Complex Analysis: Theory and Applications · Problem 5.4.16

Complex Integration — Howell & Matthews

Mathematics — Complex Analysis · complex-trig-identities

Problem

Establish the following identities. (a) eiz=cos⁡z+isin⁡ze^{iz} = \cos z + i\sin z. (b) cos⁡z=cos⁡xcosh⁡y−isin⁡xsinh⁡y\cos z = \cos x \cosh y - i\sin x \sinh y. (c) sin⁡(z1+z2)=sin⁡z1cos⁡z2+cos⁡z1sin⁡z2\sin(z_1 + z_2) = \sin z_1 \cos z_2 + \cos z_1 \sin z_2. (d) ∣cos⁡z∣2=cos⁡2x+sinh⁡2y|\cos z|^2 = \cos^2 x + \sinh^2 y. (e) cosh⁡z=cosh⁡xcos⁡y+isinh⁡xsin⁡y\cosh z = \cosh x \cos y + i\sinh x \sin y. (f) cosh⁡2z−sinh⁡2z=1\cosh^2 z - \sinh^2 z = 1. (g) cosh⁡(z1+z2)=cosh⁡z1cosh⁡z2+sinh⁡z1sinh⁡z2\cosh(z_1 + z_2) = \cosh z_1 \cosh z_2 + \sinh z_1 \sinh z_2.

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