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 Sep 17 comment Show that $\int\nolimits^{\infty}_{0} x^{-1} \sin x dx = \frac\pi2$ Yup! :) ... I'm using it to appetize my tummy. :) ... Once I finish the chapter I plan on using the standard textbooks I have. Sep 17 revised Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$? Minor spelling, punctuation and formatting edits. Sep 17 suggested approved edit on Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$? Sep 17 comment Show that $\int\nolimits^{\infty}_{0} x^{-1} \sin x dx = \frac\pi2$ @J.M.: Thanks! :) -cheers Sep 17 revised Show that $\int\nolimits^{\infty}_{0} x^{-1} \sin x dx = \frac\pi2$ added 105 characters in body Sep 17 revised Show that $\int\nolimits^{\infty}_{0} x^{-1} \sin x dx = \frac\pi2$ edited title Sep 17 comment Show that $\int\nolimits^{\infty}_{0} x^{-1} \sin x dx = \frac\pi2$ @Ragib: You needn't worry about my background. I will keep the response as a record for perusal as I learn the subject more. ... (However, cursorily, I've encountered the concept- albeit in a very sketchy manner - in Penrose's book.) Sep 17 revised Show that $\int\nolimits^{\infty}_{0} x^{-1} \sin x dx = \frac\pi2$ added 204 characters in body Sep 17 comment Show that $\int\nolimits^{\infty}_{0} x^{-1} \sin x dx = \frac\pi2$ No. But I'm currently reading through Ch. 7 of Penrose's book: there's a modest but useful bit of info on the topic. Plus, I have several ugrad CA books- Complex Analysis, 3rd. [J. Bak, D. J. Newman] (Springer, 2010) amongst them. Sep 17 comment Show that $\int\nolimits^{\infty}_{0} x^{-1} \sin x dx = \frac\pi2$ Far far?: Null. I need a full, correct answer to pore over to get an idea of the underlying concepts. (I'm using this as a pedagogical question. [Ref: R. Penrose, The Road to Reality, Ch. 7, p. 129, Problem [7.5]]). Sep 17 comment Show that $\int\nolimits^{\infty}_{0} x^{-1} \sin x dx = \frac\pi2$ How can I demonstrate the equality using the approach given. Sep 17 asked Show that $\int\nolimits^{\infty}_{0} x^{-1} \sin x dx = \frac\pi2$ Sep 16 comment Historical basis and mathematical significance of Riemann surfaces [Correction 2: "all has some link with the Riemann surface concept." --> "all have some link with the Riemann surface concept."] Sep 16 comment Historical basis and mathematical significance of Riemann surfaces [Correction: "complex analytic ideas are central (if not per-se the -only- available set of ideas)" --> "complex analytic ideas are central (if not the -only- available set of ideas)"] Sep 16 comment Historical basis and mathematical significance of Riemann surfaces which link all these together [Algebra-Geometry-Analysis (and also Number Theory, but not sure what / how Riemann surfaces apply to here / are applied here.] Sep 16 comment Historical basis and mathematical significance of Riemann surfaces ... (i.e. the realization that the Parallel Postulate is not absolutely essential) - leading to non-Euclidean geometries of various sorts, and the use of the concept of "manifold" (which, - if I'm not mistaken again - came about via the notion of Riemann surface) as a fundamental object to defining and studying these various geometries; and c) the study of analysis - from real analysis to complex analysis to whatever the field of Analysis has developed to in its current stage - all has some link with the Riemann surface concept. --- I want to to get some idea of the mathematical threads ... Sep 16 comment Historical basis and mathematical significance of Riemann surfaces @Zhen, please elaborate: "the theory of Riemann surfaces sits at the confluence of complex analysis, differential geometry, and algebraic geometry"... From what I understand- a) within Algebra, the problem of finding solutions to quadratic equations led to the introduction of the complex number $i$; progressively, this led to the field of "complex" analysis. .. Now, complex analytic ideas are central (if not per-se the -only- available set of ideas) to help prove the Fundamental Theorem of Algebra; b) the realization that there may exist geometries other than that of Euclid... Sep 16 accepted Primer on complex analysis and Riemann surfaces for undergraduate physics / theoretical physics majors Sep 16 awarded Scholar Sep 15 revised Historical basis and mathematical significance of Riemann surfaces added 112 characters in body; edited tags