Dirac Delta Function Cheat Sheet
1. Basic Property (Sifting Property)
∫ f(t) δ(t − a) dt = f(a)
Works only if "a" lies within the limits of integration.
2. Delta Function Value
- δ(t) = 0 for t ≠ 0
- ∫ δ(t) dt = 1
3. Shifting Property
δ(t − a) shifts the impulse to t = a
4. Scaling Property
δ(at) = (1 / |a|) δ(t)
5. Multiplication Property
f(t) δ(t − a) = f(a) δ(t − a)
6. Convolution with Delta
x(t) * δ(t) = x(t)
7. Common Exam Trick
∫ₐᵇ f(t) δ(t − c) dt = 0 if c is outside [a, b]
8. Quick Examples
- ∫₋₁² t² δ(t) dt = 0
- ∫₋₁² (t² + 5) δ(t) dt = 5