Necessary and Sufficient Conditions for Differentiating under the Integral Sign

an 2 00 1 Necessary and sufficient conditions for differentiating under the integral sign

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...

Some Divergent Trigonometric Integrals

1. Introduction. Browsing through an integral table on a dull Sunday afternoon some time ago, I came across four divergent trigonometric inte-grals. (See (1) and (2) below.) I was intrigued as to how these divergent integrals ended up in a respectable table. Tracing their history, it turned out they were originally " evaluated " when some convergent integrals, (5) and (6), were differentiated u...

Adaptive and intelligent control of permanent magnet synchronous motor (PMSM) using a combination of fuzzy logic and gray wolf algorithm under fault condition

Nowadays, permanent magnet synchronous motors have been widely used in industry due to the elimination of excitation losses, longer life and higher efficiency. Errors in engine and drive systems are unavoidable during operation. Therefore, a suitable scenario should be considered for when these systems fail. If the necessary predictions and control algorithms are not considered for the error co...

Double Integral Characterization for Bergman Spaces

&lrm;In this paper we characterize Bergman spaces with&lrm; &lrm;respect to double integral of the functions $|f(z)&lrm; &lrm;-f(w)|/|z-w|$,&lrm; &lrm;$|f(z)&lrm; -&lrm;f(w)|/rho(z,w)$ and $|f(z)&lrm; &lrm;-f(w)|/beta(z,w)$,&lrm; &lrm;where $rho$ and $beta$ are the pseudo-hyperbolic and hyperbolic metrics&lrm;. &lrm;We prove some necessary and sufficient conditions that implies a function to be...

Differentiating Under the Integral Sign