relation to power series

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11: 22.15 Inverse Functions

… ►The inverse Jacobian elliptic functions can be defined in an analogous manner to the inverse trigonometric functions (§4.23). … ►Equations (22.15.1) and (22.15.4), for

\operatorname{arcsn}\left(x,k\right)

, are equivalent to (22.15.12) and also to … ►

§22.15(ii) Representations as Elliptic Integrals

… ►For power-series expansions see Carlson (2008).

12: 7.17 Inverse Error Functions

… ►

§7.17(ii) Power Series

… ►

7.17.2

\operatorname{inverf}x=t+\tfrac{1}{3}t^{3}+\tfrac{7}{30}t^{5}+\tfrac{127}{630}% t^{7}+\cdots=\sum_{m=0}^{\infty}a_{m}t^{2m+1},

|x|<1

ⓘ

Symbols:: $\operatorname{inverf}\NVar{x}$ : inverse error function and $x$ : real variable
Referenced by:: Erratum (V1.1.7) for Additions
Permalink:: http://dlmf.nist.gov/7.17.E2
Encodings:: pMML, png, TeX
Addition (effective with 1.1.7):: The equality “ $=\sum_{m=0}^{\infty}a_{m}t^{2m+1}$ ” was added to this equation.
See also:: Annotations for §7.17(ii), §7.17 and Ch.7

… ►As

x\to 0

… ►

7.17.5

u=-2/\ln\left(\pi x^{2}\ln\left(1/x\right)\right),

ⓘ

Defines:: $u$ : expansion variable (locally)
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/7.17.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §7.17(iii), §7.17 and Ch.7

… ►

7.17.6

v=\ln\left(\ln\left(1/x\right)\right)-2+\ln\pi.

ⓘ

Defines:: $v$ : expansion variable (locally)
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/7.17.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §7.17(iii), §7.17 and Ch.7

…

13: 18.23 Hahn Class: Generating Functions

… ►

Hahn

…

14: 7.18 Repeated Integrals of the Complementary Error Function

… ► … ►

§7.18(iv) Relations to Other Functions

… ►

Hermite Polynomials

… ►

Confluent Hypergeometric Functions

… ►

Parabolic Cylinder Functions

…

15: 33.23 Methods of Computation

… ►The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii

\rho

and

r

, respectively, and may be used to compute the regular and irregular solutions. … ►Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. … ►

§33.23(iv) Recurrence Relations

… ►Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. … ►Noble (2004) obtains double-precision accuracy for

W_{-\eta,\mu}\left(2\rho\right)

for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7). …

16: 28.34 Methods of Computation

… ►

(b)

Representations for $w_{\mbox{\tiny I}}(\pi;a,\pm q)$ with limit formulas for special solutions of the recurrence relations §28.4(ii) for fixed $a$ and $q$ ; see Schäfke (1961a).

… ►

(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.

… ►

(d)

Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

… ►

(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 $z$ .

…

17: 33.14 Definitions and Basic Properties

… ►

§33.14(ii) Regular Solution $f\left(\epsilon,\ell;r\right)$

… ►This includes

\epsilon=0

, hence

f\left(\epsilon,\ell;r\right)

can be expanded in a convergent power series in

\epsilon

in a neighborhood of

\epsilon=0

(§33.20(ii)). ►

§33.14(iii) Irregular Solution $h\left(\epsilon,\ell;r\right)$

►For nonzero values of

\epsilon

and

r

the function

h\left(\epsilon,\ell;r\right)

is defined by … ►When

\epsilon<0

and

\ell>(-\epsilon)^{-1/2}

the quantity

A(\epsilon,\ell)

may be negative, causing

s\left(\epsilon,\ell;r\right)

and

c\left(\epsilon,\ell;r\right)

to become imaginary. …

18: 20.11 Generalizations and Analogs

… ►

§20.11(ii) Ramanujan’s Theta Function and $q$ -Series

… ►In the case

z=0

identities for theta functions become identities in the complex variable

q

, with

\left|q\right|<1

, that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … ►However, in this case

q

is no longer regarded as an independent complex variable within the unit circle, because

k

is related to the variable

\tau=\tau(k)

of the theta functions via (20.9.2). … ►For applications to rapidly convergent expansions for

\pi

see Chudnovsky and Chudnovsky (1988), and for applications in the construction of elliptic-hypergeometric series see Rosengren (2004). … ►Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas. …

19: 1.9 Calculus of a Complex Variable

… ►

Powers

… ►

§1.9(v) Infinite Sequences and Series

… ►

§1.9(vi) Power Series

… ►

Operations

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

20: 15.19 Methods of Computation

… ►The Gauss series (15.2.1) converges for

|z|<1

. … ►Large values of

|a|

|b|

, for example, delay convergence of the Gauss series, and may also lead to severe cancellation. … ►A comprehensive and powerful approach is to integrate the hypergeometric differential equation (15.10.1) by direct numerical methods. … ►

§15.19(iv) Recurrence Relations

►The relations in §15.5(ii) can be used to compute

F\left(a,b;c;z\right)

, provided that care is taken to apply these relations in a stable manner; see §3.6(ii). …