Complex Analysis: Theory and Applications · Problem 9.4.9

Applications of Harmonic Functions — Howell & Matthews

Mathematics — Complex Analysis · trigonometric-mapping-exercise

Problem

Show that w=sin⁡zw = \sin z maps the rectangle R={x+iy:−π2<x<π2, 0<y<b}R = \{x + iy : -\frac{\pi}{2} < x < \frac{\pi}{2},\, 0 < y < b\} one-to-one and onto the portion of the upper half-plane Im⁡(w)>0\operatorname{Im}(w) > 0 that lies inside the ellipse u2cosh⁡2b+v2sinh⁡2b=1.\frac{u^2}{\cosh^2 b} + \frac{v^2}{\sinh^2 b} = 1.

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