Why is $\Re(e^{(\lambda e^{i\theta}-\lambda)})\neq e^{(\lambda(\Re (e^{i\theta}-1)))}$ ?
I always thought, that $\Re$ is linear, but if I compute LHS;
$\Re(e^{(\lambda e^{i\theta}-\lambda)})=\Re(e^{\lambda(e^{i\theta-1})})=\Re(e^{\lambda(\cos\theta-1)}e^{i\lambda\sin\theta})$
$=e^{\lambda(\cos\theta-1)}\Re(e^{i\lambda\sin\theta})=e^{\lambda(\cos\theta-1)}\cos(\lambda\sin\theta)$
but RHS gives only $e^{\lambda(\cos\theta-1)}$
where does it fail ?
Thanks for your help