# Prove the commutative properties of the convolution integral f * g = g * f

Prove the commutative properties of the convolution integral f * g = g * f
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Arham Warner
Jeffrey Jordon

a) First prove the commutative property as follows:

$\left(f\cdot g\right)={\int }_{0}^{t}f\left(\tau \right)g\left(t-\tau \right)d\tau$

Now change the integration variable as follows

Let $\sigma =t-\tau$ Then $d\sigma =-dt$

$\left(f\cdot g\right)\left(t\right)={\int }_{t}^{0}f\left(t-\sigma \right)f\left(\sigma \right)\left(-1\right)d\sigma$

$={\int }_{t}^{0}f\left(t-\sigma \right)g\left(\sigma \right)d\sigma$

$=\left(g\cdot f\right)\left(t\right)$

Therefore, $\left(f\cdot g\right)=\left(g\cdot f\right)$