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Does anyone know, where I can find the proof of necessary and sufficient conditions for differentiating under the integral sign in case of Henstock integral? Here are the theorems but not all the proofs:

Necessary and sufficient conditions for differentiating under the integral sign
Erik Talvila
arXiv:math/0101012

We give necessary and sufficient conditions for differentiating under the integral sign an integral that depends on a parameter. The conditions require the equality of two iterated integrals and depend on being able to integrate every derivative. The Henstock integral is thus used in an essential way.

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Bartle's A modern theory of integration has it on pp. 199-200.

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  • $\begingroup$ Is it really a Henstock integral in the book? The theorem looks different. $\endgroup$ – green113 May 14 '15 at 8:49
  • $\begingroup$ @green113: Yes, it is. Scroll to page 1, "The purpose of this monograph is to present an exposition of a relatively new theory of the integral (variously called the "generalized Riemann integral", the "gauge integral", the "Henstock-Kurzweil integral", etc.)". $\endgroup$ – Francois Ziegler May 14 '15 at 11:34

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