divergent integrals

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1: 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. …

2: 36.14 Other Physical Applications

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§36.14(i) Caustics

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

§36.14(ii) Optics

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§36.14(iii) Quantum Mechanics

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§36.14(iv) Acoustics

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3: 2.7 Differential Equations

… ►We cannot take $f=x$ and $g=\ln x$ because $\int g{f}^{-1/2}dx$ would diverge as $x\to +\mathrm{\infty }$. …

4: 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(iv) Path and Line Integrals

… ►The path integral of a continuous function $f(x,y,z)$ is … ►The integral of a continuous function $f(x,y,z)$ over a surface $S$ is … ►

Gauss’s (or Divergence) Theorem

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5: 2.11 Remainder Terms; Stokes Phenomenon

… ►For divergent expansions the situation is even more difficult. … ►As an example consider … ►However, regardless whether we can bound the remainder, the accuracy achievable by direct numerical summation of a divergent asymptotic series is always limited. … ►From §8.19(i) the generalized exponential integral is given by … ►The transformations in §3.9 for summing slowly convergent series can also be very effective when applied to divergent asymptotic series. …

6: Bibliography S

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Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, pp. 1–42.

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External Links:

MathReview (J. Kuntzmann), ZentralBlatt

Referenced by:

§17.18.

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Gaussian-transform method for molecular integrals. I. Formulation for energy integrals. J. Chem. Phys. 43 (2), pp. 398–414.

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External Links:

Document, MathReview (R. K. Nesbet)

Referenced by:

§8.24(i).

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Algorithm AS 239. Chi-squared and incomplete gamma integral. Appl. Statist. 37 (3), pp. 466–473.

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Notes:

Double-precision Fortran.

External Links:

FullText, Code

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5th item.

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Non-linear integral equations for Heun functions. Proc. Edinburgh Math. Soc. (2) 16, pp. 281–289.

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External Links:

Document, MathReview (J. D. DePree), ZentralBlatt

Referenced by:

§31.10(ii).

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The Use of Integral Transforms. McGraw-Hill, New York.

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External Links:

ISBN 00-705-9436-8, ZentralBlatt

Referenced by:

§1.14(v), §10.43(v), §14.31(ii).

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7: 22.19 Physical Applications

… ►The period is $4K\left(\sin \left(\frac{1}{2}\alpha \right)\right)$. … ►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. … ►

§22.19(iv) Tops

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8: 22.3 Graphics

… ►The period diverges logarithmically as $k\to 1-$; see §19.12. … ►

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Figure 22.3.16: $\mathrm{sn}(x+\mathrm{i}y,k)$ for $k=0.99$, $-3K\le x\le 3K$, $0\le y\le 4K^{\prime }$. $K=3.3566\mathrm{\dots }$, $K^{\prime }=1.5786\mathrm{\dots }$. Magnify 3D Help

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Figure 22.3.17: $\mathrm{cn}(x+\mathrm{i}y,k)$ for $k=0.99$, $-3K\le x\le 3K$, $0\le y\le 4K^{\prime }$. $K=3.3566\mathrm{\dots }$, $K^{\prime }=1.5786\mathrm{\dots }$. Magnify 3D Help

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Figure 22.3.18: $\mathrm{dn}(x+\mathrm{i}y,k)$ for $k=0.99$, $-3K\le x\le 3K$, $0\le y\le 4K^{\prime }$. $K=3.3566\mathrm{\dots }$, $K^{\prime }=1.5786\mathrm{\dots }$. Magnify 3D Help

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Figure 22.3.19: $\mathrm{cd}(x+\mathrm{i}y,k)$ for $k=0.99$, $-3K\le x\le 3K$, $0\le y\le 4K^{\prime }$. $K=3.3566\mathrm{\dots }$, $K^{\prime }=1.5786\mathrm{\dots }$. Magnify 3D Help

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9: Bibliography H

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Une $q$-intégrale de Selberg et Askey. SIAM J. Math. Anal. 19 (6), pp. 1475–1489.

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External Links:

ISSN 0036-1410, Document, MathReview (R. A. Askey), ZentralBlatt

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§17.13.

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Certain integrals that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1975 (2), pp. 86–88, 95 (Russian).

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External Links:

MathReview (C. Tanaka)

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§7.14(iii).

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Elliptic Integrals. Dover Publications Inc., New York.

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Notes:

Unaltered reprint of original edition published by Wiley, New York, 1917.

External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§19.8(ii).

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Divergent Series. Clarendon Press, Oxford.

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Notes:

Reprinted by the American Mathematical Society in 2000.

External Links:

MathReview (R. P. Agnew), ZentralBlatt

Referenced by:

§1.15(ii), §1.15(viii).

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Definitions of Stieltjes Integrals of the Riemann Type. Amer. Math. Monthly 45 (5), pp. 265–278.

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External Links:

ISSN 0002-9890, Document, Link, MathReview Entry

Referenced by:

§1.16(ix).

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10: 8.25 Methods of Computation

… ►For large $|z|$ the corresponding asymptotic expansions (generally divergent) are used instead. … ►Stable recursive schemes for the computation of $E_p\left(x\right)$ are described in Miller (1960) for $x>0$ and integer $p$. …