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.
* Riemann-Stieltjes integral:
The Stieltjes integral of a bounded function f on [a, b]
with respect to another bounded function α is the limit of the Riemann
sum of f(ξi)
[α(xi+1) −
α(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;
Wikipedia page.
* Lebesgue-Stieltjes integral: see
Wikipedia page.
@ Texts: Whitney 57;
Bourbaki 63;
Royden 63;
Taylor 65;
Bartle 66;
Descombes 73;
Halmos 74;
Marle 74;
McShane 83;
Carathéodory 86 [algebraic theory];
Swartz 94;
Priestley 97;
Väth 02 [III];
Kurtz & Swartz 11.
@ 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. trigonometry.
* Integrals of exponential functions: \(\int_0^\infty
{\rm d}x\,x^n\,{\rm e}^{-ax} = n! / a^{n+1}\); > s.a. Beta
Function; Fresnel Integrals;
Gamma Function; gaussian integrals.
* Poisson's second integral:
The expression for the Bessel functions, which can be derived from Sonine's integral,
Jn(z) = (i/2)n / [Γ(n+1/2) Γ(1/2)] ∫0π dθ cos(z cosθ) sin2nθ ;
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-en [Kontsevich integral];
Shakirov TMP(10),
Stoyanovsky a1005
[integrals of exponentials of polynomials, and generalized hypergeometric functions];
Mathai & Haubold a1109 [a versatile integral].
> Integrals of exponential functions: see
Beta Function; Fresnel Integrals;
Gamma Function; gaussian integrals.
> Other integrals: see bessel functions;
Borwein Integrals; Hypergeometric Functions;
Special Functions.
Special Techniques and Related Topics > s.a. gauss-bonnet theorem;
Integral Transforms; vector [Green's theorem].
* Steepest-descent approximation:
A method for calculating integrals over \(\mathbb R\), in which one approximates
the measure dσ(t) = exp[−f(t)]
dt by dσsd(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.
* Numerical methods: Some of the common
methods are the Newton-Cotes class of methods (including the trapezoid method and its
Romberg method variant, and Simpson's 1/3 and 3/8 rules, for which the error scales
as a power of the bin width), and the Monte Carlo approach; For 1D integrals, the
Newton-Cotes methods tend to be more efficient, for higher-dimensional integrals,
the Monte Carlo method is.
* Finite-part integration: A method of
evaluating convergent integrals by means of the finite part of divergent integrals.
@ General references:
González & Moll a0812,
González et al a1004 [method of brackets];
Harnad & Orlov TMP(09) [fermionic approach, rational symmetric functions];
Kempf et al JPCS(15)-a1507 [integrating by differentiating];
Temme 14 [asymptotic methods];
Valean 19 [derivations and difficult cases];
Villanueva & Galapon a2012 [finite-part integration].
@ Steepest-descent approximation:
Koshkarov TMP(95) [for path integrals];
> s.a. Wikipedia page.
> Applications:
see minisuperspace quantum cosmology.
> Numerical methods:
see MathWorld page on Newton-Cotes methods;
Wikipedia page on Newton-Cotes methods;
montecarlo method.
> Related topics:
see Cauchy Principal Value; Finite Part Integral;
Hilbert Transform; Reynolds Theorem.
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),
comment 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].
@ Functional integrals: LaChapelle a1308,
a1501 [proposed definition];
Grangé & Werner a1812 [on paracompact manifolds];
> s.a. path integrals.
@ Related topics: Henstock 88 [generalized Riemann];
Novak JoC(01)qp/00 [quantum algorithms];
Suzuki a0806 [negative-dimensional integration];
Gudder RPMP(10),
RPMP(12)-a1105 [quantum integrals and examples].
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