The complex exponential function

Define $e^z = e^{x+iy} =e^xe^{iy} =e^x(cos(y) + isin(y)) = e^xcos(y) + ie^xsin(y)$

From the definition, it is clear that $e^{z + 2\pi i} = e^x(cos(y+2 \pi) + isin(y + 2 \pi)) = e^x (cos(y) + i sin(y)) = e^z$

Notice that if we fix $Re(z) = x_0$ then $e^{z} = e^{x_0}e^{yi}$ which is just a circle of radius $e^{x_0}$ . So lines of constant real part are taken to circles. If we hold $y = y_0$ (the imaginary part of $z$ then $e^{x + i y_0}$ is a ray of argument $y_0$ .