The last expression is called the *Euler identity*.
**Exercise** Check that this function obeys
$$\exp(z_1) \exp(z_2) = \exp(z_1 + z_2).$$ You need sum- and difference
formulas of cosine and sine.
### The polar form
A complex number can be represented by two real numbers, $a$ and $b$
which represent the real and imaginary part of the complex number. An
alternative representation is a *vector* in the complex plane, whose
horizontal component is the real, and vertical component the imaginary
part. However, it is also possible to characterize that vector by its
*length* and *direction*, where the latter can be represented by the
angle the vector makes with the horizontal axis:
The angle with the horizontal axis is denoted by $\varphi$, just as in
the case of polar coordinates. In the context of complex numbers, this
angle is denoted as the *argument*. We have:
> A complex number can be represented either by its real and imaginary
> part, corresponding to the Cartesian coordinates in the complex plane,
> or by its *norm* and its *argument*, corresponding to polar
> coordinates. The norm is the length of the vector, and the argument is
> the angle it makes with the horizontal axis.
From our previous discussion on polar coordinates we can conclude that
for a complex number $z = a + b {\rm i}$, its real and imaginary parts
can be expressed as $$a = |z| \cos\varphi$$ $$b = |z| \sin\varphi$$ The
inverse equations are $$|z| = \sqrt{a^2 + b^2}$$
$$\varphi = \arctan(b/a)$$ for $a>0$. In general:
$$\varphi = \begin{cases} \arctan(b/a) &{\rm for ~} a>0; \\
\pi + \arctan(b/a) & {\rm for ~} a<0{\rm~and~}b>0;\\
-\pi + \arctan(b/a) &{\rm for ~} a<0 {\rm ~ and ~} b<0.
\end{cases}$$
It turns out that using this magnitude $|z|$ and phase $\varphi$, we can write any complex number as
$$z = |z| e^{{\rm i} \varphi}$$
When increasing $\varphi$ with $2 \pi$, we make a full circle and reach the same point on the complex plane. In other words, when adding $2 \pi$ to our argument, we get the same complex number!
As a result, the argument $\varphi$ is defined up to $2 \pi$, and we are free to make any choice we like, such as
Some useful values of the complex exponential to know by heart are $e^{2{\rm i } \pi} = 1 $, $e^{{\rm i} \pi} = -1 $ and $e^{{\rm i} \pi/2} = {\rm i}$.
From the first expression, it also follows that
$$e^{{\rm i} (y + 2\pi n)} = e^{{\rm i}\pi} {\rm ~ for ~} n \in \mathbb{Z}$$
As a result, $y$ is only defined up to $2\pi$.
Furthermore, we can define the sine and cosine in terms of complex exponentials:
We see that during multiplication, the norm of the new number is the *product* of the norms of the multiplied numbers, and its argument is the *sum* of the arguments of the multiplied numbers. In the complex plane, this looks as follows:
**Example** Find all solutions solving $z^4 = 1$.
Of course, we know that $z = \pm 1$ are two solutions, but which other solutions are possible? We take a systematic approach:
We only consider differentiation and integration over *real* variables. We can then regard the complex ${\rm i}$ as another constant, and use our usual differentiation and integration rules:
1. [:grinning:] Given $a=1+2\rm i$ and $b=-3+4\rm i$, calculate and draw in the
complex plane the numbers $a+b$, $ab$, and $b/a$.
2. [:grinning:] Evaluate (a) $\rm i^{1/4}$, (b)
$\left(1+\rm i \sqrt{3}\right)^{1/2}$, (c) $\exp(2\rm i^3)$.
3. [:grinning:] Find the three 3rd roots of $1$ and ${\rm i}$ (i.e. all possible solutions to the equations $x^3 = 1$ and $x^3 = {\rm i}$, respectively).
4. [:grinning:] (a) Find the real and imaginary part of
5. [:sweat:] For any given complex number $z$, we can take the inverse $\frac{1}{z}$. Visualize taking the inverse in the complex plane. What geomtric operation does taking the inverse correspond to? (Hint: first consider what geometric operation $\frac{1}{z^*}$ corresponds to.)
6. [:grinning:] Compute (a)
$$\frac{d}{dt} e^{{\rm i} (kx-\omega t)},$$
and (b) calculate the real part of
$$\int_0^\infty e^{-\gamma t +\rm i \omega t} dt$$($k$, $x$, $\omega$, $t$ and
