an 2 00 1 Necessary and sufficient conditions for differentiating under the integral sign
نویسنده
چکیده
a f(x, y) dy, it is often important to know when F is differentiable and when F (x) = ∫ b a f1(x, y) dy. A sufficient condition for differentiating under the integral sign is that ∫ b a f1(x, y) dy converges uniformly; see [6, p. 260]. When we have absolute convergence, the condition |f1(x, y)| ≤ g(y) with ∫ b a g(y) dy < ∞ suffices (Weierstrass M-test and Lebesgue Dominated Convergence). If we use the Henstock integral, then it is not difficult to give necessary and sufficient conditions for differentiating under the integral sign. The conditions depend on being able to integrate every derivative. If g : [a, b] → R is continuous on [a, b] and differentiable on (a, b) it is not always the case that g is Riemann or Lebesgue integrable over [a, b]. However, the Henstock integral integrates all derivatives and thus leads to the most complete version of the Fundamental Theorem of Calculus. The Henstock integral’s definition in terms of Riemann sums is only slightly more complicated than for the Riemann integral (simpler than the improper Riemann integral), yet it includes the Riemann, improper Riemann, and Lebesgue integrals as special cases. Using the very strong version of the Fundamental Theorem we can formulate necessary and sufficient conditions for differentiating under the integral sign.
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