# Math Help - real part of a function

1. ## real part of a function

hi, this is a stupid question but is the real part of the function

$f(z) = e^{iz}$

$cos z$ ?

hi, this is a stupid question but is the real part of the function

$f(z) = e^{iz}$

$cos z$ ?
No, it's not.

Remember that $z$ is a complex number.

In other words, $z = x + iy$.

You need to write $f(z) = e^{iz}$ as $u + iv$, where $u, v$ are real functions of $x$ and $y$.

So $f(z) = e^{iz}$

$= e^{i(x + iy)}$

$= e^{ix + i^2y}$

$= e^{-y + ix}$

$= e^{-y}e^{ix}$

$= e^{-y}(\cos{x} + i\sin{x})$ by Euler's Rule

$= e^{-y}\cos{x} + i\,e^{-y}\sin{x}$.

$= u + iv$.

So the real part of $e^{iz}$ is $e^{-y}\cos{x}$ and the imaginary part of $e^{iz}$ is $e^{-y}\sin{x}$.

3. ohh thank you!! you made it so clear for me