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Math Help - desperately need help with integration with infinite limit and step function

  1. #1
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    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.

    desperately need help with integration with infinite limit and step function-capture.png

    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.
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  2. #2
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    Re: desperately need help with integration with infinite limit and step function

    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.
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  3. #3
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    Re: desperately need help with integration with infinite limit and step function

    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!
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