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...

The Method of Differentiating under the Integral Sign

The result was that, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn’t do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a ...

Necessary and Sufficient Conditions for Differentiating under the Integral Sign

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...

On Diffraction Fresnel Transforms for Boehmians

and Applied Analysis 3 Proof. Let ξ be fixed. If φ t is in S, then its diffraction Fresnel transform certainly exists. Moreover, differentiating the right-hand side of 2.3 with respect to ξ, under the integral sign, ktimes, yields a sum of polynomials, pk t ξ , say of combinations of t and ξ. That is, ∣ ∣ ∣ ∣ ∣ d dtk Fd ( φ ) ξ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ pk t ξ φ t exp ( i ( α1t 2 − 2tξ α2ξ ) 2γ1 )∣ ∣...