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)