# divergent

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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 $\mathbf{F}=F_{1}\mathbf{i}+F_{2}\mathbf{j}+F_{3}\mathbf{k}$ is
1.6.21 $\operatorname{div}\mathbf{F}=\nabla\cdot\mathbf{F}=\frac{\partial F_{1}}{% \partial x}+\frac{\partial F_{2}}{\partial y}+\frac{\partial F_{3}}{\partial z}.$
1.6.27 $\nabla\cdot(\nabla\times\mathbf{F})=\operatorname{div}\operatorname{curl}% \mathbf{F}=0,$
##### 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
• E. J. Weniger (1989) Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Computer Physics Reports 10 (5-6), pp. 189–371.
• E. J. Weniger (1996) 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.
• ##### 9: 16.2 Definition and Analytic Properties
On the circle $|z|=1$ the series (16.2.1) is absolutely convergent if $\Re\gamma_{q}>0$, convergent except at $z=1$ if $-1<\Re\gamma_{q}\leq 0$, and divergent if $\Re\gamma_{q}\leq-1$, where … In general the series (16.2.1) diverges for all nonzero values of $z$. …
##### 10: 1.9 Calculus of a Complex Variable
###### §1.9(v) Infinite Sequences and Series
The series is divergent if $s_{n}$ does not converge. …A series $\sum^{\infty}_{n=0}z_{n}$ converges (diverges) absolutely when $\lim\limits_{n\to\infty}{\left|z_{n}\right|}^{1/n}<1$ ($>1$), or when $\lim\limits_{n\to\infty}\left|\ifrac{z_{n+1}}{z_{n}}\right|<1$ ($>1$). … For a series $\sum^{\infty}_{n=0}a_{n}(z-z_{0})^{n}$ there is a number $R$, $0\leq R\leq\infty$, such that the series converges for all $z$ in $\left|z-z_{0}\right| 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. …