Toronto Math Forum
APM3462015F => APM346Misc => Textbook errors => Topic started by: Bruce Wu on November 18, 2015, 02:06:20 PM

In the proof it says $$\int\left(e^{ikx}\right)'f(x)dx=ik\hat{f}(k)$$
However, $\left(e^{ikx}\right)'=ike^{ikx}$, so shouldn't it be $ik\hat{f}(k)$? So is it the rule that is wrong or is it the proof that is wrong? Recall that the rule is $$g(x)=f'(x)\Rightarrow \hat{g}(k)=ik\hat{f}(k)$$
Should it be instead $$\hat{g}(k)=ik\hat{f}(k)$$
Which makes sense since in quantum mechanics $\hat{p}=i\hbar\frac{d}{dx}$

Actually, never mind. I found the error, it's in the proof. Integration by parts was used but the negative sign in front of the integral was neglected. The rule stands corrected.

Great insight, Fei Fan Wu! Just for anyone else wondering, the link to this error is here:
http://www.math.toronto.edu/courses/apm346h1/20159/PDEtextbook/Chapter5/S5.2.html

OK. Fixed