# questions in understanding the textbook about fourier transform

I am confused about if the text book set $$z = x-y$$, then why there is no $$-$$ after variable change? I think it is obvious that $$dy = - dz$$

They have compensated for this negative sign, but it's hidden in their $$\int_R$$ notation. Substitution affects both the $$d$$ term and the bounds of an integral, but we can't see that here. Writing it with explicit bounds, and inserting an intermediate calculation, we get $$\int_{-\infty}^\infty [\cdots]dy=-\int_\infty^{-\infty}[\cdots]dz\\ =\int_{-\infty}^\infty [\cdots ]dz$$ Hopefully this is clearer.
• @Cooper Look at any formula you have for substitution in a definite integral, in any book you may have. It will be there. Basically, the original integral was for when $y$ went from $-\infty$ to $\infty$. That corresponds to $z$ going from $\infty$ to $-\infty$. Commented Nov 25, 2019 at 7:13
$$-\int_{+\infty}^{-\infty}=\int_{-\infty}^{+\infty}= \int_{\mathbb R}$$
• No it is the opposite of it, $\int_a^b=-\int_b^a$ Commented Nov 25, 2019 at 6:30