Mathematical Tools for Physics

15—Fourier Analysis 458

If there are several frequencies, the result is a sum.

g(ω) =

∑

n

Ane−(ω−ωn)

(^2) /σ (^2) n
⇐⇒ f(t) =

∑

n

An

σn

2

√

π

e−iωnte−σ

(^2) nt (^2) / 4
In a more common circumstance you will have the time series,f(t), and will want to obtain the frequency decom-
position,g(ω), though for this example I worked backwards. The function of time is real, but the transformed
functiongis complex. Becausefis real, it follows thatgsatisfiesg(−ω) =g*(ω). See problem 13.
f
Real
Imag
g
This example has four main peaks in the frequency spectrum. The real part ofgis an even function and
the imaginary part is odd.
f
Real
g Imag
This is another example with four main peaks.
In either case, if you simply look at the function of time on the left it isn’t obvious what sort of frequencies
are present. That’s why there are standard, well-developed computer programs to to the Fourier analysis.
15.4 Derivatives
There are a few simple, but important relations involving differentiation. What is the Fourier transform of the
derivative of a function? Do some partial integration.
F(f ̇) =

∫

dteiωt

df

dt

=eiωtf(t)

∣

∣

∣

∞

−∞

−iω

∫

dteiωtf(t) =−iωF(f) (11)