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**Textbook errors / 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 $\hat{p}=-i\hbar\frac{d}{dx}$

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 $\hat{p}=-i\hbar\frac{d}{dx}$