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.
