If a d‐dimensional integral involves an integrand of the functional form F( f1(x1)+f2(x2)+⋅⋅⋅), then one can introduce an integral transform (Fourier or Laplace or variants on those) which allows all the integrals over the coordinates xi to factor. Thus a d‐dimensional integral is reduced to a one‐dimensional integral over the transform variable. This is shown to be a very powerful and practical numerical approach to a number of problems of interest. Among the examples studied is the computation of the volume of phase space for an arbitrary collection of relativistic particles. One important aspect of the approach involves numerical integration along various contours in the complex plane.
REFERENCES
1.
2.
3.
G. M. Fichtenholz, Differential‐und Integralrechnung, translated from the Russian (Deutscher Verlag, Berlin, 1964), Vol. III, pp. 383–407. Almost all of the multidimensional integrals listed by Gradshteyn and Ryzhik in their “Tables” come from this source.
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5.
There were some earlier attempts to use integral transform techniques for the phase space problem but they apparently did not succeed very far. See
M.
Kretzschmar
, Ann. Rev. Nucl. Sci.
11
, 1
(1961
).6.
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
7.
8.
P. J. Davis and P. Rabinowitz, Methods of Numerical Integration (Academic, New York, 1975), Chap. 5.
9.
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© 1985 American Institute of Physics.
1985
American Institute of Physics
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