power-series expansions in ϵ

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21: 7.18 Repeated Integrals of the Complementary Error Function

… ►

7.18.2

\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=\int_{z}^{\infty}\mathop{% \mathrm{i}^{n-1}\mathrm{erfc}}\left(t\right)\,\mathrm{d}t=\frac{2}{\sqrt{\pi}}% \int_{z}^{\infty}\frac{(t-z)^{n}}{n!}e^{-t^{2}}\,\mathrm{d}t.

ⓘ

Defines:: $\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(\NVar{z}\right)$ : repeated integrals of the complementary error function
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $z$ : complex variable and $n$ : nonnegative integer
A&S Ref:: 7.2.1 (in modified form) 7.2.3
Permalink:: http://dlmf.nist.gov/7.18.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §7.18(i), §7.18 and Ch.7

… ► … ►The confluent hypergeometric function on the right-hand side of (7.18.10) is multivalued and in the sectors

\tfrac{1}{2}\pi<\left|\operatorname{ph}z\right|<\pi

one has to use the analytic continuation formula (13.2.12). … ►

§7.18(vi) Asymptotic Expansion

►

7.18.14

\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)\sim\frac{2}{\sqrt{\pi}}% \frac{e^{-z^{2}}}{(2z)^{n+1}}\sum_{m=0}^{\infty}\frac{(-1)^{m}(2m+n)!}{n!m!(2z% )^{2m}},

z\to\infty

|\operatorname{ph}z|\leq\tfrac{3}{4}\pi-\delta(<\tfrac{3}{4}\pi)

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\operatorname{ph}$ : phase, $\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(\NVar{z}\right)$ : repeated integrals of the complementary error function, $z$ : complex variable and $n$ : nonnegative integer
A&S Ref:: 7.2.14
Permalink:: http://dlmf.nist.gov/7.18.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §7.18(vi), §7.18 and Ch.7

22: 23.17 Elementary Properties

… ►

§23.17(ii) Power and Laurent Series

… ►In (23.17.5) for terms up to

q^{48}

see Zuckerman (1939), and for terms up to

q^{100}

see van Wijngaarden (1953). …

23: 2.1 Definitions and Elementary Properties

… ►Most operations on asymptotic expansions can be carried out in exactly the same manner as for convergent power series. …

24: 23.9 Laurent and Other Power Series

§23.9 Laurent and Other Power Series

… ►Explicit coefficients

c_{n}

in terms of

c_{2}

and

c_{3}

are given up to

c_{19}

in Abramowitz and Stegun (1964, p. 636). ►For

j=1,2,3

, and with

e_{j}

as in §23.3(i), …Also, Abramowitz and Stegun (1964, (18.5.25)) supplies the first 22 terms in the reverted form of (23.9.2) as

1/\wp\left(z\right)\to 0

. ►For

z\in\mathbb{C}

…

25: 30.4 Functions of the First Kind

… ►

30.4.1

\int_{-1}^{1}\left(\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)\right)^{2}\,% \mathrm{d}x=\frac{2}{2n+1}\frac{(n+m)!}{(n-m)!},

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind, $x$ : real variable, $m$ : nonnegative integer, $n\geq m$ : integer degree and $\gamma^{2}$ : real parameter
A&S Ref:: 21.7.11
Referenced by:: §30.4(iii)
Permalink:: http://dlmf.nist.gov/30.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §30.4(i), §30.4 and Ch.30

… ►

\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)

has exactly

n-m

zeros in the interval

-1<x<1

. ►

§30.4(iii) Power-Series Expansion

… ►The expansion (30.4.7) converges in the norm of

L^{2}(-1,1)

, that is, …It is also equiconvergent with its expansion in Ferrers functions (as in (30.4.2)), that is, the difference of corresponding partial sums converges to 0 uniformly for

-1\leq x\leq 1

. …

26: 7.6 Series Expansions

§7.6 Series Expansions

►

§7.6(i) Power Series

►

7.6.1

\operatorname{erf}z=\frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n% +1}}{n!(2n+1)},

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erf}\NVar{z}$ : error function, $!$ : factorial (as in $n!$ ), $z$ : complex variable and $n$ : nonnegative integer
A&S Ref:: 7.1.5
Referenced by:: §7.6(i)
Permalink:: http://dlmf.nist.gov/7.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §7.6(i), §7.6 and Ch.7

… ►The series in this subsection and in §7.6(ii) converge for all finite values of

|z|

. ►

§7.6(ii) Expansions in Series of Spherical Bessel Functions

…

27: 19.36 Methods of Computation

… ►The incomplete integrals

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

and

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

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). … ►in agreement with (19.36.5). … ►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}

. … ►For series expansions of Legendre’s integrals see §19.5. Faster convergence of power series for

K\left(k\right)

and

E\left(k\right)

can be achieved by using (19.5.1) and (19.5.2) in the right-hand sides of (19.8.12). …

28: 19.5 Maclaurin and Related Expansions

§19.5 Maclaurin and Related Expansions

… ► … ►Series expansions of

F\left(\phi,k\right)

and

E\left(\phi,k\right)

are surveyed and improved in Van de Vel (1969), and the case of

F\left(\phi,k\right)

is summarized in Gautschi (1975, §1.3.2). For series expansions of

\Pi\left(\phi,\alpha^{2},k\right)

when

|\alpha^{2}|<1

see Erdélyi et al. (1953b, §13.6(9)). …

29: 30.16 Methods of Computation

… ►For small

|\gamma^{2}|

we can use the power-series expansion (30.3.8). …If

|\gamma^{2}|

is large we can use the asymptotic expansions in §30.9. … ►The eigenvalues of

\mathbf{A}

can be computed by methods indicated in §§3.2(vi), 3.2(vii). … ►If

|\gamma^{2}|

is large, then we can use the asymptotic expansions referred to in §30.9 to approximate

\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)

. … ►A fourth method, based on the expansion (30.8.1), is as follows. …

30: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

… ►For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).