Quantum Mechanics for Mathematicians

C+

C−

ω=−i

Figure 12.2: Evaluatingθ(t) via contour integration.

Fort > 0 , one instead closes the path usingC−in the lower half-plane, and
finds that the integral can be evaluated in terms of the residue of the pole at
ω=−i(with the minus sign coming from orientation of the curve), giving

θ(t) =− 2 πi

(

i

2 π

)

= 1

By similar arguments one can show thatθ(−t) has (as a distribution) Fourier
transform

lim

→ 0 +

−

i

√

2 π

1

ω−i

and the integral representation

θ(−t) = lim

→ 0 +

−

i

2 π

∫+∞

−∞

1

ω−i

e−iωtdω

Taking 1/

√

2 πtimes the sum of the Fourier transforms forθ(t) andθ(−t) gives
the distribution

lim

→ 0 +

i

2 π

(

1

ω+i

−

1

ω−i

)

= lim

→ 0 +

i

2 π

− 2 i

ω^2 +^2

= lim

→ 0 +

1

π



ω^2 +^2

=δ(ω) (12.11)