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Phenomena was released theatrically in Italy on 31 January 1985 with a 116-minute running time. [24] This version of Phenomena is often referred to as the "integral cut". [24] A shorter version of the film was prepared for international release that had a 110-minute running time. [24]
the latter integral being defined by the preceding construction. If g is of bounded variation, then it is possible to write = () where g 1 (x) = V x a g is the total variation of g in the interval [a, x], and g 2 (x) = g 1 (x) − g(x). Both g 1 and g 2 are monotone non-decreasing.
However, its integral along a closed path, the Berry phase , is gauge-invariant up to an integer multiple of . Thus, e i γ n {\displaystyle e^{i\gamma _{n}}} is absolutely gauge-invariant, and may be related to physical observables.
The remaining two half-lines in the two X s can be linked to each other in two ways, so that the total number of ways to form the diagram is 4 × 3 × 4 × 3 × 2 × 2, while the denominator is 4! × 4! × 2!. The total symmetry factor is 2, and the contribution of this diagram is divided by 2.
One-loop diagrams are usually computed as the integral over one independent momentum that can "run in the cycle". The Casimir effect , Hawking radiation and Lamb shift are examples of phenomena whose existence can be implied using one-loop Feynman diagrams, especially the well-known "triangle diagram":
In 1882, Gustav Kirchhoff analyzed Fresnel's theory in a rigorous mathematical formulation, as an approximate form of an integral theorem. [ 3 ] : 375 Very few rigorous solutions to diffraction problems are known however, and most problems in optics are adequately treated using the Huygens-Fresnel principle.
If A(p) and B(p) are linear functions of p, then the last integral can be evaluated using substitution. More generally, using the Dirac delta function δ {\displaystyle \delta } : [ 2 ]
Functional integrals arise in probability, in the study of partial differential equations, and in the path integral approach to the quantum mechanics of particles and fields. In an ordinary integral (in the sense of Lebesgue integration ) there is a function to be integrated (the integrand) and a region of space over which to integrate the ...