trigonometric series expansions

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11: 6.16 Mathematical Applications

… ►Consider the Fourier series … ►The first maximum of $\frac{1}{2}\mathrm{Si}\left(x\right)$ for positive $x$ occurs at $x=\pi$ and equals $(1.1789\mathrm{\dots })\times \frac{1}{4}\pi$; compare Figure 6.3.2. …Compare Figure 6.16.1. … ►It occurs with Fourier-series expansions of all piecewise continuous functions. … …

12: 28.6 Expansions for Small $q$

§28.6 Expansions for Small $q$

►

§28.6(i) Eigenvalues

… ►Leading terms of the of the power series for $m=7,8,9,\mathrm{\dots }$ are: … ►Leading terms of the power series for the normalized functions are: … ►For the corresponding expansions of ${\mathrm{se}}_m(z,q)$ for $m=3,4,5,\mathrm{\dots }$ change $\cos$ to $\sin$ everywhere in (28.6.26). …

13: 11.10 Anger–Weber Functions

… ►

§11.10(iii) Maclaurin Series

… ►These expansions converge absolutely for all finite values of $z$. … ►where …For the Fresnel integrals $C$ and $S$ see §7.2(iii). … ►

§11.10(viii) Expansions in Series of Products of Bessel Functions

…

14: 28.11 Expansions in Series of Mathieu Functions

… ►

28.11.7 $$\sin (2m+2)z=\sum _{n=0}^{\mathrm{\infty }}{B}_{2m+2}^{2n+2}(q){\mathrm{se}}_{2n+2}(z,q).$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E7

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Annotations for §28.11, §28.11 and Ch.28

15: 13.24 Series

§13.24 Series

►

§13.24(i) Expansions in Series of Whittaker Functions

►For expansions of arbitrary functions in series of $M_{\kappa ,\mu }\left(z\right)$ functions see Schäfke (1961b). ►

§13.24(ii) Expansions in Series of Bessel Functions

… ►For other series expansions see Prudnikov et al. (1990, §6.6). …

16: 29.6 Fourier Series

§29.6 Fourier Series

… ►

29.6.1 $${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)=\frac{1}{2}{A}_0+\sum _{p=1}^{\mathrm{\infty }}{A}_{2p}\cos \left(2p\varphi \right).$$

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Symbols:

${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $\cos z$: cosine function, $m$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $\varphi$: change of variable and $A_{2p}$: coefficients

Referenced by:

§29.15(i), §29.6(i)

Permalink:

http://dlmf.nist.gov/29.6.E1

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Annotations for §29.6(i), §29.6 and Ch.29

… ►An alternative version of the Fourier series expansion (29.6.1) is given by … ►

29.6.16 $${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)=\sum _{p=0}^{\mathrm{\infty }}{A}_{2p+1}\cos \left((2p+1)\varphi \right).$$

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Symbols:

${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $\cos z$: cosine function, $m$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $\varphi$: change of variable and $A_{2p}$: coefficients

Referenced by:

§29.15(i)

Permalink:

http://dlmf.nist.gov/29.6.E16

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Annotations for §29.6(ii), §29.6 and Ch.29

… ►

29.6.31 $${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)=\sum _{p=0}^{\mathrm{\infty }}{B}_{2p+1}\sin \left((2p+1)\varphi \right).$$

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Symbols:

${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\sin z$: sine function, $m$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $\varphi$: change of variable and $B_{2p+1}$: coefficents

Referenced by:

§29.15(i)

Permalink:

http://dlmf.nist.gov/29.6.E31

Encodings:

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See also:

Annotations for §29.6(iii), §29.6 and Ch.29

…

17: 6.10 Other Series Expansions

§6.10 Other Series Expansions

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§6.10(i) Inverse Factorial Series

… ►

§6.10(ii) Expansions in Series of Spherical Bessel Functions

… ►

6.10.4 $$\mathrm{Si}\left(z\right)=z\sum _{n=0}^{\mathrm{\infty }}{\left({𝗃}_n\left(\frac{1}{2}z\right)\right)}^2,$$

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Symbols:

$\mathrm{Si}\left(z\right)$: sine integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§6.10(ii)

Permalink:

http://dlmf.nist.gov/6.10.E4

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Annotations for §6.10(ii), §6.10 and Ch.6

… ►An expansion for $E_1\left(z\right)$ can be obtained by combining (6.2.4) and (6.10.8).

18: 27.14 Unrestricted Partitions

… ►Multiplying the power series for $f\left(x\right)$ with that for $1/f\left(x\right)$ and equating coefficients, we obtain the recursion formula … ►Rademacher (1938) derives a convergent series that also provides an asymptotic expansion for $p\left(n\right)$: ►

27.14.9 $$p\left(n\right)=\frac{1}{\pi \sqrt{2}}\sum _{k=1}^{\mathrm{\infty }}\sqrt{k}{A}_k(n){\left[\frac{d}{dt}\frac{\sinh \left(K\sqrt{t}/k\right)}{\sqrt{t}}\right]}_{t=n-(1/24)},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sinh z$: hyperbolic sine function, $p\left(n\right)$: total number of partitions of $n$ , $k$: positive integer, $n$: positive integer, $K$: change of variable and $A_k(n)$: coefficients

A&S Ref:

24.2.1 I.C

Referenced by:

§27.20

Permalink:

http://dlmf.nist.gov/27.14.E9

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Annotations for §27.14(iii), §27.14 and Ch.27

… ►The 24th power of $\eta \left(\tau \right)$ in (27.14.12) with ${\mathrm{e}}^{2\pi \mathrm{i}\tau }=x$ is an infinite product that generates a power series in $x$ with integer coefficients called Ramanujan’s tau function $\tau \left(n\right)$: …

19: 14.15 Uniform Asymptotic Approximations

… ►In other words, the convergent hypergeometric series expansions of ${𝖯}_{\nu }^{-\mu }\left(\pm x\right)$ are also generalized (and uniform) asymptotic expansions as $\mu \to \mathrm{\infty }$, with scale $1/\mathrm{\Gamma }\left(j+1+\mu \right)$, $j=0,1,2,\mathrm{\dots }$; compare §2.1(v). … ►For asymptotic expansions and explicit error bounds, see Dunster (2003b) and Gil et al. (2000). … ►For convergent series expansions see Dunster (2004). … ►See also Olver (1997b, pp. 311–313) and §18.15(iii) for a generalized asymptotic expansion in terms of elementary functions for Legendre polynomials $P_n\left(\cos \theta \right)$ as $n\to \mathrm{\infty }$ with $\theta$ fixed. … ►For asymptotic expansions and explicit error bounds, see Boyd and Dunster (1986). …

20: 7.24 Approximations

… ►

§7.24(ii) Expansions in Chebyshev Series

… ►

Schonfelder (1978) gives coefficients of Chebyshev expansions for $x^{-1}\mathrm{erf}x$ on $0\le x\le 2$, for $x{\mathrm{e}}^{x^2}\mathrm{erfc}x$ on $[2,\mathrm{\infty })$, and for ${\mathrm{e}}^{x^2}\mathrm{erfc}x$ on $[0,\mathrm{\infty })$ (30D).

►

Shepherd and Laframboise (1981) gives coefficients of Chebyshev series for $(1+2x){\mathrm{e}}^{x^2}\mathrm{erfc}x$ on $(0,\mathrm{\infty })$ (22D).

►

§7.24(iii) Padé-Type Expansions

►

Luke (1969b, vol. 2, pp. 422–435) gives main diagonal Padé approximations for $F\left(z\right)$, $\mathrm{erf}z$, $\mathrm{erfc}z$, $C\left(z\right)$, and $S\left(z\right)$; approximate errors are given for a selection of $z$-values.