I'm working on a project, which requires the following numerical integration:
$$\frac{a^j}{\Gamma(j)}\int_{-\infty}^t f(s) e^{-a(t-s)}(t-s)^{j-1}ds$$ for some bounded continuous function $f$. Now, the problem at hand is really to evaluate $$\frac{a^j}{\Gamma(j)}\int_{-\infty}^te^{-a(t-s)}(t-s)^{j-1}ds$$ because since $f$ is bounded, I don't really care about it for now.
I want to use the change of variables $$u=e^{-\frac{a}{j}(t-s)}$$
but when I do, I must be making an error somewhere, because after the change of variable, the transformed integral does not give the same value as the one I started out with. Here is what I was doing:
$$u=e^{-\frac{a}{j}(t-s)} \Leftrightarrow -\frac{j}{a}\log(u) = t-s.$$ Also,
$$ du = \frac{a}{j}e^{-\frac{a}{j}(t-s)}ds \Rightarrow ds = \frac{j}{a}e^{\frac{a}{j}(t-s)}du$$
Then, having determined what $ds$ is in terms of $u$, and making the appropriate substitutions, I get that \begin{align*} && &\frac{a^j}{\Gamma(j)}\int_{-\infty}^t e^{-a(t-s)}(t-s)^{j-1}ds&&\\ && &= \frac{a^j}{\Gamma(j)}\int_{0}^{1}e^{-a(-\frac{j}{a}\log(u))} \left(\frac{-j}{a}\log(u)\right)^{j-1} \frac{j}{a} e^{\frac{a}{j}\left(\frac{-j}{a}\right)\log(u)}du &&\\ && &= \frac{j^j}{\Gamma(j)}\int^1_0 u^{j-1}(-\log(u))^{j-1}du&& \end{align*}
Except this is wrong...and I don't know why. Punching in the value of the original integral into Wolfram Alpha, I know what the integral I have performed the change of variables on should evaluate to, except it doesn't evaluate to what it should.
I don't think it's an algebraic manipulation error, but rather something to do with how I'm changing variables. I guess I am missing something in my reasoning with respect to this.
I appreciate your help!