# Arfken, Weber, and Harris Fourier transform properties inconsistent?

Does anybody know where to find the erratum page for Arfken, Weber, and Harris' Mathematical Methods in the Physical Sciences seventh edition?

In Arfken, Weber, and Harris' Mathematical Methods in the Physical Sciences seventh edition, the Fourier transform is defined on page 966 as:

$$g (\omega) =\frac{1}{\sqrt{2 \pi }}\underset{-\infty }{\overset{\infty }{\int }} \text{dt} f (t) e^{i t \omega }. \tag{20.10}$$

Then, on page 981, an important property of the Fourier transform is
given as :

$$\left[\frac{d^n f(t)}{\text{dt}^n}\right]^T(\omega) =g (\omega) (-\text{i\omega })^n. \tag{20.56}$$

Where the $$[\ \ ]^T (\omega)$$ notation means "the Fourier transform of the thing inside, which is a function of $$(\omega)$$".I do not think that (20.56) is consistent with the definition and should instead be :

$$\left[\frac{d^n f(t)}{\text{dt}^n}\right]^T(\omega) =g (\omega) (+\text{i\omega })^n \ \ \ \ \ \ \ (new \ 20.56)[goes\ with\ 20.10]$$

Unless they were to have defined the Fourier transform alternately:

$$g (\omega) =\frac{1}{\sqrt{2 \pi} }\overset{\infty }{\underset{-\infty }{\int }}\text{dt} f (t) e^{-\text{i\omega t}} \ \ \ \ \ \ \ (alternative\ 20.10)[goes\ with\ 20.56]$$

Have I made a sign error?

More detail: the way to derive it is to use [tabular] integration by parts:

$$\left( \begin{array}{cc} e^{i\omega t} & f' (t) \\ e^{i\omega t} (i\omega) & f (t) \\ \end{array} \right)$$

Where $$f(t)$$ must be zero at the endpoints.

• the integration by parts brings a minus sign with it each time hence the minus in the bracket. Btw there are a lot of errors in the book if i remember correctly (which is no shame given the extent of the book). Sep 21, 2020 at 16:45
• Does your modification intentionally contain $2\pi$ instead of $\sqrt{2\pi}$, or is that difference in addition to the change $i\to-i$?
– rob
Sep 21, 2020 at 16:57
• Aha, I did in fact make a sign error along with the radical error. I will probably have to delete my question. Sep 21, 2020 at 18:02
• just in case you don't get a better recommendation, there was a errata on studocu when I worked through the book. But ever since I found a full copy of a book which cost me 80euro to buy on there I'm a bit skeptical about the site's operation model... Sep 22, 2020 at 7:08

The question was answered in the comments by (ctsmd) [thank you]. The integration by parts brings a minus sign each time. Also, thank you to (rob) for pointing out the missing $$\sqrt{ \ }$$ symbol (which I have since edited).