1 8 2 Show that cos z cos x sinh y where x iy Find all zeros

Last updated: 4/16/2023

1 8 2 Show that cos z cos x sinh y where x iy Find all zeros and periods of cos z Solution Use trigonometric formulas from page 29 and 30 in CA we have cos z cos x iy cos rcos iy sin r sin iy cos x cosh y i sin x sinh y Now take the modulus squared and use cosh y 1 sinh y cos 2 cos r cosh y sin x sinh y cos r 1 sinh y sin r sinh y cos r cos r sin r sinh y cos r sinhy The identity for cos 22 shows that the only zeros of cos z are the zeros of cos on the real axis because cos z 0 cos x 0 r 1 ma m 0 1 2 Translation by any period A of cos z sends zeros to zeros Thus any period is an integral multiple of and since odd integral multiples are not periods the only periods of cos z are 2an 0 n