By Sonia Mazzucchi

Even if greater than 60 years have handed because their first visual appeal, Feynman course integrals have not begun to lose their fascination and luster. they aren't just a bold tool of theoretical physics, but additionally a mathematical problem; actually, numerous mathematicians within the final forty years have dedicated their efforts to the rigorous mathematical definition of Feynman's principles. This quantity presents an in depth, self-contained description of the mathematical problems in addition to the prospective ideas used to unravel those problems. specifically, it provides an entire review of the mathematical consciousness of Feynman direction integrals by way of well-defined useful integrals, that's, the endless dimensional oscillatory integrals. It comprises the normal effects at the subject in addition to the more moderen advancements got through the writer. Mathematical Feynman course Integrals and Their purposes is dedicated to either mathematicians and physicists, graduate scholars and researchers who're drawn to the matter of mathematical foundations of Feynman direction integrals

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N−1 ) (r) , with Θ(r) = 1 for r ≥ 0 and Θ(r) = 0 for r < 0. Let us introduce the notation k ≡ |k|fk (φ1 , . . , φn−1 ), aj ≡ aj (φ1 , . . , φn−1 ), j = 0, . . , 2M , j P (r) = 2M ∈ C, = | |eiφ , with −π ≤ φ ≤ 0. j=0 aj r and Let us consider the complex plane and set z = ρeiθ . 20) is absolutely convergent, while if ∈ R \ {0} it needs a regularization. If ∈ R, > 0 we have +∞ eik r e i P (r) n−1 r ↓0 0 while if +∞ 0 eik z e dr = lim i P (z) n−1 z dz z=ρei <0 eik r e i P (r) n−1 r eik z e dr = lim ↓0 i P (z) n−1 z dz .

N−1 ) ∈ Sn−1 . The function fk : Sn−1 → [−1, 1] is defined by k x = fk (φ1 , . . , φn−1 ) . ,φn−1 ) (r) , with Θ(r) = 1 for r ≥ 0 and Θ(r) = 0 for r < 0. Let us introduce the notation k ≡ |k|fk (φ1 , . . , φn−1 ), aj ≡ aj (φ1 , . . , φn−1 ), j = 0, . . , 2M , j P (r) = 2M ∈ C, = | |eiφ , with −π ≤ φ ≤ 0. j=0 aj r and Let us consider the complex plane and set z = ρeiθ . 20) is absolutely convergent, while if ∈ R \ {0} it needs a regularization. If ∈ R, > 0 we have +∞ eik r e i P (r) n−1 r ↓0 0 while if +∞ 0 eik z e dr = lim i P (z) n−1 z dz z=ρei <0 eik r e i P (r) n−1 r eik z e dr = lim ↓0 i P (z) n−1 z dz .

P (x) = A2M (x, . . ,x) ), δ > 0, as t → ∞. g(|x|), March 27, 2009 10:32 World Scientific Book - 9in x 6in 30 MathematicalFeynman Mathematical Feynman Path Integrals and Applications Then the limit of regularized integrals: lim ↓0 e i P (xei ) f (xei )dx, 0 < < π/4M, >0 is given by: einπ/4M e i P (eiπ/4M x) f (eiπ/4M x)dx . 27) Rn The latter integral is absolutely convergent and it is understood in Lebesgue sense. 3 includes for instance the polynomials of any degree and the exponentials. 1. This is, however, not necessarily true for all f ∈ Sλ∞ satisfying (1) and (2).

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