In this section, we consider the important complex function

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

substituting $z$ with its definition,

$$f(z)=e^{x+iy}=e^x e^{iy}$$

The $e^x$ part is easy to understand since it is the ordinary exponential function with a real argument, but what about the second exponential?  What does it mean for an exponent to be imaginary?  To find out, we can taylor expand the term $e^{iy}$ about $y=0$:

$$\displaystyle e^{iy}=1+iy-\frac{y^2}{2}-i\frac{y^3}{3!}+\frac{y^4}{4!}+i\frac{y^5}{5!}-\frac{y^6}{6!}-i\frac{y^7}{7!}+\cdots$$

grouping the real and imaginary parts,

$$\displaystyle e^{iy}=\left(1-\frac{y^2}{2}+\frac{y^4}{4!}-\frac{y^6}{6!}+\cdots\right) + i\left( y-\frac{y^3}{3!}+\frac{y^5}{5!}-\frac{y^7}{7!}+\cdots\right)$$

The real part is composed of the even terms of the expansion while the imaginary part is composed of the odd terms.  It is easy to verify that

$$\displaystyle \cos(y)=1-\frac{y^2}{2}+\frac{y^4}{4!}-\frac{y^6}{6!}+\cdots$$

and

$$\displaystyle \sin(y)= y-\frac{y^3}{3!}+\frac{y^5}{5!}-\frac{y^7}{7!}+\cdots$$

Thus

$$e^{iy}=\cos y+i\sin y$$

This is known as Euler's formula. If we set $y=\pi$, we find that

$$e^{i\pi}=-1$$

or

$$e^{i\pi}+1=0$$

This is known as Euler's identity.  It significant because it relates the fundamental numbers $e$,  $\pi$, $i$, $1$, and $0$ using multiplication, exponentiation, addition, and equality. Euler's formula also gives us another way to write the imaginary unit:

$$e^{i\pi/2}=i$$

Returning to the original expression,

$$e^{z}=(\cos y + i\sin y)e^x$$