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Glasser's Master Theorem

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The identity

PVint_(-infty)^inftyF(phi(x))dx=PVint_(-infty)^inftyF(x)dx
(1)

holds for any integrable function F(x) and phi(x) of the form

phi(x)=|a|x-sum_(n=1)^N(|alpha_n|)/(x-beta_n),
(2)

with a, {alpha_n}_(n=1)^N, and {beta_n}_(n=1)^N arbitrary constants (Glasser 1983). Here, PV denotes a Cauchy principal value. This generalized the result known to Cauchy that

PVint_(-infty)^inftyF(u)dx=int_(-infty)^inftyF(x)dx,
(3)

where u=x-1/x.

See also

Ramanujan's Master Theorem

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References

Glasser, M. L. "A Remarkable Property of Definite Integrals." Math. Comput. 40, 561-563, 1983.

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Glasser's Master Theorem

Cite this as:

Weisstein, Eric W. "Glasser's Master Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GlassersMasterTheorem.html

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