Mathematical Tools for Physics

15—Fourier Analysis 461

∫∞

−∞

dω

2 π

e−iω(t−t

′)

−mω^2 −ibω+k

=− 2 πi

∑

ω±

Res

C 3

C 4

(15)

The denominator in Eq. ( 14 ) is−m(ω−ω+)(ω−ω−). Use this form to compute the residues. Leave the 1 / 2 π
aside for the moment and you have

e−iω(t−t

′)

−mω^2 −ibω+k

=

e−iω(t−t

′)

−m(ω−ω+)(ω−ω−)

The residues of this atω±are the coefficients of these first order poles.

atω+:

e−iω+(t−t

′)

−m(ω+−ω−)

and atω−:

e−iω−(t−t

′)

−m(ω−−ω+)

The explicit values ofω±are

ω+=

−ib+

√

−b^2 + 4km

2 m

and ω−=

−ib−

√

−b^2 + 4km

2 m

Let ω′=

√

−b^2 + 4km

2 m

and γ=

b

2 m

The difference that appears in the preceding equation is then

ω+−ω−= (ω′−iγ)−(−ω′−iγ) = 2ω′