Complex Analysis: Theory and Applications · Problem 5.4.5

Complex Integration — Howell & Matthews

Mathematics — Complex Analysis · complex-trig-identities

Problem

Show that, for all zz, (a) sin⁡(π−z)=sin⁡z\sin(\pi - z) = \sin z. (b) sin⁡(π2−z)=cos⁡z\sin\left(\dfrac{\pi}{2} - z\right) = \cos z. (c) sinh⁡(z+iπ)=−sinh⁡z\sinh(z + i\pi) = -\sinh z. (d) tanh⁡(z+iπ)=tanh⁡z\tanh(z + i\pi) = \tanh z. (e) sin⁡(iz)=isinh⁡z\sin(iz) = i\sinh z. (f) cosh⁡(iz)=cos⁡z\cosh(iz) = \cos z.

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