### Author Topic: Section 5.2 Theorem 3.c)  (Read 1044 times)

#### Bruce Wu ##### Section 5.2 Theorem 3.c)
« 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 $\hat{p}=-i\hbar\frac{d}{dx}$

#### Bruce Wu ##### Re: Section 5.2 Theorem 3.c)
« Reply #1 on: November 18, 2015, 02:07:37 PM »
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.

#### Emily Deibert ##### Re: Section 5.2 Theorem 3.c)
« Reply #2 on: November 18, 2015, 02:12:13 PM »
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/PDE-textbook/Chapter5/S5.2.html

#### Victor Ivrii ##### Re: Section 5.2 Theorem 3.c)
« Reply #3 on: November 19, 2015, 09:41:11 AM »
OK. Fixed