trigonometric series expansions

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21: 2.11 Remainder Terms; Stokes Phenomenon

… ►Secondly, the asymptotic series represents an infinite class of functions, and the remainder depends on which member we have in mind. … ► … ►

§2.11(iii) Exponentially-Improved Expansions

… ►In this way we arrive at hyperasymptotic expansions. … ►The transformations in §3.9 for summing slowly convergent series can also be very effective when applied to divergent asymptotic series. …

22: 18.18 Sums

… ►

§18.18(i) Series Expansions of Arbitrary Functions

… ►

Legendre

… ►

Laguerre

… ►

Hermite

… ►

Ultraspherical

…

23: 22.10 Maclaurin Series

§22.10 Maclaurin Series

►

§22.10(i) Maclaurin Series in $z$

… ►The full expansions converge when

|z|<\min\left(K\left(k\right),{K^{\prime}}\left(k\right)\right)

. ►

§22.10(ii) Maclaurin Series in $k$ and $k^{\prime}$

… ►

24: 1.8 Fourier Series

§1.8 Fourier Series

… ► … ►

Uniqueness of Fourier Series

… ►For collections of Fourier-series expansions see Prudnikov et al. (1986a, v. 1, pp. 725–740), Gradshteyn and Ryzhik (2000, pp. 45–49), and Oberhettinger (1973).

25: 3.10 Continued Fractions

… ►

§3.10(ii) Relations to Power Series

►Every convergent, asymptotic, or formal series … ►We say that it corresponds to the formal power series … ►We say that it is associated with the formal power series

f(z)

in (3.10.7) 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^{2n-1}

n=1,2,3,\dots

. … ►

Forward Series Recurrence Algorithm

…

26: 7.6 Series Expansions

§7.6 Series Expansions

►

§7.6(i) Power Series

… ►

7.6.5

C\left(z\right)=\cos\left(\tfrac{1}{2}\pi z^{2}\right)\sum_{n=0}^{\infty}\frac% {(-1)^{n}\pi^{2n}}{1\cdot 3\cdots(4n+1)}z^{4n+1}+\sin\left(\tfrac{1}{2}\pi z^{% 2}\right)\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n+1}}{1\cdot 3\cdots(4n+3)}z^{% 4n+3}.

ⓘ

Symbols:: $C\left(\NVar{z}\right)$ : Fresnel integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $n$ : nonnegative integer
A&S Ref:: 7.3.12
Referenced by:: §11.10(vi), §7.6(i)
Permalink:: http://dlmf.nist.gov/7.6.E5
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: 28.5 Second Solutions $\operatorname{fe}_{n}$ , $\operatorname{ge}_{n}$

… ►As

q\to 0

with

n\neq 0

C_{n}(q)\to 0

S_{n}(q)\to 0

C_{n}(q)f_{n}(z,q)\to\sin nz

, and

S_{n}(q)g_{n}(z,q)\to\cos nz

. … ►

\operatorname{fe}_{n}\left(z,0\right)=\sin nz,

►

\operatorname{ge}_{n}\left(z,0\right)=\cos nz

n=1,2,3,\dots

;

… ►For further information on

C_{n}(q)

S_{n}(q)

, and expansions of

f_{n}(z,q)

g_{n}(z,q)

in Fourier series or in series of

\operatorname{ce}_{n}

\operatorname{se}_{n}

functions, see McLachlan (1947, Chapter VII) or Meixner and Schäfke (1954, §2.72). …

28: 5.7 Series Expansions

§5.7 Series Expansions

►

§5.7(i) Maclaurin and Taylor Series

… ►For 15D numerical values of

c_{k}

see Abramowitz and Stegun (1964, p. 256), and for 31D values see Wrench (1968). … ►For 20D numerical values of the coefficients of the Maclaurin series for

\Gamma\left(z+3\right)

see Luke (1969b, p. 299). ►

§5.7(ii) Other Series

…

29: 14.18 Sums

… ►

§14.18(i) Expansion Theorem

►For expansions of arbitrary functions in series of Legendre polynomials see §18.18(i), and for expansions of arbitrary functions in series of associated Legendre functions see Schäfke (1961b). … ►

Zonal Harmonic Series

… ►

Dougall’s Expansion

…

30: 28.2 Definitions and Basic Properties

… ►With

\zeta={\sin}^{2}z

we obtain the algebraic form of Mathieu’s equation …With

\zeta=\cos z

we obtain another algebraic form: … ►

\cos\left(\pi\nu\right)

is an entire function of

a,q^{2}

. … ►The Fourier series of a Floquet solution … ►Near

q=0

a_{n}\left(q\right)

and

b_{n}\left(q\right)

can be expanded in power series in

q

(see §28.6(i)); elsewhere they are determined by analytic continuation (see §28.7). …