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The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics.It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.
Feynman parametrization. Feynman parametrization is a technique for evaluating loop integrals which arise from Feynman diagrams with one or more loops. However, it is sometimes useful in integration in areas of pure mathematics as well.
Since the series converges uniformly on the support of the integration path, we are allowed to exchange integration and summation. The series of the path integrals then collapses to a much simpler form because of the previous computation. So now the integral around C of every other term not in the form cz −1 is zero, and the integral is ...
The method of steepest descent is a method to approximate a complex integral of the form for large , where and are analytic functions of . Because the integrand is analytic, the contour can be deformed into a new contour without changing the integral. In particular, one seeks a new contour on which the imaginary part, denoted , of is constant ...
Integral of the Gaussian function, equal to sqrt (π) A graph of the function and the area between it and the -axis, (i.e. the entire real line) which is equal to . The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function over the entire real line. Named after the German mathematician Carl ...
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. [1][2][3] Contour integration is closely related to the calculus of residues, [4] a method of complex analysis. One use for contour integrals is the evaluation of integrals along the real line that are ...
Path integration. Path integration sums the vectors of distance and direction traveled from a start point to estimate current position, and so the path back to the start. Path integration is the method thought to be used by animals for dead reckoning.
In the calculus of variations and classical mechanics, the Euler–Lagrange equations[1] are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange.