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1: 9.14 Incomplete Airy Functions
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. …
2: 1.10 Functions of a Complex Variable
§1.10(viii) Functions Defined by Contour Integrals
3: 2.10 Sums and Sequences
The asymptotic behavior of entire functions defined by Maclaurin series can be approached by converting the sum into a contour integral by use of the residue theorem and applying the methods of §§2.4 and 2.5. …
4: 21.7 Riemann Surfaces
Then the matrix defined by …is a Riemann matrix and it is used to define the corresponding Riemann theta function. … Define the holomorphic differential …Then the prime form on the corresponding compact Riemann surface Γ is defined by … Define the operation …
5: 1.9 Calculus of a Complex Variable
§1.9(iii) Integration
If x ( t ) and y ( t ) are continuous and x ( t ) and y ( t ) are piecewise continuous, then z ( t ) defines a contour. … A simple closed contour is a simple contour, except that z ( a ) = z ( b ) . …
Winding Number
6: Errata
  • Other Changes


    • In Equations (15.6.1)–(15.6.9), the Olver hypergeometric function F ( a , b ; c ; z ) , previously omitted from the left-hand sides to make the formulas more concise, has been added. In Equations (15.6.1)–(15.6.5), (15.6.7)–(15.6.9), the constraint | ph ( 1 - z ) | < π has been added. In (15.6.6), the constraint | ph ( - z ) | < π has been added. In Section 15.6 Integral Representations, the sentence immediately following (15.6.9), “These representations are valid when | ph ( 1 - z ) | < π , except (15.6.6) which holds for | ph ( - z ) | < π .”, has been removed.

    • In Subsection 25.2(ii) Other Infinite Series, it is now mentioned that (25.2.5), defines the Stieltjes constants γ n . Consequently, γ n in (25.2.4), (25.6.12) are now identified as the Stieltjes constants.

    • In (25.11.36) we have emphasized the link with the Dirichlet L -function, and used the fact that χ ( k ) = 0 . A sentence just below (25.11.36) was added indicating that one should make a comparison with (25.15.1) and (25.15.3).

    • Additional keywords are being added to formulas (an ongoing project); these are visible in the associated ‘info boxes’ linked to the icons to the right of each formula, and provide better search capabilities.

  • Changes


    • There have been extensive changes in the notation used for the integral transforms defined in §1.14. These changes are applied throughout the DLMF. The following table summarizes the changes.

      Transform New Abbreviated Old
      Notation Notation Notation
      Fourier ( f ) ( x ) f ( x )
      Fourier Cosine c ( f ) ( x ) c f ( x )
      Fourier Sine s ( f ) ( x ) s f ( x )
      Laplace ( f ) ( s ) f ( s ) ( f ( t ) ; s )
      Mellin ( f ) ( s ) f ( s ) ( f ; s )
      Hilbert ( f ) ( s ) f ( s ) ( f ; s )
      Stieltjes 𝒮 ( f ) ( s ) 𝒮 f ( s ) 𝒮 ( f ; s )

      Previously, for the Fourier, Fourier cosine and Fourier sine transforms, either temporary local notations were used or the Fourier integrals were written out explicitly.

    • Several changes have been made in §1.16(vii) to

      1. (i)

        make consistent use of the Fourier transform notations ( f ) , ( ϕ ) and ( u ) where f is a function of one real variable, ϕ 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).

    • An entire new Subsection 1.16(viii) Fourier Transforms of Special Distributions, was contributed by Roderick Wong.

    • The validity constraint | ph z | < 1 6 π was added to (9.5.6). Additionally, specific source citations are now given in the metadata for all equations in Chapter 9 Airy and Related Functions.

    • The relation between Clebsch-Gordan and 3 j symbols was clarified, and the sign of m 3 was changed for readability. The reference Condon and Shortley (1935) for the Clebsch-Gordan coefficients was replaced by Edmonds (1974) and Rotenberg et al. (1959) and the references for 3 j , 6 j , 9 j symbols were made more precise in §34.1.

    • The website’s icons and graphical decorations were upgraded to use SVG, and additional icons and mouse-cursors were employed to improve usability of the interactive figures.

  • Figures 36.3.18, 36.3.19, 36.3.20, 36.3.21


    The scaling error reported on 2016-09-12 by Dan Piponi also applied to contour and density plots for the phase of the hyperbolic umbilic canonical integrals. Scales were corrected in all figures. The interval - 8.4 x - y 2 8.4 was replaced by - 12.0 x - y 2 12.0 and - 12.7 x + y 2 4.2 replaced by - 18.0 x + y 2 6.0 . All plots and interactive visualizations were regenerated to improve image quality.

    See accompanying text See accompanying text
    (a) Contour plot. (b) Density plot.

    Figure 36.3.18: Phase of hyperbolic umbilic canonical integral ph Ψ ( H ) ( x , y , 0 ) .

    See accompanying text See accompanying text
    (a) Contour plot. (b) Density plot.

    Figure 36.3.19: Phase of hyperbolic umbilic canonical integral ph Ψ ( H ) ( x , y , 1 ) .

    See accompanying text See accompanying text
    (a) Contour plot. (b) Density plot.

    Figure 36.3.20: Phase of hyperbolic umbilic canonical integral ph Ψ ( H ) ( x , y , 2 ) .

    See accompanying text See accompanying text
    (a) Contour plot. (b) Density plot.

    Figure 36.3.21: Phase of hyperbolic umbilic canonical integral ph Ψ ( H ) ( x , y , 3 ) .

    Reported 2016-09-28.

  • Other Changes


    • 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 α was more precisely identified as the Riemann-Liouville fractional integral operator of order α , 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).

    • Changes to §8.18(ii)–§8.11(v): A sentence was added in §8.18(ii) to refer to Nemes and Olde Daalhuis (2016). Originally §8.11(iii) was applicable for real variables a and x = λ a . It has been extended to allow for complex variables a and z = λ a (and we have replaced x with z in the subsection heading and in Equations (8.11.6) and (8.11.7)). Also, we have added two paragraphs after (8.11.9) to replace the original paragraph that appeared there. Furthermore, the interval of validity of (8.11.6) was increased from 0 < λ < 1 to the sector 0 < λ < 1 , | ph a | π 2 - δ , and the interval of validity of (8.11.7) was increased from λ > 1 to the sector λ > 1 , | ph a | 3 π 2 - δ . A paragraph with reference to Nemes (2016) has been added in §8.11(v), and the sector of validity for (8.11.12) was increased from | ph z | π - δ to | ph z | 2 π - δ . Two new Subsections 13.6(vii), 13.18(vi), both entitled Coulomb Functions, were added to note the relationship of the Kummer and Whittaker functions to various forms of the Coulomb functions. A sentence was added in both §13.10(vi) and §13.23(v) noting that certain generalized orthogonality can be expressed in terms of Kummer functions.

    • Four of the terms in (14.15.23) were rewritten for improved clarity.

    • In §15.6 it was noted that (15.6.8) can be rewritten as a fractional integral.

    • In applying changes in Version 1.0.12 to (16.15.3), an editing error was made; it has been corrected.

    • In §34.1, the reference for Clebsch-Gordan coefficients, Condon and Shortley (1935), was replaced by Edmonds (1974) and Rotenberg et al. (1959). The references for 3 j , 6 j , 9 j symbols were made more precise.

    • Images in Figures 36.3.1, 36.3.2, 36.3.3, 36.3.4, 36.3.5, 36.3.6, 36.3.7, 36.3.8 and Figures 36.3.13, 36.3.14, 36.3.15, 36.3.16, 36.3.17 were resized for consistency.

    • Meta.Numerics (website) was added to the Software Index.

  • Equation (22.19.6)

    22.19.6
    x ( t ) = cn ( t 1 + 2 η , k )

    Originally the term 1 + 2 η was given incorrectly as 1 + η in this equation and in the line above. Additionally, for improved clarity, the modulus k = 1 / 2 + η - 1 has been defined in the line above.

    Reported 2014-05-02 by Svante Janson.

  • 7: 9.13 Generalized Airy Functions
    For properties of the zeros of the functions defined in this subsection see Laforgia and Muldoon (1988) and references given therein. …
    §9.13(ii) Generalizations from Integral Representations
    Reid (1972) and Drazin and Reid (1981, Appendix) introduce the following contour integrals in constructing approximate solutions to the Orr--Sommerfeld equation for fluid flow: … Further properties of these functions, and also of similar contour integrals containing an additional factor ( ln t ) q , q = 1 , 2 , , in the integrand, are derived in Reid (1972), Drazin and Reid (1981, Appendix), and Baldwin (1985). …
    8: 9.17 Methods of Computation
    A comprehensive and powerful approach is to integrate the defining differential equation (9.2.1) by direct numerical methods. …
    §9.17(iii) Integral Representations
    Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing Ai ( z ) in the complex plane, once values of this function can be generated on the positive real axis. … In the first method the integration path for the contour integral (9.5.4) is deformed to coincide with paths of steepest descent (§2.4(iv)). …The second method is to apply generalized Gauss–Laguerre quadrature (§3.5(v)) to the integral (9.5.8). …
    9: 3.3 Interpolation
    where C is a simple closed contour in D described in the positive rotational sense and enclosing the points z , z 1 , z 2 , , z n . … and A k n are the Lagrangian interpolation coefficients defined by … Let c n be defined by … where ω n + 1 ( ζ ) is given by (3.3.3), and C is a simple closed contour in D described in the positive rotational sense and enclosing z 0 , z 1 , , z n . … For interpolation of a bounded function f on the cardinal function of f is defined by …
    10: 15.9 Relations to Other Functions
    The Jacobi transform is defined as
    15.9.12 f ~ ( λ ) = 0 f ( t ) ϕ λ ( α , β ) ( t ) ( 2 sinh t ) 2 α + 1 ( 2 cosh t ) 2 β + 1 d t ,
    15.9.13 f ( t ) = 1 2 π i - i i f ~ ( i λ ) Φ i λ ( α , β ) ( t ) Γ ( 1 2 ( α + β + 1 + λ ) ) Γ ( 1 2 ( α - β + 1 + λ ) ) Γ ( α + 1 ) Γ ( λ ) 2 α + β + 1 - λ d λ ,
    where the contour of integration is located to the right of the poles of the gamma functions in the integrand, and … It is defined by: …