To extract the formula for number 7 from the formula in number 1, let's first understand the structure of both:

1. Formula 1 on the left is a piecewise function defined as:

   1 & \text{if } -b < x < b \\ 0 & \text{otherwise} \end{cases}$$ And the corresponding Fourier transform is written as: $$f(w) = \frac{2 \sin(bw)}{\pi w}$$

2. Formula 7 on the left is: $f(x) = e^{iax}, \quad \text{for } -b < x < b$

   The Fourier transform is: $f(w) = \frac{e^{i a w} - e^{-i a w}}{\sqrt{2 \pi}} = \frac{2 i \sin(a w)}{\sqrt{2 \pi}}$

Extraction Process:

From the form of Formula 1, you can deduce that the Fourier transform is obtained by integrating the function over the specified range.

For the function $e^{iax}$, a similar process applies, but instead of a simple 1, the exponential function's transform leads to the result for Formula 7. Essentially, it's the result of calculating the Fourier transform for the given piecewise function where the function is $e^{iax}$.

Thus, the Fourier transform of $e^{iax}$ leads to the final formula: $f(w) = \frac{2i \sin(a w)}{\sqrt{2 \pi}}$

Would you like more details on how this Fourier transform is computed or have any further questions?