Some interdependencies of sum and product integrals

Helton, Jon C.

doi:10.1090/S0002-9939-1973-0308340-5

Some interdependencies of sum and product integrals
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by Jon C. Helton

Proc. Amer. Math. Soc. 37 (1973), 201-206

DOI: https://doi.org/10.1090/S0002-9939-1973-0308340-5

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Abstract:

In a recent paper Davis and Chatfield show if $\smallint _a^b{G^2} = 0$, then $\smallint _a^bG$ exists if and only if $\prod \nolimits _a^b {(1 + G)}$ exists and is not zero. In this paper we extend that result and prove if $\beta > 0,|G| < 1 - \beta$ on $[a,b]$ and $\smallint _a^b{G^2}$ exists, then $\smallint _a^bG$ exists if and only if $\prod \nolimits _a^b {(1 + G)}$ exists and is not zero. Furthermore, if $\prod \nolimits _x^y {(1 + G)} = \exp \smallint _x^yG$ for $a \leqq x < y \leqq b$, then $\smallint _a^b{G^2} = 0$.

References

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