power-series expansions in ρ

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1: 33.6 Power-Series Expansions in $\rho$

§33.6 Power-Series Expansions in $\rho$

…

2: 28.6 Expansions for Small $q$

… ►

§28.6(i) Eigenvalues

… ►Leading terms of the of the power series for

m=7,8,9,\dots

are: … ►Numerical values of the radii of convergence

\rho_{n}^{(j)}

of the power series (28.6.1)–(28.6.14) for

n=0,1,\dots,9

are given in Table 28.6.1. … ►

§28.6(ii) Functions $\operatorname{ce}_{n}$ and $\operatorname{se}_{n}$

►Leading terms of the power series for the normalized functions are: …

3: 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. Cancellation errors increase with increases in

\rho

and

|r|

, and may be estimated by comparing the final sum of the series with the largest partial sum. … ►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. … ►Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions (§3.11(iv)) for the analytic continuations of Coulomb functions. ►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). …

4: 12.14 The Function $W\left(a,x\right)$

… ►

§12.14(v) Power-Series Expansions

… ►In the following expansions, obtained from Olver (1959),

\mu

is large and positive, and

\delta

is again an arbitrary small positive constant. ►

Positive $a$ , $2\sqrt{a}<x<\infty$

… ►The expansions for the derivatives corresponding to (12.14.25), (12.14.26), and (12.14.31) may be obtained by formal term-by-term differentiation with respect to

t

; compare the analogous results in §§12.10(ii)–12.10(v). ►

Airy-type Uniform Expansions

…

5: 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). … ►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}

. … ►This method loses significant figures in

\rho

\alpha^{2}

and

k^{2}

are nearly equal unless they are given exact values—as they can be for tables. … ►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). …

6: 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: …

7: 3.10 Continued Fractions

… ►

§3.10(ii) Relations to Power Series

… ►We say that it corresponds to the formal power series …if the expansion of its

n

th convergent

C_{n}

in ascending powers of

z

agrees with (3.10.7) up to and including the term in

z^{n-1}

n=1,2,3,\dots

. … ►For several special functions the

S

-fractions are known explicitly, but in any case the coefficients

a_{n}

can always be calculated from the power-series coefficients by means of the quotient-difference algorithm; see Table 3.10.1. … ►We say that it is associated with the formal power series

f(z)

in (3.10.7) if the expansion of its

n