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21: 27.4 Euler Products and Dirichlet Series

§27.4 Euler Products and Dirichlet Series

… ►if the series on the left is absolutely convergent. … ►Euler products are used to find series that generate many functions of multiplicative number theory. … ►The Riemann zeta function is the prototype of series of the form …called Dirichlet series with coefficients $f(n)$. …

22: 1.8 Fourier Series

§1.8 Fourier Series

… ► … ►

Uniqueness of Fourier Series

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§1.8(ii) Convergence

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23: 28.19 Expansions in Series of ${\mathrm{me}}_{\nu +2n}$ Functions

§28.19 Expansions in Series of ${\mathrm{me}}_{\nu +2n}$ Functions

… ►The series (28.19.2) converges absolutely and uniformly on compact subsets within $S$. …

24: 28.30 Expansions in Series of Eigenfunctions

§28.30 Expansions in Series of Eigenfunctions

… ►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 …

25: 5.7 Series Expansions

§5.7 Series Expansions

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§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 $\mathrm{\Gamma }\left(z+3\right)$ see Luke (1969b, p. 299). ►

§5.7(ii) Other Series

…

26: 8.7 Series Expansions

§8.7 Series Expansions

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8.7.6 $$\mathrm{\Gamma }(a,x)=x^a{\mathrm{e}}^{-x}\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(a)}\left(x\right)}{n+1},$$ $x>0$, .

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{\Re }$: real part, $x$: real variable, $z$: complex variable, $a$: parameter and $n$: nonnegative integer

Proof sketch:

Integrate (18.12.13) from $z=0$ to $z=1$ and for the integral on the left-hand side use the substitution $z=\frac{t}{1+t}$. The resulting integral is (8.6.5). The constraint follows from (18.15.14).

Referenced by:

Erratum (V1.1.8) for Equation (8.7.6)

Permalink:

http://dlmf.nist.gov/8.7.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.1.8):

The constraint was updated to include .

Suggested 2022-10-14 by Walter Gautschi

See also:

Annotations for §8.7 and Ch.8

►For an expansion for $\gamma (a,\mathrm{i}x)$ in series of Bessel functions $J_n\left(x\right)$ that converges rapidly when $a>0$ and $x$ ($\ge 0$) is small or moderate in magnitude see Barakat (1961).

27: 22.10 Maclaurin Series

§22.10 Maclaurin Series

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§22.10(i) Maclaurin Series in $z$

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

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28: 9.19 Approximations

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§9.19(ii) Expansions in Chebyshev Series

… ►The constants $a$ and $b$ are chosen numerically, with a view to equalizing the effort required for summing the series. … ►

Corless et al. (1992) describe a method of approximation based on subdividing $ℂ$ into a triangular mesh, with values of $\mathrm{Ai}\left(z\right)$, ${\mathrm{Ai}}^{\prime }\left(z\right)$ stored at the nodes. $\mathrm{Ai}\left(z\right)$ and ${\mathrm{Ai}}^{\prime }\left(z\right)$ are then computed from Taylor-series expansions centered at one of the nearest nodes. The Taylor coefficients are generated by recursion, starting from the stored values of $\mathrm{Ai}\left(z\right)$, ${\mathrm{Ai}}^{\prime }\left(z\right)$ at the node. Similarly for $\mathrm{Bi}\left(z\right)$, ${\mathrm{Bi}}^{\prime }\left(z\right)$.

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MacLeod (1994) supplies Chebyshev-series expansions to cover $\mathrm{Gi}\left(x\right)$ for and $\mathrm{Hi}\left(x\right)$ for . The Chebyshev coefficients are given to 20D.

29: 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. …

30: 31.11 Expansions in Series of Hypergeometric Functions

§31.11 Expansions in Series of Hypergeometric Functions

… ►Such series diverge for Fuchs–Frobenius solutions. …Every Heun function can be represented by a series of Type II. ►

§31.11(v) Doubly-Infinite Series

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