About the Project
NIST

hyperbolic series for squares

AdvancedHelp

(0.001 seconds)

7 matching pages

1: 22.11 Fourier and Hyperbolic Series
A related hyperbolic series is …
2: 22.16 Related Functions
Fourier Series
See Figure 22.16.2. …
22.16.26 ( x , k ) = - 0 x ( cs 2 ( t , k ) - t - 2 ) d t + x - 1 - cn ( x , k ) ds ( x , k ) .
3: 4.38 Inverse Hyperbolic Functions: Further Properties
§4.38 Inverse Hyperbolic Functions: Further Properties
§4.38(i) Power Series
§4.38(ii) Derivatives
In the following equations square roots have their principal values. … All square roots have either possible value.
4: Bibliography S
  • F. W. Schäfke and A. Finsterer (1990) On Lindelöf’s error bound for Stirling’s series. J. Reine Angew. Math. 404, pp. 135–139.
  • I. J. Schwatt (1962) An Introduction to the Operations with Series. 2nd edition, Chelsea Publishing Co., New York.
  • T. C. Scott, G. Fee, and J. Grotendorst (2013) Asymptotic series of generalized Lambert W function. ACM Commun. Comput. Algebra 47 (3), pp. 75–83.
  • B. L. Shea (1988) Algorithm AS 239. Chi-squared and incomplete gamma integral. Appl. Statist. 37 (3), pp. 466–473.
  • S. K. Suslov (2003) An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.
  • 5: 4.45 Methods of Computation
    Then we take square roots repeatedly until | y | is sufficiently small, where …
    Hyperbolic and Inverse Hyperbolic Functions
    The hyperbolic functions can be computed directly from the definitions (4.28.1)–(4.28.7). …For arccsch , arcsech , and arccoth we have (4.37.7)–(4.37.9). … Similarly for the hyperbolic and inverse hyperbolic functions; compare (4.28.1)–(4.28.7), §4.37(iv), and (4.37.7)–(4.37.9). …
    6: 19.36 Methods of Computation
    The reductions in §19.29(i) represent x , y , z as squares, for example x = U 12 2 in (19.29.4). … The incomplete integrals R F ( x , y , z ) and R G ( x , y , z ) can be computed by successive transformations in which two of the three variables converge quadratically to a common value and the integrals reduce to R C , accompanied by two quadratically convergent series in the case of R G ; compare Carlson (1965, §§5,6). … The step from n to n + 1 is an ascending Landen transformation if θ = 1 (leading ultimately to a hyperbolic case of R C ) or a descending Gauss transformation if θ = - 1 (leading to a circular case of R C ). … For series expansions of Legendre’s integrals see §19.5. Faster convergence of power series for K ( k ) and E ( k ) can be achieved by using (19.5.1) and (19.5.2) in the right-hand sides of (19.8.12). …
    7: Errata
  • Figures 36.3.9, 36.3.10, 36.3.11, 36.3.12


    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) Density plot. (b) 3D plot.

    Figure 36.3.9: Modulus of hyperbolic umbilic canonical integral function | Ψ ( H ) ( x , y , 0 ) | .

    See accompanying text See accompanying text
    (a) Density plot. (b) 3D plot.

    Figure 36.3.10: Modulus of hyperbolic umbilic canonical integral function | Ψ ( H ) ( x , y , 1 ) | .

    See accompanying text See accompanying text
    (a) Density plot. (b) 3D plot.

    Figure 36.3.11: Modulus of hyperbolic umbilic canonical integral function | Ψ ( H ) ( x , y , 2 ) | .

    See accompanying text See accompanying text
    (a) Density plot. (b) 3D plot.

    Figure 36.3.12: Modulus of hyperbolic umbilic canonical integral function | Ψ ( H ) ( x , y , 3 ) | .

    Reported 2016-09-12 by Dan Piponi.

  • 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.

  • Section 4.43


    The first paragraph has been rewritten to correct reported errors. The new version is reproduced here.

    Let p ( 0 ) and q be real constants and

    4.43.1
    A = ( - 4 3 p ) 1 / 2 ,
    B = ( 4 3 p ) 1 / 2 .

    The roots of

    4.43.2 z 3 + p z + q = 0

    are:

    1. (a)

      A sin a , A sin ( a + 2 3 π ) , and A sin ( a + 4 3 π ) , with sin ( 3 a ) = 4 q / A 3 , when 4 p 3 + 27 q 2 0 .

    2. (b)

      A cosh a , A cosh ( a + 2 3 π i ) , and A cosh ( a + 4 3 π i ) , with cosh ( 3 a ) = - 4 q / A 3 , when p < 0 , q < 0 , and 4 p 3 + 27 q 2 > 0 .

    3. (c)

      B sinh a , B sinh ( a + 2 3 π i ) , and B sinh ( a + 4 3 π i ) , with sinh ( 3 a ) = - 4 q / B 3 , when p > 0 .

    Note that in Case (a) all the roots are real, whereas in Cases (b) and (c) there is one real root and a conjugate pair of complex roots. See also §1.11(iii).

    Reported 2014-10-31 by Masataka Urago.

  • Table 3.5.21


    The correct corner coordinates for the 9-point square, given on the last line of this table, are ( ± 3 5 h , ± 3 5 h ) . Originally they were given incorrectly as ( ± 3 5 h , 0 ) , ( ± 3 5 h , 0 ) .

    Diagram ( x j , y j ) w j R
    \begin{picture}(2.4,3.0)(-1.2,-1.55)\put(0.0,0.0){\line(1,0){0.05}}\put(0.1,0.% 0){\line(1,0){0.05}}\put(0.2,0.0){\line(1,0){0.05}}\put(0.3,0.0){\line(1,0){0.% 05}}\put(0.4,0.0){\line(1,0){0.05}}\put(0.5,0.0){\line(1,0){0.05}}\put(0.6,0.0% ){\line(1,0){0.05}}\put(0.7,0.0){\line(1,0){0.05}}\put(0.8,0.0){\line(1,0){0.0% 5}}\put(0.9,0.0){\line(1,0){0.05}} \put(0.0,0.0){\line(0,1){0.05}}\put(0.0,0.1){\line(0,1){0.05}}\put(0.0,0.2){% \line(0,1){0.05}}\put(0.0,0.3){\line(0,1){0.05}}\put(0.0,0.4){\line(0,1){0.05}% }\put(0.0,0.5){\line(0,1){0.05}}\put(0.0,0.6){\line(0,1){0.05}}\put(0.0,0.7){% \line(0,1){0.05}}\put(0.0,0.8){\line(0,1){0.05}}\put(0.0,0.9){\line(0,1){0.05}% } \put(0.0,0.0){\line(-1,0){0.05}}\put(-0.1,0.0){\line(-1,0){0.05}}\put(-0.2,0.0% ){\line(-1,0){0.05}}\put(-0.3,0.0){\line(-1,0){0.05}}\put(-0.4,0.0){\line(-1,0% ){0.05}}\put(-0.5,0.0){\line(-1,0){0.05}}\put(-0.6,0.0){\line(-1,0){0.05}}\put% (-0.7,0.0){\line(-1,0){0.05}}\put(-0.8,0.0){\line(-1,0){0.05}}\put(-0.9,0.0){% \line(-1,0){0.05}} \put(0.0,0.0){\line(0,-1){0.05}}\put(0.0,-0.1){\line(0,-1){0.05}}\put(0.0,-0.2% ){\line(0,-1){0.05}}\put(0.0,-0.3){\line(0,-1){0.05}}\put(0.0,-0.4){\line(0,-1% ){0.05}}\put(0.0,-0.5){\line(0,-1){0.05}}\put(0.0,-0.6){\line(0,-1){0.05}}\put% (0.0,-0.7){\line(0,-1){0.05}}\put(0.0,-0.8){\line(0,-1){0.05}}\put(0.0,-0.9){% \line(0,-1){0.05}} \put(-1.0,1.0){\line(1,0){2.0}} \put(-1.0,1.0){\line(0,-1){2.0}} \put(1.0,-1.0){\line(-1,0){2.0}} \put(1.0,-1.0){\line(0,1){2.0}} \put(0.0,0.0){\circle*{0.15}}\put(0.7746,0.0){\circle*{0.15}}\put(-0.7746,0.0)% {\circle*{0.15}}\put(0.0,0.7746){\circle*{0.15}}\put(0.0,-0.7746){\circle*{0.1% 5}}\put(0.7746,0.7746){\circle*{0.15}}\put(-0.7746,0.7746){\circle*{0.15}}\put% (0.7746,-0.7746){\circle*{0.15}}\put(-0.7746,-0.7746){\circle*{0.15}}\end{picture}
    ( 0 , 0 ) 16 81 O ( h 6 )
    ( ± 3 5 h , 0 ) , ( 0 , ± 3 5 h ) 10 81
    ( ± 3 5 h , ± 3 5 h ) 25 324

    Reported 2014-01-13 by Stanley Oleszczuk.

  • Table 22.5.4


    Originally the limiting form for sc ( z , k ) in the last line of this table was incorrect ( cosh z , instead of sinh z ).

    sn ( z , k ) tanh z cd ( z , k ) 1 dc ( z , k ) 1 ns ( z , k ) coth z
    cn ( z , k ) sech z sd ( z , k ) sinh z nc ( z , k ) cosh z ds ( z , k ) csch z
    dn ( z , k ) sech z nd ( z , k ) cosh z sc ( z , k ) sinh z cs ( z , k ) csch z

    Reported 2010-11-23.