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11: Bibliography C
  • B. C. Carlson (1999) Toward symbolic integration of elliptic integrals. J. Symbolic Comput. 28 (6), pp. 739–753.
  • B. C. Carlson (1998) Elliptic Integrals: Symmetry and Symbolic Integration. In Tricomi’s Ideas and Contemporary Applied Mathematics (Rome/Turin, 1997), Atti dei Convegni Lincei, Vol. 147, pp. 161–181.
  • A. D. Chave (1983) Numerical integration of related Hankel transforms by quadrature and continued fraction expansion. Geophysics 48 (12), pp. 1671–1686.
  • C. W. Clenshaw and A. R. Curtis (1960) A method for numerical integration on an automatic copmputer. Numer. Math. 2 (4), pp. 197–205.
  • H. S. Cohl (2010) Derivatives with respect to the degree and order of associated Legendre functions for | z | > 1 using modified Bessel functions. Integral Transforms Spec. Funct. 21 (7-8), pp. 581–588.
  • 12: 19.25 Relations to Other Functions
    §19.25 Relations to Other Functions
    All terms on the right-hand sides are nonnegative when k 2 0 , 0 k 2 1 , or 1 k 2 c , respectively. … ( F 1 and F D are equivalent to the R -function of 3 and n variables, respectively, but lack full symmetry.)
    13: 4.23 Inverse Trigonometric Functions
    In (4.23.1) and (4.23.2) the integration paths may not pass through either of the points t = ± 1 . …In (4.23.3) the integration path may not intersect ± i . … The principal values (or principal branches) of the inverse sine, cosine, and tangent are obtained by introducing cuts in the z -plane as indicated in Figures 4.23.1(i) and 4.23.1(ii), and requiring the integration paths in (4.23.1)–(4.23.3) not to cross these cuts. …The principal branches are denoted by arcsin z , arccos z , arctan z , respectively. … are respectively
    14: 3.5 Quadrature
    §3.5(iii) Romberg Integration
    Further refinements are achieved by Romberg integration. … For these cases the integration path may need to be deformed; see §3.5(ix). … A second example is provided in Gil et al. (2001), where the method of contour integration is used to evaluate Scorer functions of complex argument (§9.12). … The standard Monte Carlo method samples points uniformly from the integration region to estimate the integral and its error. …
    15: 33.23 Methods of Computation
    The 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. …
    §33.23(iii) Integration of Defining Differential Equations
    When 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). … Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. …
    16: Errata
  • Paragraph Inversion Formula (in §35.2)

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

  • 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.

  • 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 γ ( 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.

  • 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.

  • Subsections 14.5(ii), 14.5(vi)

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

  • 17: 8.21 Generalized Sine and Cosine Integrals
    Furthermore, 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. … (obtained from (5.2.1) by rotation of the integration path) is also needed. … In 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. …
    §8.21(v) Special Values
    When z with | ph z | π - δ ( < π ), …
    18: 1.8 Fourier Series
    Let f ( x ) be an absolutely integrable function of period 2 π , and continuous except at a finite number of points in any bounded interval. …
    §1.8(iii) Integration and Differentiation
    when f ( x ) and g ( x ) are square-integrable and a n , b n and a n , b n are their respective Fourier coefficients. … Suppose that f ( x ) is twice continuously differentiable and f ( x ) and | f ′′ ( x ) | are integrable over ( - , ) . … Suppose also that f ( x ) is integrable on [ 0 , ) and f ( x ) 0 as x . …
    19: 18.2 General Orthogonal Polynomials
    A system (or set) of polynomials { p n ( x ) } , n = 0 , 1 , 2 , , is said to be orthogonal on ( a , b ) with respect to the weight function w ( x ) ( 0 ) if …Here w ( x ) is continuous or piecewise continuous or integrable, and such that 0 < a b x 2 n w ( x ) d x < for all n . … Then a system of polynomials { p n ( x ) } , n = 0 , 1 , 2 , , is said to be orthogonal on X with respect to the weights w x if … The orthogonality relations (18.2.1)–(18.2.3) each determine the polynomials p n ( x ) uniquely up to constant factors, which may be fixed by suitable normalization. … Conversely, if a system of polynomials { p n ( x ) } satisfies (18.2.10) with a n - 1 c n > 0 ( n 1 ), then { p n ( x ) } is orthogonal with respect to some positive measure on (Favard’s theorem). …
    20: 1.10 Functions of a Complex Variable
    and the integration contour is described once in the positive sense. … where N and P are respectively the numbers of zeros and poles, counting multiplicity, of f within C , and Δ C ( ph f ( z ) ) is the change in any continuous branch of ph ( f ( z ) ) as z passes once around C in the positive sense. … each location again being counted with multiplicity equal to that of the corresponding zero or pole. … is analytic in D and its derivatives of all orders can be found by differentiating under the sign of integration. … For each t [ a , b ) , f ( z , t ) is analytic in D ; f ( z , t ) is a continuous function of both variables when z D and t [ a , b ) ; the integral (1.10.18) converges at b , and this convergence is uniform with respect to z in every compact subset S of D . …