desperately need help with integration with infinite limit and step function

Hi,

This is my first post, I'm not sure if this is right place to ask. I have this derivation to do for my research and I'm having problem to understand a integration in a paper. The equations are in in the attachment with this post and also in the picture below. Sorry, I couldn't find a way a way to directly write the equations here.

Attachment 24731

Could anyone, please, help me understand how the two sides in the second equation are equal? Shouldn't the first limit of integration in the right-hand side be 'x' instead of '-ve infinity'?

Thanks in advance.

I don't know the context, but that sure looks like a consequence of u being a step function - and I've often seen them written with the name "u".
If u is the unit step function at 0 (i.e. u(s) = 1 for s>=0, and u(s) = 0 for s<0), then your formula drops out, since then h(t,x)=a(x) when x<t, and h(t,x)=0 when x>t.

Yes, u(t-x) is a step function. so, u(t-x)=1 when t>x and 0 otherwise. But, in the integration, x is the variable! I was still thinking of t as the variable. Thanks! That explains it! Thank you so very much!