find-zeros-function-f-z-1-cosz-

Question Number 145822 by Mrsof last updated on 08/Jul/21

f i n d z e r o s f u n c t i o n f (z) = 1 - c o s z

Commented by Mrsof last updated on 08/Jul/21

? ? ? ? ? ? ?

Answered by mathmax by abdo last updated on 09/Jul/21

z_{k} = 2 k π are roots

f (z) = 0 \Rightarrow 1 - cosz = 0 \Rightarrow cosz = 1 \Rightarrow \frac{e^{iz} + e^{- iz}}{2} = 1 \Rightarrow e^{iz} + e^{- iz} - 2 = 0

\Rightarrow u + u^{- 1} - 2 = 0 (u = e^{iz}) \Rightarrow u^{2} + 1 - 2 u = 0 \Rightarrow {(u - 1)}^{2} = 0 \Rightarrow u = 1

e^{iz} = 1 = e^{i (2 k π)} \Rightarrow z_{k} = 2 k π (k \in Z) are roots of f