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A Generalisation of Dirichlet's Multiple Integral

Published online by Cambridge University Press:  20 January 2009

Henry Jack
Henry Jack
Affiliation:
QUeen'S College, Dundee
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A previous note (2) showed how the integral of f(1x1+2x2++nxn) over the interior of a simplex could be reduced to a contour integral. The same idea is applied here in Theorems 1 and 2 to give a generalisation of Dirichlet's multiple integral ((1), pp. 169172). These results are then used in Theorem 3 to reduce an integral over all real n-dimensional space to a contour integral. In Theorem 4 an integral over the group of all 33 orthogonal matrices of determinant 1 is reduced to a contour integral. This result can be extended formally to the case of 44 matrices; beyond this it seems difficult to go.

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 14 , Issue 3 , June 1965 , pp. 233 - 237
Copyright
Copyright Edinburgh Mathematical Society 1965

References

REFERENCES

(1) Edwards, J., A Treatise on the Integral Calculus, Vol. II (London, 1922).Google Scholar
(2) Jack, H., An integral over the interior of a simplex, Proc. Edin. Math. Soc. (2), 13 (1962), 167171.CrossRefGoogle Scholar
(3) Ponting, F. W. and Potter, H. S. A., The volume of orthogonal and unitary space, Quart. J. Math. (Oxford), 20 (1949), 146154CrossRefGoogle Scholar