Legendre Polynomials on Chen Kai Bloghttps://www.chenk.top/en/tags/legendre-polynomials/Recent content in Legendre Polynomials on Chen Kai BlogHugoenThu, 07 Sep 2023 09:00:00 +0000Ordinary Differential Equations (5): Power Series and Special Functionshttps://www.chenk.top/en/ode/05-laplace-transform/Thu, 07 Sep 2023 09:00:00 +0000https://www.chenk.top/en/ode/05-laplace-transform/<p><strong>Some ODEs have no solutions in terms of familiar functions.</strong> The Bessel equation, the Legendre equation, the Airy equation — all arise naturally in physics (heat conduction in cylinders, gravitational fields of planets, quantum tunneling). Their solutions <em>define</em> entirely new functions. This chapter shows you how to find them using power series, why the Frobenius extension is forced upon us at singular points, and why the same handful of &ldquo;special functions&rdquo; keeps appearing across physics and engineering.</p>