Complex Analysis: Theory and Applications · Problem 9.4.1

The tangent function is a conformal map that wraps regions of the complex plane into new shapes by exploiting its periodic structure and poles. The key insight is to track where the boundary of your strip maps: the vertical lines at $x = -\frac{\pi}{4}$ and $x = 0$ become curves in the $w$-plane (found by evaluating $\tan(-\frac{\pi}{4} + iy)$ and $\tan(iy)$), while the real axis at $y = 0$ maps to the real axis itself. The interior of the strip then fills the region between these image curves, so you need to determine what those boundaries look like and which side of them the image occupies.

The tangent function is a conformal map that wraps regions of the complex plane into new shapes by exploiting its periodic structure and poles. The key insight is to track where the boundary of your strip maps: the vertical lines at $x = -\frac{\pi}{4}$ and $x = 0$ become curves in the $w$-plane (found by evaluating $\tan(-\frac{\pi}{4} + iy)$ and $\tan(iy)$), while the real axis at $y = 0$ maps to the real axis itself. The interior of the strip then fills the region between these image curves, so you need to determine what those boundaries look like and which side of them the image occupies.