# divergent integrals

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## 1—10 of 16 matching pages

##### 1: 2.6 Distributional Methods
###### §2.6(i) DivergentIntegrals
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. …
##### 3: 2.7 Differential Equations
We cannot take $f=x$ and $g=\ln x$ because $\int gf^{-1/2}\mathrm{d}x$ would diverge as $x\to+\infty$. …
##### 4: 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(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 …
##### 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 $4K\left(\sin\left(\frac{1}{2}\alpha\right)\right)$. … 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. …
##### 8: 22.3 Graphics
The period diverges logarithmically as $k\to 1-$; see §19.12. … 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 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 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 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 $E_{p}\left(x\right)$ are described in Miller (1960) for $x>0$ and integer $p$. …