Question Number 1797 by 112358 last updated on 29/Sep/15
$${Let}\:{f}\:{be}\:{a}\:{function}\:{defined}\:{on} \\ $$$${the}\:{domain}\:{D}_{{f}} =\mathbb{C}\:{and}\:{f}\left({z}\right)={cosz} \\ $$$${for}\:{z}\in{D}_{{f}} \:.\:{Is}\:{f}\left({z}\right)\:{a}\:{member} \\ $$$${of}\:\mathbb{C}?\:{Compute}\:{for}\:{example} \\ $$$${f}\left(\mathrm{2}+\mathrm{6}{i}\right). \\ $$$$ \\ $$
Answered by 123456 last updated on 29/Sep/15
$${z}={x}+{y}\imath,\left({x},{y}\right)\in\mathbb{R}^{\mathrm{2}} \\ $$$$\mathrm{cos}\:{z}=\mathrm{cos}\:{x}\:\mathrm{cosh}\:{y}−\imath\:\mathrm{sin}\:{x}\:\mathrm{sinh}\:{y} \\ $$$$\mathrm{sin}\:\left({y}\imath\right)=\imath\:\mathrm{sinh}\:{y} \\ $$$$\mathrm{cos}\:\left({y}\imath\right)=\mathrm{cosh}\:{y} \\ $$$$..... \\ $$