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Convolution Theorem
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Diff Eq
Convolution
Convolution Theorem L{f*g} = F(s)G(s)
Compute (f*g)(t) = ∫₀ᵗ f(τ)g(t−τ)dτ and verify that L{f*g} = F(s)·G(s).
Live Calculator · Step-by-Step · Differential Equations
Function Pair
Select f and g
f=1, g=1 → f*g = t
f=t, g=1 → f*g = t²/2
f=eᵃᵗ, g=eᵇᵗ → f*g = (eᵃᵗ−eᵇᵗ)/(a−b)
f=sin(t), g=sin(t) → f*g = (sin t − t cos t)/2
f=eᵗ, g=sin(t) → f*g = (eᵗ−sin t−cos t)/2
Parameter a (for eᵃᵗ)
Parameter b (for eᵇᵗ, b ≠ a)
Compute Convolution
Results
f(t)
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g(t)
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(f*g)(t) — Convolution
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Verification: F(s)·G(s)
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Step-by-Step Work
Select a pair and compute.
f(t), g(t), and (f*g)(t)
f(t)
g(t)
(f*g)(t)