Understanding the Convolution Property in Physics: Derivative Inside an Integral

[radiator]

something I often see without justification in my physics books. What is the justification for the following convolution property (pulling the derivative inside the integral)
$$(f*g)^\prime = \frac{d}{dx} \int f(x) g(x-u) dx = \int f(x) \frac{d}{dx} g(x-u) dx = f(x) * g(x)^\prime$$

 

[jbunniii]

I think you mean
$$(f*g)^\prime(x) = \frac{d}{dx} \int f(u) g(x-u) du = \int f(u) \frac{d}{dx} g(x-u) du = (f * g^\prime)(x)$$
This is a special case of a more general situation. Suppose
$$F(x) = \int_{a}^{b} f(x,y) dy$$
where $a$ and $b$ are constants not dependent on $x$. Then if $f$ is sufficiently nice, we have
$$F'(x) = \int_{a}^{b} \frac{\partial}{\partial x} f(x,y) dy$$
Sufficiently nice means, for example, that $f$ is continuous and that $\partial f /\partial x$ exists and is continuous. No point writing out a proof here; see this Wikipedia entry:

http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign#Proof_of_Theorem

 

[radiator]

Thanks for the correction. I guess I understand the justification.