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1: 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. …
2: 36.14 Other Physical Applications
§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
We cannot take f = x and g = ln x because g f - 1 / 2 d x would diverge as x + . …
4: 1.6 Vectors and Vector-Valued Functions
The divergence of a differentiable vector-valued function F = F 1 i + F 2 j + F 3 k 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
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
  • D. Shanks (1955) Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, pp. 1–42.
  • I. Shavitt and M. Karplus (1965) Gaussian-transform method for molecular integrals. I. Formulation for energy integrals. J. Chem. Phys. 43 (2), pp. 398–414.
  • B. L. Shea (1988) Algorithm AS 239. Chi-squared and incomplete gamma integral. Appl. Statist. 37 (3), pp. 466–473.
  • B. D. Sleeman (1969) Non-linear integral equations for Heun functions. Proc. Edinburgh Math. Soc. (2) 16, pp. 281–289.
  • I. N. Sneddon (1972) The Use of Integral Transforms. McGraw-Hill, New York.
  • 7: 22.19 Physical Applications
    The period is 4 K ( sin ( 1 2 α ) ) . … As a 1 / β from below the period diverges since a = ± 1 / β are points of unstable equilibrium. … As a 2 / β from below the period diverges since x = 0 is a point of unstable equlilibrium. …As | a | 1 / β from above the period again diverges. …
    §22.19(iv) Tops
    8: 22.3 Graphics
    The period diverges logarithmically as k 1 - ; see §19.12. …
    See accompanying text
    Figure 22.3.16: sn ( x + i y , k ) for k = 0.99 , - 3 K x 3 K , 0 y 4 K . K = 3.3566 , K = 1.5786 . Magnify 3D Help
    See accompanying text
    Figure 22.3.17: cn ( x + i y , k ) for k = 0.99 , - 3 K x 3 K , 0 y 4 K . K = 3.3566 , K = 1.5786 . Magnify 3D Help
    See accompanying text
    Figure 22.3.18: dn ( x + i y , k ) for k = 0.99 , - 3 K x 3 K , 0 y 4 K . K = 3.3566 , K = 1.5786 . Magnify 3D Help
    See accompanying text
    Figure 22.3.19: cd ( x + i y , k ) for k = 0.99 , - 3 K x 3 K , 0 y 4 K . K = 3.3566 , K = 1.5786 . Magnify 3D Help
    9: Bibliography H
  • L. Habsieger (1988) Une q -intégrale de Selberg et Askey. SIAM J. Math. Anal. 19 (6), pp. 1475–1489.
  • P. I. Hadži (1975a) Certain integrals that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1975 (2), pp. 86–88, 95 (Russian).
  • H. Hancock (1958) Elliptic Integrals. Dover Publications Inc., New York.
  • G. H. Hardy (1949) Divergent Series. Clarendon Press, Oxford.
  • I. D. Hill (1973) Algorithm AS66: The normal integral. Appl. Statist. 22 (3), pp. 424–427.
  • 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 ( x ) are described in Miller (1960) for x > 0 and integer p . …