About the Project
NIST

asymptotic expansions of integrals

AdvancedHelp

(0.006 seconds)

1—10 of 92 matching pages

1: 6.12 Asymptotic Expansions
§6.12(i) Exponential and Logarithmic Integrals
For the function χ see §9.7(i). …
§6.12(ii) Sine and Cosine Integrals
2: 16.25 Methods of Computation
Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. …
3: 6.13 Zeros
Values of c 1 and c 2 to 30D are given by MacLeod (1996b). …
6.13.2 c k , s k α + 1 α - 16 3 1 α 3 + 1673 15 1 α 5 - 5 07746 105 1 α 7 + ,
4: 8.20 Asymptotic Expansions of E p ( z )
§8.20 Asymptotic Expansions of E p ( z )
§8.20(i) Large z
Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii). For an exponentially-improved asymptotic expansion of E p ( z ) see §2.11(iii).
§8.20(ii) Large p
5: 7.12 Asymptotic Expansions
§7.12(ii) Fresnel Integrals
The asymptotic expansions of C ( z ) and S ( z ) are given by (7.5.3), (7.5.4), and
7.12.2 f ( z ) 1 π z m = 0 ( - 1 ) m ( 1 2 ) 2 m ( π z 2 / 2 ) 2 m ,
They are bounded by | csc ( 4 ph z ) | times the first neglected terms when 1 8 π | ph z | < 1 4 π . …
§7.12(iii) Goodwin–Staton Integral
6: 9.15 Mathematical Applications
Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …
7: 12.18 Methods of Computation
These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …
8: 2.3 Integrals of a Real Variable
For the Fourier integral
§2.3(ii) Watson’s Lemma
2.3.12 0 f ( x t ) q ( t ) d t s = 0 f ( s + λ μ ) a s x ( s + λ ) / μ , x + ,
§2.3(iii) Laplace’s Method
Then …
9: 2.6 Distributional Methods
2.6.3 0 t - s - ( 1 / 3 ) x + t d t , s = 1 , 2 , 3 , .
§2.6(ii) Stieltjes Transform
§2.6(iii) Fractional Integrals
If both f and g in (2.6.34) have asymptotic expansions of the form (2.6.9), then the distribution method can also be used to derive an asymptotic expansion of the convolution f g ; see Li and Wong (1994). … The method of distributions can be further extended to derive asymptotic expansions for convolution integrals: …
10: 7.13 Zeros
As n the x n and y n corresponding to the zeros of C ( z ) satisfy … For an asymptotic expansion of the zeros of 0 z exp ( 1 2 π i t 2 ) d t ( = ( 0 ) - ( z ) = C ( z ) + i S ( z ) ) see Tuẑilin (1971).