Mathematical Tools for Physics

15—Fourier Analysis 459

Here I’ve introduced the occasionally useful notation thatF(f)is the Fourier transform off. The boundary
terms in the partial integration will go to zero if you assume that the functionfapproaches zero at infinity.
Thenthtime derivative simply give you more factors:(−iω)non the transformed function.

15.5 Green’s Functions
This technique showed up in the chapter on ordinary differential equations, section4.5, as a method to solve
the forced harmonic oscillator. In that instance I said that you can look at a force as a succession of impulses,
as if you’re looking at the atomic level and visualizing a force as many tiny collisions by atoms. Here I’ll get
to the same sort of result as an application of transform methods. The basic technique is to Fourier transform
everything in sight.
The damped, forced harmonic oscillator differential equation is

m

d^2 x

dt^2

+b

dx

dt

+kx=F 0 (t) (12)

Multiply byeiωtand integrate over all time. You do the transforms of the derivatives by partial integration as in
Eq. ( 11 ).
∫∞

−∞

dteiωt[Eq. ( 12 )]=−mω^2 x ̃−ibω ̃x+k ̃x=F ̃ 0 , where x ̃(ω) =

∫∞

−∞

dteiωtx(t)

This is an algebraic equation that I can solve for the function ̃x(ω).

x ̃(ω) =

F ̃ 0 (ω)

−mω^2 −ibω+k

Now use the inverse transform to recover the functionx(t).

x(t) =

∫∞

−∞

dω

2 π

e−iωt ̃x(ω) =

∫

dω

2 π

e−iωt

F ̃ 0 (ω)

−mω^2 −ibω+k

=

∫

dω

2 π

e−iωt

−mω^2 −ibω+k

∫

dt′F 0 (t′)eiωt

′

=

∫

dt′F 0 (t′)

∫

dω

2 π

e−iωt

−mω^2 −ibω+k

eiωt

′

(13)