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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