# divergent

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## 1—10 of 20 matching pages

##### 1: 25.17 Physical Applications

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►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

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►These are the structurally stable focal singularities (envelopes) of families of rays, on which the intensities of the geometrical (ray) theory diverge.
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##### 3: 1.6 Vectors and Vector-Valued Functions

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►The

*divergence*of a differentiable vector-valued function $\mathbf{F}={F}_{1}\mathbf{i}+{F}_{2}\mathbf{j}+{F}_{3}\mathbf{k}$ is ►
1.6.21
$$\mathrm{div}\mathbf{F}=\nabla \cdot \mathbf{F}=\frac{\partial {F}_{1}}{\partial x}+\frac{\partial {F}_{2}}{\partial y}+\frac{\partial {F}_{3}}{\partial z}.$$

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

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###### Gauss’s (or Divergence) Theorem

…##### 4: 8.25 Methods of Computation

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►For large $|z|$ the corresponding asymptotic expansions (generally divergent) are used instead.
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##### 5: 9.17 Methods of Computation

##### 6: 22.19 Physical Applications

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►As $a\to \sqrt{1/\beta}$ from below the period diverges since $a=\pm \sqrt{1/\beta}$ are points of unstable equilibrium.
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►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.
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##### 7: 2.6 Distributional Methods

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###### §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

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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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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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##### 9: 16.2 Definition and Analytic Properties

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►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
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►In general the series (16.2.1) diverges for all nonzero values of $z$.
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##### 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*. …