trigonometric series expansions

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31: 10.23 Sums

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§10.23(iii) Series Expansions of Arbitrary Functions

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Other Series Expansions

►For other types of expansions of arbitrary functions in series of Bessel functions, see Watson (1944, Chapters 17–19) and Erdélyi et al. (1953b, §§ 7.10.2–7.10.4). … … ►For collections of sums of series involving Bessel or Hankel functions see Erdélyi et al. (1953b, §7.15), Gradshteyn and Ryzhik (2000, §§8.51–8.53), Hansen (1975), Luke (1969b, §9.4), Prudnikov et al. (1986b, pp. 651–691 and 697–700), and Wheelon (1968, pp. 48–51).

32: 4.33 Maclaurin Series and Laurent Series

§4.33 Maclaurin Series and Laurent Series

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4.33.1 $$\sinh z=z+\frac{z^3}{3!}+\frac{z^5}{5!}+\mathrm{\cdots },$$

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Symbols:

$!$: factorial (as in $n!$), $\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

4.5.62

Permalink:

http://dlmf.nist.gov/4.33.E1

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Annotations for §4.33 and Ch.4

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4.33.2 $$\cosh z=1+\frac{z^2}{2!}+\frac{z^4}{4!}+\mathrm{\cdots }.$$

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Symbols:

$!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function and $z$: complex variable

A&S Ref:

4.5.63

Permalink:

http://dlmf.nist.gov/4.33.E2

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Annotations for §4.33 and Ch.4

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4.33.3 $$\tanh z=z-\frac{z^3}{3}+\frac{2}{15}{z}^5-\frac{17}{315}{z}^7+\mathrm{\cdots }+\frac{2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}{z}^{2n-1}+\mathrm{\cdots },$$ .

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\tanh z$: hyperbolic tangent function, $n$: integer and $z$: complex variable

A&S Ref:

4.5.64

Permalink:

http://dlmf.nist.gov/4.33.E3

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Annotations for §4.33 and Ch.4

… ►For expansions that correspond to (4.19.4)–(4.19.9), change $z$ to $\mathrm{i}z$ and use (4.28.8)–(4.28.13).

33: 24.19 Methods of Computation

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Buhler et al. (1992) uses the expansion

24.19.3 $$\frac{t^2}{\cosh t-1}=-2\sum _{n=0}^{\mathrm{\infty }}(2n-1)B_{2n}\frac{t^{2n}}{(2n)!},$$

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Symbols:

$B_n$: Bernoulli numbers, $!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function, $n$: integer and $t$: real or complex

Referenced by:

§24.19(ii)

Permalink:

http://dlmf.nist.gov/24.19.E3

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Annotations for §24.19(ii), §24.19 and Ch.24

and computes inverses modulo $p$ of the left-hand side. Multisectioning techniques are applied in implementations. See also Crandall (1996, pp. 116–120).

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34: 4.45 Methods of Computation

… ►The function $\ln x$ can always be computed from its ascending power series after preliminary scaling. … ►

Trigonometric Functions

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Inverse Trigonometric Functions

►The function $\arctan x$ can always be computed from its ascending power series after preliminary transformations to reduce the size of $x$. … ►Initial approximations are obtainable, for example, from the power series (4.13.6) (with $t\ge 0$) when $x$ is close to $-1/\mathrm{e}$, from the asymptotic expansion (4.13.10) when $x$ is large, and by numerical integration of the differential equation (4.13.4) (§3.7) for other values of $x$. …

35: Bibliography S

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Very high accuracy Chebyshev expansions for the basic trigonometric functions. Math. Comp. 34 (149), pp. 237–244.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview (Claude Carasso), ZentralBlatt

Referenced by:

§4.47(i).

Encodings:

BibTeX

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FGH, a code for the calculation of Coulomb radial wave functions from series expansions. Comput. Phys. Comm. 146 (2), pp. 250–253.

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Notes:

Single-precision Fortran code.

External Links:

Document, Code, ZentralBlatt

Referenced by:

10th item.

Encodings:

BibTeX

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On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.

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External Links:

ZentralBlatt

Referenced by:

§12.13(i).

Encodings:

BibTeX

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On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.

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External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

Encodings:

BibTeX

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The expansion of Lamé functions into series of associated Legendre functions of the second kind. Proc. Cambridge Philos. Soc. 62, pp. 441–452.

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External Links:

Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.17(i).

Encodings:

BibTeX

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36: 5.11 Asymptotic Expansions

§5.11 Asymptotic Expansions

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Terminology

►The expansion (5.11.1) is called Stirling’s series (Whittaker and Watson (1927, §12.33)), whereas the expansion (5.11.3), or sometimes just its leading term, is known as Stirling’s formula (Abramowitz and Stegun (1964, §6.1), Olver (1997b, p. 88)). … ►For re-expansions of the remainder terms in (5.11.1) and (5.11.3) in series of incomplete gamma functions with exponential improvement (§2.11(iii)) in the asymptotic expansions, see Berry (1991), Boyd (1994), and Paris and Kaminski (2001, §6.4). … ►

37: 10.35 Generating Function and Associated Series

§10.35 Generating Function and Associated Series

… ►Jacobi–Anger expansions: for $z,\theta \in ℂ$, ►

10.35.2 $${\mathrm{e}}^{z\cos \theta }=I_0\left(z\right)+2\sum _{k=1}^{\mathrm{\infty }}{I}_k\left(z\right)\cos \left(k\theta \right),$$

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Symbols:

$\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.6.34

Referenced by:

§10.35

Permalink:

http://dlmf.nist.gov/10.35.E2

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Annotations for §10.35 and Ch.10

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10.35.3 $${\mathrm{e}}^{z\sin \theta }=I_0\left(z\right)+2\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k{I}_{2k+1}\left(z\right)\sin \left((2k+1)\theta \right)+2\sum _{k=1}^{\mathrm{\infty }}{(-1)}^k{I}_{2k}\left(z\right)\cos \left(2k\theta \right).$$

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Symbols:

$\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\sin z$: sine function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.6.35

Referenced by:

§10.35

Permalink:

http://dlmf.nist.gov/10.35.E3

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Annotations for §10.35 and Ch.10

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$$\sinh z=2I_1\left(z\right)+2I_3\left(z\right)+2I_5\left(z\right)+\mathrm{\dots }.$$

38: 10.12 Generating Function and Associated Series

§10.12 Generating Function and Associated Series

… ►Jacobi–Anger expansions: for $z,\theta \in ℂ$, ►

$\cos \left(z\sin \theta \right)=J_0\left(z\right)+2{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{J}_{2k}\left(z\right)\cos \left(2k\theta \right),$

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$\sin \left(z\sin \theta \right)=2{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{J}_{2k+1}\left(z\right)\sin \left((2k+1)\theta \right),$

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$\cos \left(z\cos \theta \right)=J_0\left(z\right)+2{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{(-1)}^k{J}_{2k}\left(z\right)\cos \left(2k\theta \right),$

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39: 36.8 Convergent Series Expansions

§36.8 Convergent Series Expansions

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${\mathrm{\Psi }}_K\left(𝐱\right)={\displaystyle \frac{2}{K+2}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\mathrm{i}}^n\cos \left({\displaystyle \frac{\pi (n(K+1)-1)}{2(K+2)}}\right)\mathrm{\Gamma }\left({\displaystyle \frac{n+1}{K+2}}\right)a_n(𝐱),$ $K$ odd,

… ►For multinomial power series for ${\mathrm{\Psi }}_K\left(𝐱\right)$, see Connor and Curtis (1982). …

40: 11.9 Lommel Functions

… ►where $A$, $B$ are arbitrary constants, $s_{\mu ,\nu }\left(z\right)$ is the Lommel function defined by …and … ►

§11.9(ii) Expansions in Series of Bessel Functions

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§11.9(iii) Asymptotic Expansion

►For fixed $\mu$ and $\nu$, …