However, the rescaled function
\( f(z) = \frac{\frac{4}{\pi} \tan^{-1}(z) + 1}{2} = \frac{2}{\pi} \tan^{-1}(z) + \frac{1}{2} \)
might work. This maps the right half of the u.c. to \( 1 + ix \) and the left half to \( ix \), thus allowing us to create a basis for the function at the imaginary axis.
\( f(z) = \frac{\frac{4}{\pi} \tan^{-1}(z) + 1}{2} = \frac{2}{\pi} \tan^{-1}(z) + \frac{1}{2} \)
might work. This maps the right half of the u.c. to \( 1 + ix \) and the left half to \( ix \), thus allowing us to create a basis for the function at the imaginary axis.