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21: 28.34 Methods of Computation

… ►

(a)

Summation of the power series in §§28.6(i) and 28.15(i) when $\left|q\right|$ is small.

… ►

(a)

Summation of the power series in §§28.6(ii) and 28.15(ii) when $\left|q\right|$ is small.

►

(b)

Use of asymptotic expansions and approximations for large $q$ (§§28.8(ii)–28.8(iv)).

… ►

(a)

Numerical summation of the expansions in series of Bessel functions (28.24.1)–(28.24.13). These series converge quite rapidly for a wide range of values of $q$ and $z$ .

… ►

(c)

Use of asymptotic expansions for large $z$ or large $q$ . See §§28.25 and 28.26.

22: 33.20 Expansions for Small $|\epsilon|$

… ►

§33.20(i) Case $\epsilon=0$

… ►

§33.20(ii) Power-Series in $\epsilon$ for the Regular Solution

… ►where

A(\epsilon,\ell)

is given by (33.14.11), (33.14.12), and … ►

§33.20(iv) Uniform Asymptotic Expansions

… ►These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders

2\ell+1

and

2\ell+2

23: 19.19 Taylor and Related Series

§19.19 Taylor and Related Series

… ►The number of terms in

T_{N}

can be greatly reduced by using variables

\mathbf{Z}=\boldsymbol{{1}}-(\mathbf{z}/A)

with

A

chosen to make

E_{1}(\mathbf{Z})=0

. Then

T_{N}

has at most one term if

N\leq 5

in the series for

R_{F}

. … ►Special cases are given in (19.36.1) and (19.36.2).

24: 20.6 Power Series

§20.6 Power Series

… ►where

z_{m,n}

is given by (20.2.5) and the minimum is for

m,n\in\mathbb{Z}

, except

m=n=0

. …In the double series the order of summation is important only when

j=1

. For further information on

\delta_{2j}

see §23.9: since the double sums in (20.6.6) and (23.9.1) are the same, we have

\delta_{2n}=c_{n}/(2n-1)

when

n\geq 2

25: 6.18 Methods of Computation

… ►For small or moderate values of

x

and

|z|

, the expansion in power series (§6.6) or in series of spherical Bessel functions (§6.10(ii)) can be used. … ►For large

x

and

\left|z\right|

, expansions in inverse factorial series (§6.10(i)) or asymptotic expansions (§6.12) are available. The attainable accuracy of the asymptotic expansions can be increased considerably by exponential improvement. … ►Power series, asymptotic expansions, and quadrature can also be used to compute the functions

\mathrm{f}\left(z\right)

and

\mathrm{g}\left(z\right)

. … ►Zeros of

\operatorname{Ci}\left(x\right)

and

\operatorname{si}\left(x\right)

can be computed to high precision by Newton’s rule (§3.8(ii)), using values supplied by the asymptotic expansion (6.13.2) as initial approximations. …

26: 4.24 Inverse Trigonometric Functions: Further Properties

… ►

§4.24(i) Power Series

… ►

4.24.2

\operatorname{arccos}z=(2(1-z))^{1/2}\*\left(1+\sum_{n=1}^{\infty}\frac{1\cdot 3% \cdot 5\cdots(2n-1)}{2^{2n}(2n+1)n!}(1-z)^{n}\right),

|1-z|\leq 2

ⓘ

Symbols:: $!$ : factorial (as in $n!$ ), $\operatorname{arccos}\NVar{z}$ : arccosine function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.4.41 (which is stated differently and the constraint on $z$ is more restrictive.)
Permalink:: http://dlmf.nist.gov/4.24.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(i), §4.24 and Ch.4

… ►

4.24.5

\operatorname{arctan}z=\frac{z}{z^{2}+1}\*\left(1+\frac{2}{3}\frac{z^{2}}{1+z^% {2}}+\frac{2\cdot 4}{3\cdot 5}\left(\frac{z^{2}}{1+z^{2}}\right)^{2}+\cdots% \right),

\Re\left(z^{2}\right)>-\tfrac{1}{2}

ⓘ

Symbols:: $\operatorname{arctan}\NVar{z}$ : arctangent function, $\Re$ : real part and $z$ : complex variable
A&S Ref:: 4.4.42 (has an error in the conditions on $z$ .)
Permalink:: http://dlmf.nist.gov/4.24.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(i), §4.24 and Ch.4

… ►The above equations are interpreted in the sense that every value of the left-hand side is a value of the right-hand side and vice versa. …

27: 23.17 Elementary Properties

… ►

§23.17(ii) Power and Laurent Series

►When

|q|<1

►

23.17.4

\lambda\left(\tau\right)=16q(1-8q+44q^{2}+\cdots),

ⓘ

Symbols:: $\lambda\left(\NVar{\tau}\right)$ : elliptic modular function, $q$ : nome and $\tau$ : complex variable
Permalink:: http://dlmf.nist.gov/23.17.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §23.17(ii), §23.17 and Ch.23

… ►In (23.17.5) for terms up to

q^{48}

see Zuckerman (1939), and for terms up to

q^{100}

see van Wijngaarden (1953). … ►with

q^{1/12}=e^{i\pi\tau/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: 30.3 Eigenvalues

… ►In equation (30.3.5) we can also use … ►

§30.3(iv) Power-Series Expansion

…

30: 1.9 Calculus of a Complex Variable

… ►

§1.9(v) Infinite Sequences and Series

… ►

§1.9(vi) Power Series

… ►

Operations

… ►

1.9.57

\ln f(z)=q_{1}z+q_{2}z^{2}+q_{3}z^{3}+\cdots,

ⓘ

Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, $z$ : variable and $q_{j}$ : coefficients
Referenced by:: §1.9(vi)
Permalink:: http://dlmf.nist.gov/1.9.E57
Encodings:: pMML, png, TeX
See also:: Annotations for §1.9(vi), §1.9(vi), §1.9 and Ch.1

… ►Lastly, a power series can be differentiated any number of times within its circle of convergence: …