The method of differentiating under the integral sign

Differentiating Under the Integral Sign

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

buckling of viscoelastic composite plates using the finite strip method

در سال های اخیر، تقاضای استفاده از تئوری خطی ویسکوالاستیسیته بیشتر شده است. با افزایش استفاده از کامپوزیت های پیشرفته در صنایع هوایی و همچنین استفاده روزافزون از مواد پلیمری، اهمیت روش های دقیق طراحی و تحلیل چنین ساختارهایی بیشتر شده است. این مواد جدید از خودشان رفتارهای مکانیکی ارائه می دهند که با تئوری های الاستیسیته و ویسکوزیته، نمی توان آن ها را توصیف کرد. این مواد، خواص ویسکوالاستیک دارند....

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...