21.11 Problem. Let a > 0 and let f(x) = ea². Show that f is integrable. [Hint: if |x|≥ 1, then |f(x)|≤e-all. How does this help? u(x,t) = √ √ √ û(k, t)e³k² dk 1 ikx √2π La F (k)e-1² teikz dk. (22.1) 2πT This may well be a valid candidate for a solution formula! 22.4 Problem. (i) Suppose that f: RC is continuous with |f| integrable. Fix t > 0 and x E R. Prove that g(k) = e-k²teik is integrable, so the integral on the right in (22.1) converges. [Hint: Problem 21.11.] =e (ii) Assume that we may differentiate under the integral on the right in (22.1) with respect to x and t as much as we want for x R and t> 0. Show that u as defined by (22.1) satisfies ut = Uxx' (iii) Show that u as defined by (22.1) meets u(x, 0) = f(x). [Hint: Untheorem 22.2.] 22.2 Untheorem. Let f: R→C be "nice." Then f(x)= = 1 √2πT Lo ƒ (k)eikz dk.

21.11 Problem. Let a > 0 and let f(x) = ea². Show that f is integrable. [Hint: if |x|≥ 1, then |f(x)|≤e-all. How does this help? u(x,t) = √ √ √ û(k, t)e³k² dk 1 ikx √2π La F (k)e-1² teikz dk. (22.1) 2πT This may well be a valid candidate for a solution formula! 22.4 Problem. (i) Suppose that f: RC is continuous with |f| integrable. Fix t > 0 and x E R. Prove that g(k) = e-k²teik is integrable, so the integral on the right in (22.1) converges. [Hint: Problem 21.11.] =e (ii) Assume that we may differentiate under the integral on the right in (22.1) with respect to x and t as much as we want for x R and t> 0. Show that u as defined by (22.1) satisfies ut = Uxx' (iii) Show that u as defined by (22.1) meets u(x, 0) = f(x). [Hint: Untheorem 22.2.] 22.2 Untheorem. Let f: R→C be "nice." Then f(x)= = 1 √2πT Lo ƒ (k)eikz dk.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.6: The Inverse Trigonometric Functions

Problem 92E

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