Basic Analysis I — Real Analysis · Problem 5.5.13

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · improper-integrals-cauchy-principal-value

Problem

a) Compute p.v.∫−111/x dxp.v.\int_{-1}^{1} 1/x \, dx. b) Compute lim⁡ϵ→0+(∫−1−ϵ1/x dx+∫2ϵ11/x dx)\lim_{\epsilon \to 0^+}\left(\int_{-1}^{-\epsilon} 1/x \, dx + \int_{2\epsilon}^{1} 1/x \, dx\right) and show it is not equal to the principal value. c) Show that if ff is integrable on [a,b][a, b], then p.v.∫abf=∫abfp.v.\int_a^b f = \int_a^b f (for an arbitrary c∈(a,b)c \in (a,b)). d) Suppose f:[−1,1]→Rf : [-1,1] \to \mathbb{R} is an odd function (f(−x)=−f(x)f(-x) = -f(x)) that is integrable on [−1,−ϵ][-1, -\epsilon] and [ϵ,1][\epsilon, 1] for all ϵ>0\epsilon > 0. Prove that p.v.∫−11f=0p.v.\int_{-1}^{1} f = 0. e) Suppose f:[−1,1]→Rf : [-1,1] \to \mathbb{R} is continuous and differentiable at 00. Show that p.v.∫−11f(x)x dxp.v.\int_{-1}^{1} \frac{f(x)}{x} \, dx exists. (Recall: It is sometimes desirable to assign a value to integrals that normally cannot be interpreted even as improper integrals, e.g. ∫−111/x dx\int_{-1}^{1} 1/x \, dx. Suppose f:[a,b]→Rf : [a,b] \to \mathbb{R} is a function and a<c<ba < c < b, where ff is Riemann integrable on the intervals [a,c−ϵ][a, c-\epsilon] and [c+ϵ,b][c+\epsilon, b] for all ϵ>0\epsilon > 0. Define the Cauchy principal value of ∫abf\int_a^b f as p.v.∫abf≔lim⁡ϵ→0+(∫ac−ϵf+∫c+ϵbf),p.v.\int_a^b f \coloneqq \lim_{\epsilon \to 0^+} \left( \int_a^{c-\epsilon} f + \int_{c+\epsilon}^b f \right), if the limit exists.)

Related theory

Browse the problem library