Given this equation:
$$\frac{-1}{4c}[\int_{ -\infty}^{\infty}g(\varpi)Sgn(x - ct - \varpi).d\varpi -\int_{-\infty}^{\infty}g(\varpi)Sgn(x+ct - \varpi ).d\varpi ]$$ Where sgn is the signum function, how do I reduce it to
$$ \frac{1}{2}\int_{x-ct}^{x+ct}g(\varpi )d\varpi $$?
While I can see how the bounds changed, I am unable to see how the function within the integral simplified.
Any help would be appreciated
The signum function is the "sign" function. It is $+1$ if the argument is positive, $-1$ if the argument is negative and $0$ if the argument is $0$. Think of $x-ct$ as fixed, which it is for the purposes of the integral with respect to $w$. Then $\mbox{Sgn}(x-ct-w)$ is positive if $x-ct > w$ and is negative if $x-ct < w$. So divide the line into $$++++++(x-ct)--------------$$ The above represents how the funtion is $+1$ if $w < x-ct$ and is $-1$ if $w > x-ct$. Similarly, the second function $\mbox{Sgn}(x+ct-w)$ looks like $$----------------(x+ct)+++++$$ If you add those two, then you get what you want. However, you're subtracting the two, which makes me suspicious of a sign error in what you wrote.