# expansions of solutions in series of

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##### 11: 16.25 Methods of Computation
Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. They are similar to those described for confluent hypergeometric functions, and hypergeometric functions in §§13.29 and 15.19. There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). This occurs when the wanted solution is intermediate in asymptotic growth compared with other solutions. In these cases integration, or recurrence, in either a forward or a backward direction is unstable. …
##### 12: 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.

• ###### §28.34(iii) Floquet Solutions
• (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 $z$.

• ##### 13: 2.9 Difference Equations
Often $f(n)$ and $g(n)$ can be expanded in seriesAs in the case of differential equations (§§2.7(iii), 2.7(iv)) recessive solutions are unique and dominant solutions are not; furthermore, one member of a numerically satisfactory pair has to be recessive. … For asymptotic expansions in inverse factorial series see Olde Daalhuis (2004a). … But there is an independent solutionFor discussions of turning points, transition points, and uniform asymptotic expansions for solutions of linear difference equations of the second order see Wang and Wong (2003, 2005). …
##### 14: 10.74 Methods of Computation
###### §10.74(i) SeriesExpansions
The power-series expansions given in §§10.2 and 10.8, together with the connection formulas of §10.4, can be used to compute the Bessel and Hankel functions when the argument $x$ or $z$ is sufficiently small in absolute value. … In other circumstances the power series are prone to slow convergence and heavy numerical cancellation. … Furthermore, the attainable accuracy can be increased substantially by use of the exponentially-improved expansions given in §10.17(v), even more so by application of the hyperasymptotic expansions to be found in the references in that subsection. … In the interval $0, $J_{\nu}\left(x\right)$ needs to be integrated in the forward direction and $Y_{\nu}\left(x\right)$ in the backward direction, with initial values for the former obtained from the power-series expansion (10.2.2) and for the latter from asymptotic expansions (§§10.17(i) and 10.20(i)). …
##### 15: 3.7 Ordinary Differential Equations
For classification of singularities of (3.7.1) and expansions of solutions in the neighborhoods of singularities, see §2.7. …
###### §3.7(iii) Taylor-Series Method: Boundary-Value Problems
General methods for boundary-value problems for ordinary differential equations are given in Ascher et al. (1995). … The method consists of a set of rules each of which is equivalent to a truncated Taylor-series expansion, but the rules avoid the need for analytic differentiations of the differential equation. …
##### 16: 28.30 Expansions in Series of Eigenfunctions
###### §28.30 ExpansionsinSeries of Eigenfunctions
Let $\widehat{\lambda}_{m}$, $m=0,1,2,\dots$, be the set of characteristic values (28.29.16) and (28.29.17), arranged in their natural order (see (28.29.18)), and let $w_{m}(x)$, $m=0,1,2,\dots$, be the eigenfunctions, that is, an orthonormal set of $2\pi$-periodic solutions; thus Then every continuous $2\pi$-periodic function $f(x)$ whose second derivative is square-integrable over the interval $[0,2\pi]$ can be expanded in a uniformly and absolutely convergent series
28.30.3 $f(x)=\sum_{m=0}^{\infty}f_{m}w_{m}(x),$
##### 17: 13.2 Definitions and Basic Properties
###### Standard Solutions
The first two standard solutions are: … The series (13.2.2) and (13.2.3) converge for all $z\in\mathbb{C}$. …
###### §13.2(v) Numerically Satisfactory Solutions
Fundamental pairs of solutions of (13.2.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are …
##### 18: 2.7 Differential Equations
All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. … Hence unless the series (2.7.8) terminate (in which case the corresponding $\Lambda_{j}$ is zero) they diverge. … Although the expansions (2.7.14) apply only in the sectors (2.7.15) and (2.7.16), each solution $w_{j}(z)$ can be continued analytically into any other sector. … Note that the coefficients in the expansions (2.7.12), (2.7.13) for the “late” coefficients, that is, $a_{s,1}$, $a_{s,2}$ with $s$ large, are the “early” coefficients $a_{j,2}$, $a_{j,1}$ with $j$ small. … In theory either pair may be used to construct any other solution
##### 19: 2.8 Differential Equations with a Parameter
The form of the asymptotic expansion depends on the nature of the transition points in $\mathbf{D}$, that is, points at which $f(z)$ has a zero or singularity. … In Cases I and II the asymptotic solutions are in terms of the functions that satisfy (2.8.8) with $\psi(\xi)=0$. … For another approach to these problems based on convergent inverse factorial series expansions see Dunster et al. (1993) and Dunster (2001a, 2004). …
###### §2.8(iv) Case III: Simple Pole
For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). …
##### 20: Bibliography L
• T. M. Larsen, D. Erricolo, and P. L. E. Uslenghi (2009) New method to obtain small parameter power series expansions of Mathieu radial and angular functions. Math. Comp. 78 (265), pp. 255–274.
• E. W. Leaver (1986) Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics. J. Math. Phys. 27 (5), pp. 1238–1265.
• J. L. López and E. Pérez Sinusía (2014) New series expansions for the confluent hypergeometric function $M(a,b,z)$ . Appl. Math. Comput. 235, pp. 26–31.
• J. L. López and N. M. Temme (2013) New series expansions of the Gauss hypergeometric function. Adv. Comput. Math. 39 (2), pp. 349–365.
• Y. L. Luke (1959) Expansion of the confluent hypergeometric function in series of Bessel functions. Math. Tables Aids Comput. 13 (68), pp. 261–271.