convergence properties

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21: 25.12 Polylogarithms

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§25.12(i) Dilogarithms

… ►For graphics see Figures 25.12.1 and 25.12.2, and for further properties see Maximon (2003), Kirillov (1995), Lewin (1981), Nielsen (1909), and Zagier (1989). … ►

§25.12(ii) Polylogarithms

… ►The series also converges when $|z|=1$, provided that $\mathrm{\Re }s>1$. … ►Further properties include …

22: 13.14 Definitions and Basic Properties

§13.14 Definitions and Basic Properties

… ►converge for all $z\in ℂ$. … ► …

23: 25.14 Lerch’s Transcendent

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§25.14(ii) Properties

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25.14.6 $$\mathrm{\Phi }(z,s,a)=\frac{1}{2}{a}^{-s}+{\int }_0^{\mathrm{\infty }}\frac{z^x}{{(a+x)}^s}dx-2{\int }_0^{\mathrm{\infty }}\frac{\sin \left(x\ln z-s\arctan \left(x/a\right)\right)}{{(a^2+x^2)}^{s/2}({\mathrm{e}}^{2\pi x}-1)}dx,$$ $\mathrm{\Re }a>0$ if ; $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $\left|z\right|=1$.

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Symbols:

$\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $\sin z$: sine function, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integral representation

Source:

Erdélyi et al. (1953a, (1.11.4), p. 28)

Notes:

For the case see Erdélyi et al. (1953a, (1.11.4), p. 28). In the case $\left|z\right|=1$ one checks that the first integral converges absolutely iff $\mathrm{\Re }s>1$.

Referenced by:

Erratum (V1.1.4) for Equation (25.14.6)

Permalink:

http://dlmf.nist.gov/25.14.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

The constraint which originally read “$\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $|z|=1,\mathrm{\Re }a>0$” has been improved to be “$\mathrm{\Re }a>0$ if ; $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $\left|z\right|=1$”.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.14(ii), §25.14 and Ch.25

►For these and further properties see Erdélyi et al. (1953a, pp. 27–31).

24: 1.10 Functions of a Complex Variable

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§1.10(ix) Infinite Products

… ►The convergence of the infinite product is uniform if the sequence of partial products converges uniformly. ►

$M$-test

… ►Many properties are a direct consequence of this representation: Taking the $x$-derivative gives us …

25: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►, ${lim}_{m,n\to \mathrm{\infty }}\Vert v_m-v_n\Vert =0$) converges in norm to some $v\in V$, i. … ►where the infinite sum means convergence in norm, … ►where the limit has to be understood in the sense of $L^2$ convergence in the mean: … ►Often circumstances allow rather stronger statements, such as uniform convergence, or pointwise convergence at points where $f(x)$ is continuous, with convergence to $(f(x_0-)+f(x_0+))/2$ if $x_0$ is an isolated point of discontinuity. … ►for $f(x)\in L^2$ and piece-wise continuous, with convergence as discussed in §1.18(ii). …

26: 28.2 Definitions and Basic Properties

§28.2 Definitions and Basic Properties

… ►Other properties are as follows. … ►A solution with the pseudoperiodic property (28.2.14) is called a Floquet solution with respect to $\nu$. … ►converges absolutely and uniformly in compact subsets of $ℂ$. … ►

Change of Sign of $q$

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27: 4.13 Lambert $W$-Function

… ►They are denoted by $W_k\left(z\right)$, $k\in \mathbb{Z}$, and have the property … ►Properties include: … ►For large enough $\left|z\right|$ the series on the right-hand side of (4.13.10) is absolutely convergent to its left-hand side. In the case of $k=0$ and real $z$ the series converges for $z\ge \mathrm{e}$. …

28: 17.2 Calculus

… ►when this product converges. … ►For properties of the function $f\left(q\right)=q^{-1/24}\eta \left(\frac{\ln q}{2\pi \mathrm{i}}\right)={(q;q)}_{\mathrm{\infty }}$ see §27.14. … ►When $n\to \mathrm{\infty }$ in (17.2.35), and when $m\to \mathrm{\infty }$ in (17.2.38), the results become convergent infinite series and infinite products (see (17.5.1) and (17.5.4)). … ►When $q\to 1$ the $q$-derivatives converge to the corresponding ordinary derivatives. … ►provided that ${\sum }_{j=-\mathrm{\infty }}^{\mathrm{\infty }}f(q^j)q^j$ converges. …

29: 23.2 Definitions and Periodic Properties

§23.2 Definitions and Periodic Properties

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§23.2(ii) Weierstrass Elliptic Functions

… ►The double series and double product are absolutely and uniformly convergent in compact sets in $ℂ$ that do not include lattice points. … ►For further quasi-periodic properties of the $\sigma$-function see Lawden (1989, §6.2).

30: 2.10 Sums and Sequences

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The first infinite integral in (2.10.2) converges.

… ►We need a “comparison function” $g(z)$ with the properties: … ►It is unnecessary for $f(z)-g(z)$ to be continuous on $|z|=r$: it suffices that the integrals in (2.10.28) converge uniformly. … ►For uniform expansions when two singularities coalesce on the circle of convergence see Wong and Zhao (2005). …