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1: 25.17 Physical Applications

… ►Quantum field theory often encounters formally divergent sums that need to be evaluated by a process of regularization: for example, the energy of the electromagnetic vacuum in a confined space (Casimir–Polder effect). It has been found possible to perform such regularizations by equating the divergent sums to zeta functions and associated functions (Elizalde (1995)).

2: 36.14 Other Physical Applications

… ►These are the structurally stable focal singularities (envelopes) of families of rays, on which the intensities of the geometrical (ray) theory diverge. …

3: 1.6 Vectors and Vector-Valued Functions

… ►The divergence of a differentiable vector-valued function $𝐅=F_1𝐢+F_2𝐣+F_3𝐤$ is ►

1.6.21 $$\mathrm{div}𝐅=\nabla \cdot 𝐅=\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}.$$

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Defines:

$\mathrm{div}$: divergence of vector-valued function

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: variable and $F_j$: vector function components

Permalink:

http://dlmf.nist.gov/1.6.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1

… ►

1.6.27 $$\nabla \cdot (\nabla \times 𝐅)=\mathrm{div}\mathrm{curl}𝐅=0,$$

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Symbols:

$\mathrm{curl}$: of vector-valued function and $\mathrm{div}$: divergence of vector-valued function

Permalink:

http://dlmf.nist.gov/1.6.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1

… ►

Gauss’s (or Divergence) Theorem

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4: 8.25 Methods of Computation

… ►For large $|z|$ the corresponding asymptotic expansions (generally divergent) are used instead. …

5: 9.17 Methods of Computation

… ►Since these expansions diverge, the accuracy they yield is limited by the magnitude of $|z|$. …

6: 22.19 Physical Applications

… ►As $a\to \sqrt{1/\beta }$ from below the period diverges since $a=\pm \sqrt{1/\beta }$ are points of unstable equilibrium. … ►As $a\to \sqrt{2/\beta }$ from below the period diverges since $x=0$ is a point of unstable equlilibrium. …As $\left|a\right|\to \sqrt{1/\beta }$ from above the period again diverges. …

7: 2.6 Distributional Methods

… ►

§2.6(i) Divergent Integrals

… ►Although divergent, these integrals may be interpreted in a generalized sense. … ►The fact that expansion (2.6.6) misses all the terms in the second series in (2.6.7) raises the question: what went wrong with our process of reaching (2.6.6)? In the following subsections, we use some elementary facts of distribution theory (§1.16) to study the proper use of divergent integrals. … ►On inserting this identity into (2.6.54), we immediately encounter divergent integrals of the form …However, in the theory of generalized functions (distributions), there is a method, known as “regularization”, by which these integrals can be interpreted in a meaningful manner. …

8: Bibliography W

… ►

Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Computer Physics Reports 10 (5-6), pp. 189–371.

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External Links:

Document

Referenced by:

§2.11(vi), §3.9(v), §3.9(vi).

Encodings:

BibTeX

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Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations. Computers in Physics 10 (5), pp. 496–503.

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External Links:

Document

Referenced by:

§2.11(vi), §2.11(vi).

Encodings:

BibTeX

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9: 16.2 Definition and Analytic Properties

… ►On the circle $|z|=1$ the series (16.2.1) is absolutely convergent if $\mathrm{\Re }{\gamma }_q>0$, convergent except at $z=1$ if , and divergent if $\mathrm{\Re }{\gamma }_q\le -1$, where … ►In general the series (16.2.1) diverges for all nonzero values of $z$. …

10: 1.9 Calculus of a Complex Variable

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§1.9(v) Infinite Sequences and Series

… ►The series is divergent if $s_n$ does not converge. …A series ${\sum }_{n=0}^{\mathrm{\infty }}{z}_n$ converges (diverges) absolutely when ($>1$), or when ($>1$). … ►For a series ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n{(z-z_0)}^n$ there is a number $R$, $0\le R\le \mathrm{\infty }$, such that the series converges for all $z$ in and diverges for $z$ in $\left|z-z_0\right|>R$. … ►If the limit exists, then the double series is convergent; otherwise it is divergent. …