Question

Let $z=x+iy$ then prove that $|e^z|=e^x$.

Answer 1

Now,

$e^z$

$=e^{x+iy}$

$=e^x(\cos y+i\sin y)$.

Then $|e^z|$

$=\sqrt{e^{2x}({\cos }^{2}y+{\sin }^{2}y)}$

$=e^x$.