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Equality of definite integrals, relation between integrands

  1. Jun 12, 2014 #1
    Suppose we are given two functions:

    [tex]f:\mathbb R \times \mathbb C \rightarrow\mathbb C[/tex]
    [tex]g:\mathbb R \times \mathbb C \rightarrow\mathbb C[/tex]

    and the equation relating the Stieltjes Integrals

    [tex]\int_a^\infty f(x,z)d\sigma(x)=\int_a^\infty g(x,z)d\rho(x)[/tex]

    where a is some real number, the distributional derivatives of [itex]\sigma[/itex] and [itex]\rho[/itex] exist almost everywhere(they have a countable number of jump discontinuities), and z is a complex number, usually seen as the parameter on which the convergence of the integrals depends. Formulation of infinite sums in this fashion is useful in a variety of areas, in particular in analytic number theory. Von Mangoldt actually used this method to verify Riemann's explicit formula for the prime counting function.

    My question is, in general, are there any useful relations between [itex]f(x)\frac{d\sigma(x)}{dx}[/itex] and [itex]g(x)\frac{d\rho(x)}{dx}[/itex] that can be obtained directly from the equality of the integrals?

    i.e. although the sums are equal, this does not in general imply term by term equality.

    But, for instance, does the equality of the integrals suggest existence of functional expansions of one integrand in terms of the other that can be obtained through a suitable inversion transform?
     
  2. jcsd
  3. Jun 13, 2014 #2

    mathman

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    Gold Member

    Are the integrals equal for just a particular value of a, or for all values of a? If for one particluar value, there is nothing you can say about the relations. On the other hand if the integrals are equal for all values of a, the expressions are equal almost everywhere.
     
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