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
… ►§36.14(iii) Quantum Mechanics
… ►§36.14(iv) Acoustics
…3: 2.7 Differential Equations
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►We cannot take and because would diverge as .
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4: 1.6 Vectors and Vector-Valued Functions
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►The divergence of a differentiable vector-valued function is
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§1.6(iv) Path and Line Integrals
… ►The path integral of a continuous function is … ►The integral of a continuous function over a surface is … ►Gauss’s (or Divergence) Theorem
…5: 2.11 Remainder Terms; Stokes Phenomenon
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►For divergent expansions the situation is even more difficult.
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►As an example consider
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►However, regardless whether we can bound the remainder, the accuracy achievable by direct numerical summation of a divergent asymptotic series is always limited.
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►From §8.19(i) the generalized exponential integral is given by
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►The transformations in §3.9 for summing slowly convergent series can also be very effective when applied to divergent asymptotic series.
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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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Gaussian-transform method for molecular integrals. I. Formulation for energy integrals.
J. Chem. Phys. 43 (2), pp. 398–414.
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Algorithm AS 239. Chi-squared and incomplete gamma integral.
Appl. Statist. 37 (3), pp. 466–473.
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Non-linear integral equations for Heun functions.
Proc. Edinburgh Math. Soc. (2) 16, pp. 281–289.
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The Use of Integral Transforms.
McGraw-Hill, New York.
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7: 22.19 Physical Applications
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►The period is .
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►As from below the period diverges since are points of unstable equilibrium.
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►As from below the period diverges since is a point of unstable equlilibrium.
…As from above the period again diverges.
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§22.19(iv) Tops
…8: 22.3 Graphics
9: Bibliography H
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Une -intégrale de Selberg et Askey.
SIAM J. Math. Anal. 19 (6), pp. 1475–1489.
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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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Elliptic Integrals.
Dover Publications Inc., New York.
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Divergent Series.
Clarendon Press, Oxford.
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Definitions of Stieltjes Integrals of the Riemann Type.
Amer. Math. Monthly 45 (5), pp. 265–278.
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