Mathematical Methods for Physicists: A concise introduction - Site Map

FOURIER SERIES AND INTEGRALS
4.10 Find the exponential form of the Fourier series of the function whose de®nition
in one period is f …x† ˆe x ; 1 < x < 1.
4.11 (a) Show that the set of functions
1; sin x
L ;
x
cos
L ;
2x
sin
L ;
2x
cos
L ;
3x
sin
L ;
3x
cos
L ; ...
form an orthogonal set in the interval …L; L†.
(b) Determine the corresponding normalizing constants for the set in (a) so
that the set is orthonormal in …L; L†.
4.12 Express f …x; y† ˆxy as a Fourier series for 0 x 1; 0 y 2.
4.13 Steady-state heat conduction in a rectangular plate: Consider steady-state
heat conduction in a ¯at plate having temperature values prescribed on the
sides (Fig. 4.22). The boundary value problem modeling this is:
@ 2 u
@ 2 x 2 ‡ @2 u
@ 2 ˆ 0; 0 < x