Integration
Theory| Integration
Theory |
In General > s.a. analysis; measure
theory.
* Riemann integral: >
see MathWorld page, Wikipedia page.
* Lebesgue integral: >
see MathWorld page, Wikipedia page.
* Stieltjes integral: The
Stieltjes integral of a bounded function f on [a, b]
with respect to another bounded function 
is the
limit of the Riemann sum of f(
i)[(xi+1) – (xi)]
for a partition a < x1 < x2 < ... < xn–1 < b of
the interval, as the width of the subintervals goes to zero, if the limit
is well-defined; > s.a. MathWorld page.
@ Texts: McShane 44; Whitney 57; Bourbak 63i; Royden 63; ATaylor 65;
Bartle 66; Descombes 73; Halmos 74; Marle 74; McShane 83; Carathéodory
86 [algebraic theory]; Swartz
94; Väth
02 [III]; Kurtz & Swartz 04.
@ Tables: Magnus & Oberhettinger 43; Petit Bois 61; Abramowitz & Stegun
ed-65; Gradshteyn & Ryzhik 93; Gradshteyn et al 00.
@ Handbook: Zwillinger 92.
Special Integrals > s.a. Fresnel
Integrals; gaussian
integrals; Hypergeometric
Functions; Special Functions;
trigonometry.
* Poisson's second integral:
The relationship, which can be derived from
Sonine's integral,
Jn(z) =
(i/2)n / [
(n+1/2)
(1/2)] 
0Pi
d
cos(z cos)
sin2n ;
Special cases: For n = 0, one gets Parseval's integral.
@ References: Tung & Jódar AML(06)mp/04 [dilogarithmic
double integrals]; Chmutov & Duzhin m.GT/05-in
[Kontsevich
integral].
Special Techniques and Related Topics > s.a. gauss-bonnet
theorem;
Integral Transforms; vector
calculus [Green's theorem].
* Steepest-descent approximation:
A method for calculating integrals
over R, in which one approximates the measure d
(t)
= exp[–f(t)]
dt by dsd(t)
= 
i exp[–f(xi) – f''(xi) t2/2]
dt, where xi are
the minima of f; > It is related to the Stationary-Phase
Approximation.
@ Steepest-descent approximation: Koshkarov TMP(95)
[for path integrals]; > s.a. Wikipedia page.
@ Other techniques: González & Moll a0812 [method
of brackets]; Harnad & Orlov TMP(09) [fermionic approach, rational symmetric
functions].
> Applications: see minisuperspace
quantum cosmology.
Generalizations > s.a. integration
on manifolds [including
Stokes' theorem]; lie groups.
* Fractional integrals: Developed by Riemann-Liouville.
@ Infinite-dimensional spaces: Kolmogorov 36; DeWitt-Morette CMP(72), CMP(74);
> s.a. space of connections.
@ Fractional integrals: Bateman 54 v2, ch XIII; Nigmatullin TMP(92),
comm Rutman TMP(94);
Rutman TMP(95) [interpretation];
Kobelev
m.CA/00 [generalization];
Podlubny FCAA(02)m.CA/01 [interpretation].
@ Henstock / Kurzweil: Kurzweil 00; Swartz 01 [& McShane].
@ Related topics: Henstock 88 [generalized Riemann]; Novak JoC(01)qp/00 [quantum
algorithms]; Suzuki a0806 [negative-dimensional integration].
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send feedback and suggestions to bombelli at olemiss.edu – modified 27
aug 2009