repeated integrals of error functions

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11—15 of 15 matching pages

11: 2.11 Remainder Terms; Stokes Phenomenon

… ►In order to guard against this kind of error remaining undetected, the wanted function may need to be computed by another method (preferably nonasymptotic) for the smallest value of the (large) asymptotic variable

x

that is intended to be used. … ►Here

\operatorname{erfc}

is the complementary error function (§7.2(i)), and … ►These answers are linked to the terms involving the complementary error function in the more powerful expansions typified by the combination of (2.11.10) and (2.11.15). … ►For error bounds see Dunster (1996c). … ►Often the process of re-expansion can be repeated any number of times. …

12: 2.4 Contour Integrals

… ►If, in addition, the corresponding integrals with

Q

and

F

replaced by their derivatives

Q^{(j)}

and

F^{(j)}

j=1,2,\dots,m

, converge uniformly, then by repeated integrations by parts … ►For error bounds see Boyd (1993). … ►Thus the right-hand side of (2.4.14) reduces to the error terms. … ►For a coalescing saddle point and a pole see Wong (1989, Chapter 7) and van der Waerden (1951); in this case the uniform approximants are complementary error functions. For a coalescing saddle point and endpoint see Olver (1997b, Chapter 9) and Wong (1989, Chapter 7); if the endpoint is an algebraic singularity then the uniform approximants are parabolic cylinder functions with fixed parameter, and if the endpoint is not a singularity then the uniform approximants are complementary error functions. …

13: 2.6 Distributional Methods

… ►

§2.6(i) Divergent Integrals

… ►To assign a distribution to the function

f_{n}(t)

, we first let

f_{n,n}(t)

denote the

n

th repeated integral (§1.4(v)) of

f_{n}

: … ►An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). … ►

f_{n,j}(t)

being the

j

th repeated integral of

f_{n}

; compare (2.6.15). … ►However, in the theory of generalized functions (distributions), there is a method, known as “regularization”, by which these integrals can be interpreted in a meaningful manner. …

14: 10.22 Integrals

… ►

§10.22(i) Indefinite Integrals

… ►

§10.22(ii) Integrals over Finite Intervals

… ►

Fractional Integral

… ►When

\alpha=m=1,2,3,\ldots

the left-hand side of (10.22.36) is the

m

th repeated integral of

J_{\nu}\left(x\right)

(§§1.4(v) and 1.15(vi)). … ►

Weber–Schafheitlin Discontinuous Integrals, including Special Cases

…

15: 19.36 Methods of Computation

… ►Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18). … ►If the iteration of (19.36.6) and (19.36.12) is stopped when

c_{s}<\sqrt{r}t_{s}

(

M

and

T

being approximated by

a_{s}

and

t_{s}

, and the infinite series being truncated), then the relative error in

R_{F}

and

R_{G}

is less than

r

if we neglect terms of order

r^{2}

. … ►The function

\operatorname{cel}\left(k_{c},p,a,b\right)

is computed by successive Bartky transformations (Bulirsch and Stoer (1968), Bulirsch (1969b)). … ►Lee (1990) compares the use of theta functions for computation of

K\left(k\right)

E\left(k\right)

, and

K\left(k\right)-E\left(k\right)

0\leq k^{2}\leq 1

, with four other methods. … ►Similarly, §19.26(ii) eases the computation of functions such as

R_{F}\left(x,y,z\right)

when

x

(

>0

) is small compared with

\min(y,z)

. …