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REVIEW OF SERIES EXPANSION
REVIEW OF SERIES EXPANSION Introduction In the second half of the course, we will focus quite a bit of attention on the use of series expan-sions in physics and mathematics. We will spend several weeks studying Fourier series (Ch. 7 in Boas) , series solutions of differential equations (Ch. 12 of Boas) as well as Legendre series (also Ch. 12 Boas).
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SERIES SOLUTIONS OF DIFFERENTIAL EQUATIONS
www.dslavsk.sites.luc.eduLet’s start with a simple differential equation: ′′− ′+y y y =2 0 (1) We recognize this instantly as a second order homogeneous constant coefficient equation. Just as instantly we realize the characteristic equation has equal roots, so we can write the solution to this equation as: x = + y e A Bx ( ) (2) where A and B are constants ...
LAPLACE’S EQUATION IN SPHERICAL COORDINATES
www.dslavsk.sites.luc.eduThe coordinate systems you will encounter most frequently are Cartesian, cylindrical and spherical polar. We investigated Laplace’s equation in Cartesian coordinates in class and just began investigating its solution in spherical coordinates. Let’s expand that discussion here. We begin with Laplace’s equation: 2V. ∇ = 0 (1)
LAPLACE’S EQUATION IN SPHERICAL COORDINATES
www.dslavsk.sites.luc.eduequation with specific boundary conditions. For this purpose, let’s use the example in Boas pp. 647-649. Without any loss of meaning, we can use talk about finding the potential inside a sphere rather than the temperature inside a sphere. So, let’s assume there is a sphere of radius
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