non-normalizable continuum, or scattering, states are mentioned, with appropriate references in what follows, focus is on the L 2 eigenfunctions corresponding to the point, or discrete, spectrum, and representing bound rather than scattering states, these former being expressed in terms of OP’s or EOP’s. … β–ΊKuijlaars 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. … β–ΊOrthogonality and normalization of eigenfunctions of this form is respect to the measure r 2 ⁒ d r ⁒ sin ⁑ ΞΈ ⁒ d ΞΈ ⁒ d Ο• . … β–Ίwith an infinite set of orthonormal L 2 eigenfunctions … β–ΊFor either sign of Z , and s chosen such that n + l + 1 + ( 2 ⁒ Z / s ) > 0 , n = 0 , 1 , 2 , , truncation of the basis to N terms, with x i N [ 1 , 1 ] , the discrete eigenvectors are the orthonormal L 2 functions …
    29: 1.14 Integral Transforms
    β–ΊIf f ⁑ ( t ) is absolutely integrable on ( , ) , then F ⁑ ( x ) is continuous, F ⁑ ( x ) 0 as x ± , and … β–ΊIn many applications f ⁑ ( t ) is absolutely integrable and f ⁑ ( t ) is continuous on ( , ) . … β–ΊThe Fourier cosine transform and Fourier sine transform are defined respectively by … β–Ί
    Differentiation and Integration
    β–ΊNote: If f ⁑ ( x ) is continuous and Ξ± and Ξ² are real numbers such that f ⁑ ( x ) = O ⁑ ( x Ξ± ) as x 0 + and f ⁑ ( x ) = O ⁑ ( x Ξ² ) as x , then x Οƒ 1 ⁒ f ⁑ ( x ) is integrable on ( 0 , ) for all Οƒ ( Ξ± , Ξ² ) . …
    30: 1.5 Calculus of Two or More Variables
    β–ΊSufficient conditions for the limit to exist are that f ⁑ ( x , y ) is continuous, or piecewise continuous, on R . … β–Ί
    Change of Order of Integration
    β–ΊIn the cases (1.5.30) and (1.5.33) they are defined by taking limits in the repeated integrals (1.5.32) and (1.5.34) in an analogous manner to (1.4.22)–(1.4.23). … β–ΊA more general concept of integrability (both finite and infinite) for functions on domains in ℝ n is Lebesgue integrability. … β–ΊAgain the mapping is one-to-one except perhaps for a set of points of volume zero. …