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title: Complex Numbers
---
# Complex numbers
# 1. Complex Numbers
The lecture on complex numbers consists of three parts, each with their own video:
- [Definition and basic operations](#definition-and-basic-operations)
- [Complex functions](#complex-functions)
- [Differentiation and integration](#differentiation-and-integration)
- [1.1. Definition and basic operations](#definition-and-basic-operations)
- [1.2. Complex functions](#complex-functions)
- [1.3. Differentiation and integration](#differentiation-and-integration)
**Total video length: 38 minutes and 53 seconds**
## Definition and basic operations
## 1.1 Definition and basic operations
<iframe width="100%" height=315 src="https://www.youtube-nocookie.com/embed/fLMdaMuEp8s?rel=0" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
Complex numbers are numbers of the form $$z = a + b {\rm i}.$$ Here
$\rm i$ is the square root of -1: $${\rm i} = \sqrt{-1},$$ or,
equivalently: $${\rm i}^2 = -1.$$
### Complex numbers
!!! info "Definition I"
Complex numbers are numbers of the form $$z = a + b {\rm i}.$$
Here $\rm i$ is the square root of -1: $${\rm i} = \sqrt{-1},$$
or equivalently: $${\rm i}^2 = -1.$$
Usual operations for numbers have their natural extension for complex
numbers as we shall see below.
Usual operations on numbers have their natural extension for complex
numbers, as we shall see below.
Some definitions:
Some useful definitions:
- For a complex number $z = a + b {{\rm i}}$, $a$ is called the *real
part*, and $b$ the *imaginary part*.
!!! info "Definition II"
For a complex number $z = a + b {{\rm i}}$, $a$ is called the *real part*, and $b$ the *imaginary part*.
- The *complex conjugate* $z^*$ of $z = a + b {{\rm i}}$ is defined as
$$z^* = a - b{{\rm i}},$$ i.e., taking the complex conjugate means
flipping the sign of the imaginary part.
!!! info "Complex conjugate"
The *complex conjugate* $z^*$ of $z = a + b {{\rm i}}$ is defined as
$$z^* = a - b{{\rm i}},$$
i.e., taking the complex conjugate means flipping the sign of the imaginary part.
### Addition
!!! info "Addition"
For two complex numbers, $z_1 = a_1 + b_1 {{\rm i}}$ and $z_2 = a_2 + b_2 {{\rm i}}$,
the sum $w = z_1 + z_2$ is given as
$$w = w_1 + w_2 {{\rm i}}= (a_1 + a_2) + (b_1 + b_2) {{\rm i}}$$
For two complex numbers, $z_1 = a_1 + b_1 {{\rm i}}$ and
$z_2 = a_2 + b_2 {{\rm i}}$, the sum $w = z_1 + z_2$ is given as
$$w = w_1 + w_2 {{\rm i}}= (a_1 + a_2) + (b_1 + b_2) {{\rm i}}$$ where
the parentheses in the rightmost expression have been added to group the
real and the imaginary part. A consequence of this definition is that
the sum of a complex number and its complex conjugate is real:
$$z + z^* = a + b {{\rm i}}+ a - b {{\rm i}}= 2a,$$ i.e., this results
in twice the real part of $z$. Similarly, subtracting $z^*$ from $z$
yields $$z - z^* = a + b {{\rm i}} - a + b {{\rm i}}= 2b{\rm i},$$ i.e.,
twice the imaginary part of $z$ (times $\rm i$).
where the parentheses in the rightmost expression have been added to group the real and the imaginary part. A consequence of this definition is that the sum of a complex number and its complex conjugate is real:
$$z + z^* = a + b {{\rm i}}+ a - b {{\rm i}}= 2a,$$ i.e., this results in twice the real part of $z$.
### Multiplication
Similarly, subtracting $z^*$ from $z$ yields $$z - z^* = a + b {{\rm i}} - a + b {{\rm i}}= 2b{\rm i},$$ i.e., twice the imaginary part of $z$ (times $\rm i$).
### Multiplication
For the same two complex numbers $z_1$ and $z_2$ as above, their product
is calculated as
$$w = z_1 z_2 = (a_1 + b_1 {{\rm i}}) (a_2 + b_2 {{\rm i}}) = (a_1 a_2 - b_1 b_2) + (a_1 b_2 + a_2 b_1) {{\rm i}},$$
where the parentheses have again be used to indicate the real and
imaginary parts.
!!! info "Multiplication"
For the same two complex numbers $z_1$ and $z_2$ as above, their product is calculated as
$$w = z_1 z_2 = (a_1 + b_1 {{\rm i}}) (a_2 + b_2 {{\rm i}}) = (a_1 a_2 - b_1 b_2) + (a_1 b_2 + a_2 b_1) {{\rm i}},$$
where the parentheses have again beèn used to indicate the real and imaginary parts.
A consequence of this definition is that the product of a complex number
$z = a + b {{\rm i}}$ with its conjugate is real:
$$z z^* = (a+b{{\rm i}})(a-b{{\rm i}}) = a^2 + b^2.$$ The square root of
this number is the *norm* $|z|$ of $z$:
$$z z^* = (a+b{{\rm i}})(a-b{{\rm i}}) = a^2 + b^2.$$
The square root of this number is called the *norm* $|z|$ of $z$:
$$|z| = \sqrt{z z^*} = \sqrt{a^2 + b^2}.$$
### Division
The quotient $z_1/z_2$ of two complex numbers $z_1$ and $z_2$ as above,
can be evaluated by multiplying the numerator and denominator by the
complex conjugate of $z_2$:
$$\frac{z_1}{z_2} = \frac{z_1 z_2^*}{z_2 z_2^*} = \frac{(a_1 a_2 + b_1 b_2) + (-a_1 b_2 + a_2 b_1) {{\rm i}}}{a_2^2 + b_2^2}.$$
Check this!
The quotient $z_1/z_2$ of two complex numbers $z_1$ and $z_2$ defined above can be evaluated by multiplying the numerator and denominator by the complex conjugate of $z_2$:
!!! info "Division"
$$\frac{z_1}{z_2} = \frac{z_1 z_2^*}{z_2 z_2^*} = \frac{(a_1 a_2 + b_1 b_2) + (-a_1 b_2 + a_2 b_1) {{\rm i}}}{a_2^2 + b_2^2}.$$
Try this yourself!
!!! check "Example:"
$$\begin{align}
......@@ -79,111 +79,124 @@ Check this!
Complex numbers can be rendered on a two-dimensional (2D) plane, the
*complex plane*. This plane is spanned by two unit vectors, one
horizontal, which represents the real number 1, whereas the vertical
unit vector represents ${\rm i}$.
horizontal representing the real number 1 and the vertical
unit vector representing ${\rm i}$.
![image](figures/complex_numbers_5_0.svg)
Note that the norm of $z$ is the length of this vector.
<figure markdown>
![image](figures/complex_numbers_5_0.svg)
<figcaption>The norm of $z$ is the length of its vector spanned in the complex plane.</figcaption>
</figure>
#### Addition in the complex plane
Adding two numbers in the complex plane corresponds to adding the
horizontal and vertical components:
![image](figures/complex_numbers_8_0.svg)
Adding two numbers in the complex plane corresponds to adding their
respective horizontal and vertical components:
We see that the sum is found as the diagonal of a parallelogram spanned
by the two numbers.
<figure markdown>
![image](figures/complex_numbers_8_0.svg)
<figcaption>The sum of two complex numbers is found as the diagonal of a parallelogram spanned by the vectors of those two numbers.</figcaption>
</figure>
## Complex functions
## 1.2. Complex functions
<iframe width="100%" height=315 src="https://www.youtube-nocookie.com/embed/7XtR_wDSqRc?rel=0" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
Real functions can (most of the times) be written in terms of a Taylor series:
Real functions can (most of the times) be written in terms of a Taylor series expanded at a point $x_{0}$:
$$f(x) = \sum \limits_{n=0}^{\infty} \frac{f^{(n)}(x_{0})}{n!} (x-x_{0})^{n}$$
We can write something similar for complex functions,
when replacing the *real* variable $x$ with its *complex* counterpart $z$:
We can write something similar for complex functions by replacing the *real* variable $x$ with its *complex* counterpart $z$:
$$f(z) = \sum \limits_{n=0}^{\infty} \frac{f^{(n)}(x_{0})}{n!} (z-x_{0})^{n}$$
For this course, the most important function is the *complex exponential function*, at which we will have a look below.
For this course, the most important function is the *complex exponential function*, at which we will have a closer look below.
### The complex exponential function
The complex exponential is used *extremely often*.
It occurs in Fourier transforms and it is very convenient for doing calculations
involving cosines and sines.
It also makes doing many common operations on complex number a lot easier.
It occurs in Fourier transforms and it is very convenient for doing calculations involving cosines and sines.
It also makes many common operations on complex number a lot easier to perform.
The exponential function $f(z) = \exp(z) = e^z$ is defined as:
$$\exp(z) = e^{x + {\rm i}y} = e^{x} e^{{\rm i} y} = e^{x} \left( \cos y + {\rm i} \sin y\right).$$
!!! info "The exponential function and Euler identity"
The exponential function $f(z) = \exp(z) = e^z$ is defined as:
$$\exp(z) = e^{x + {\rm i}y} = e^{x} e^{{\rm i} y} = e^{x} \left( \cos y + {\rm i} \sin y\right).$$
The last expression is called the *Euler identity*.
**Exercise** Check that this function obeys
$$\exp(z_1) \exp(z_2) = \exp(z_1 + z_2).$$ You need sum- and difference
formulas of cosine and sine.
!!! note "**Exercise**"
Check that this function obeys
$$\exp(z_1) \exp(z_2) = \exp(z_1 + z_2).$$
*You will need sum and difference formulas of cosine and sine.*
### The polar form
A complex number can be represented by two real numbers, $a$ and $b$
which represent the real and imaginary part of the complex number. An
alternative representation is a *vector* in the complex plane, whose
horizontal component is the real, and vertical component the imaginary
part. However, it is also possible to characterize that vector by its
*length* and *direction*, where the latter can be represented by the
angle the vector makes with the horizontal axis:
![image](figures/complex_numbers_10_0.svg)
The angle with the horizontal axis is denoted by $\varphi$, just as in
the case of polar coordinates. In the context of complex numbers, this
angle is denoted as the *argument*. We have:
> A complex number can be represented either by its real and imaginary
> part, corresponding to the Cartesian coordinates in the complex plane,
> or by its *norm* and its *argument*, corresponding to polar
> coordinates. The norm is the length of the vector, and the argument is
> the angle it makes with the horizontal axis.
From our previous discussion on polar coordinates we can conclude that
for a complex number $z = a + b {\rm i}$, its real and imaginary parts
can be expressed as $$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}$$
When increasing $\varphi$ with $2 \pi$, we make a full circle and reach the same point on the complex plane. In other words, when adding $2 \pi$ to our argument, we get the same complex number!
As a result, the argument $\varphi$ is defined up to $2 \pi$, and we are free to make any choice we like, such as
$$\begin{align}
-\pi < \varphi < \pi \textrm{ (left)} \\
-\frac{\pi}{2} < \varphi < \frac{3 \pi}{2} \textrm{ (right)} \end{align} $$
![image](figures/complex_numbers_11_0.svg)
Some useful values of the complex exponential to know by heart are $e^{2{\rm i } \pi} = 1 $, $e^{{\rm i} \pi} = -1 $ and $e^{{\rm i} \pi/2} = {\rm i}$.
From the first expression, it also follows that
$$e^{{\rm i} (y + 2\pi n)} = e^{{\rm i}y} {\rm ~ for ~} n \in \mathbb{Z}$$
As a result, $y$ is only defined up to $2\pi$.
A complex number $z$ can be represented by two real numbers, $a$ and $b$, which correspond to the real and imaginary part of the complex number.
Another representation of $z$ is a *vector* in the complex plane with a horizontal component that corresponds to the real part of $z$ and a vertical component that corresponds to the imaginary part of $z$.
It is also possible to characterize that vector by its *length* and *direction*, where the latter can be represented by the
angle that the vector makes with the horizontal axis:
<figure markdown>
![image](figures/complex_numbers_10_0.svg)
<figcaption>The angle with the horizontal axis is denoted by $\varphi$
like in the case of conventional polar coordinates,
but in the context of complex numbers, this angle is called as the <b>argument</b>.</figcaption>
</figure>
!!! info "Polar form of complex numbers"
A complex number can be represented either by its real and imaginary part
corresponding to the Cartesian coordinates in the complex plane,
or by its *norm* and its *argument* corresponding to polar coordinates.
The norm is the length of the vector, and the argument is the angle it makes with the horizontal axis.
We can conclude that for a complex number $z = a + b {\rm i}$, its real and imaginary parts
can be expressed in polar coordinates as $$a = |z| \cos\varphi$$ $$b = |z| \sin\varphi$$
!!! info "Inverse equations"
The inverse equations are $$|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}$$
It turns out that by using the magnitude $|z|$ and phase $\varphi$, we can write any complex number as
$$z = |z| e^{{\rm i} \varphi}$$
By increasing $\varphi$ by $2 \pi$, we make a full circle around the origin and reach the same point on the complex plane. In other words, by adding $2 \pi$ to the argument of $z$, we get the same complex number $z$!
As a result, the argument $\varphi$ is defined up to $2 \pi$, and we are free to make any choice we like, such as in the examples in the figure below:
<figure markdown>
![image](figures/complex_numbers_11_0.svg)
<figcaption> $-\pi < \varphi < \pi$ (left) and (right) $-\frac{\pi}{2} < \varphi < \frac{3 \pi}{2}$ </figcaption>
</figure>
Some useful values of the complex exponential to know by heart are:
!!! tip "Useful identities:"
$$e^{2{\rm i } \pi} = 1$$
$$e^{{\rm i} \pi} = -1 $$
$$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}y} {\rm ~ for ~} n \in \mathbb{Z}$$
As a result, $y$ is only defined up to $2\pi$.
Furthermore, we can define the sine and cosine in terms of complex exponentials:
$$\cos(x) = \frac{e^{{\rm i} x} + e^{-{\rm i} x}}{2}$$
$$\sin(x) = \frac{e^{{\rm i} x} - e^{-{\rm i} x}}{2i}$$
Most operations on complex numbers are easiest when converting the complex number to its *polar form*, using the exponential.
Some operations which are common in real analysis are then easily derived for their complex counterparts:
$$z^{n} = \left(r e^{{\rm i} \varphi}\right)^{n} = r^{n} e^{{\rm i} n \varphi}$$
$$\sqrt[n]{z} = \sqrt[n]{r e^{{\rm i} \varphi} } = \sqrt[n]{r} e^{{\rm i}\varphi/n} $$
$$\log(z) = log \left(r e^{{\rm i} \varphi}\right) = log(r) + {\rm i} \varphi$$
$$z_{1}z_{2} = r_{1} e^{{\rm i} \varphi_{1}} r_{2} e^{{\rm i} \varphi_{2}} = r_{1} r_{2} e^{{\rm i} (\varphi_{1} + \varphi_{2})}$$
We see that during multiplication, the norm of the new number is the *product* of the norms of the multiplied numbers, and its argument is the *sum* of the arguments of the multiplied numbers. In the complex plane, this looks as follows:
!!! info "Complex sine and cosine"
$$\cos(x) = \frac{e^{{\rm i} x} + e^{-{\rm i} x}}{2}$$
$$\sin(x) = \frac{e^{{\rm i} x} - e^{-{\rm i} x}}{2i}$$
Most operations on complex numbers become easier when complex numbers are converted to their *polar form* using the complex exponential.
Some functions and operations, which are common in real analysis, can be easily derived for their complex counterparts by sustituting the real variable $x$ with the complex variable $z$ in its polar form:
!!! info "Examples of some complex functions stated using polar form"
$$z^{n} = \left(r e^{{\rm i} \varphi}\right)^{n} = r^{n} e^{{\rm i} n \varphi}$$
$$\sqrt[n]{z} = \sqrt[n]{r e^{{\rm i} \varphi} } = \sqrt[n]{r} e^{{\rm i}\varphi/n} $$
$$\log(z) = log \left(r e^{{\rm i} \varphi}\right) = log(r) + {\rm i} \varphi$$
$$z_{1}z_{2} = r_{1} e^{{\rm i} \varphi_{1}} r_{2} e^{{\rm i} \varphi_{2}} = r_{1} r_{2} e^{{\rm i} (\varphi_{1} + \varphi_{2})}$$
Use of polar form lets us notice immediately that for example, as a result of multiplication, the norm of the new number is the *product* of the norms of the multiplied numbers and its argument is the *sum* of the arguments of the multiplied numbers.
![image](figures/complex_numbers_12_0.svg)
In the complex plane, this looks as follows:
<figure markdown>
![image](figures/complex_numbers_12_0.svg)
<figcaption></figcaption>
</figure>
!!! check "Example: Find all solutions solving $z^4 = 1$."
Of course, we know that $z = \pm 1$ are two solutions, but which other solutions are possible? We take a systematic approach:
......@@ -192,55 +205,56 @@ We see that during multiplication, the norm of the new number is the *product* o
& \Leftrightarrow \varphi = 0, \varphi = \frac{\pi}{2}, \varphi = -\frac{\pi}{2}, \varphi = \pi \\
& \Leftrightarrow z = 1, z = i, z = -i, z = -1 \end{align}$$
## Differentiation and integration
## 1.3. Differentiation and integration
<iframe width="100%" height=315 src="https://www.youtube-nocookie.com/embed/JyftSqmmVdU?rel=0" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
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:
$$\frac{d}{d\varphi} e^{{\rm i} \varphi} = e^{{\rm i} \varphi} \frac{d}{d\varphi} ({\rm i} \varphi) ={\rm i} e^{{\rm i} \varphi} .$$
$$\int_{0}^{\pi} e^{{\rm i} \varphi} = \frac{1}{{\rm i}} \left[ e^{{\rm i} \varphi} \right]_{0}^{\pi} = -{\rm i}(-1 -1) = 2 {\rm i}$$
**We only consider differentiation and integration over *real* variables.**
## Bonus: the complex exponential function and trigonometry
We can then regard the complex ${\rm i}$ as another constant, and use our usual differentiation and integration rules:
!!! info "Differentiation and Integration rules"
$$\frac{d}{d\varphi} e^{{\rm i} \varphi} = e^{{\rm i} \varphi} \frac{d}{d\varphi} ({\rm i} \varphi) ={\rm i} e^{{\rm i} \varphi} .$$
$$\int_{0}^{\pi} e^{{\rm i} \varphi} = \frac{1}{{\rm i}} \left[ e^{{\rm i} \varphi} \right]_{0}^{\pi} = -{\rm i}(-1 -1) = 2 {\rm i}$$
## 1.4. Bonus: the complex exponential function and trigonometry
Let us show some tricks where the simple properties of the exponential
function helps in re-deriving trigonometric identities.
Let us show some tricks in the folloiwing examples where the simple properties of the exponential
function help in re-deriving trigonometric identities.
1. Take $|z_1| = |z_2| = 1$, and $\arg{(z_1)} = \varphi_1$ and
$\arg{(z_2)} = \varphi_2$. Then it is easy to see that
$z_i = \exp({\rm i} \varphi_i)$, $i=1, 2$. Then:
$$z_1 z_2 = \exp[{\rm i} (\varphi_1 + \varphi_2)].$$ The left hand
side can be written as
!!! example "Properties of the complex exponential function I"
Take $|z_1| = |z_2| = 1$, and $\arg{(z_1)} = \varphi_1$ and
$\arg{(z_2)} = \varphi_2$.
It is easy to see then that $z_i = \exp({\rm i} \varphi_i)$, $i=1, 2$. Then:
$$z_1 z_2 = \exp[{\rm i} (\varphi_1 + \varphi_2)].$$
The left hand side can be written as
$$\begin{align}
z_1 z_2 & = \left[ \cos(\varphi_1) + {\rm i} \sin(\varphi_1) \right] \left[ \cos(\varphi_2) + {\rm i} \sin(\varphi_2) \right] \\
& = \cos\varphi_1 \cos\varphi_2 - \sin\varphi_1 \sin\varphi_2 + {\rm i} \left( \cos\varphi_1 \sin\varphi_2 +
\sin\varphi_1 \cos\varphi_2 \right).
\end{align}$$
On the other hand, the right
hand side can be written as
Also, the right hand side can be written as
$$\exp[{\rm i} (\varphi_1 + \varphi_2)] = \cos(\varphi_1 + \varphi_2) + {\rm i} \sin(\varphi_1 + \varphi_2).$$
Comparing the two expressions, equating their real and imaginary
parts, we find
Comparing the two expressions, equating their real and imaginary parts, we find
$$\cos(\varphi_1 + \varphi_2) = \cos\varphi_1 \cos\varphi_2 - \sin\varphi_1 \sin\varphi_2;$$
$$\sin(\varphi_1 + \varphi_2) = \cos\varphi_1 \sin\varphi_2 +
\sin\varphi_1 \cos\varphi_2.$$ Note that we used the resulting
formulas already in order to derive the properties of the
exponential function. The point is that you can use the properties
of the complex exponential to quickly find the form of gonometric
formulas which you easily forget.
2. As a final example, consider what we can learn from the derivative
of the exponential function:
\sin\varphi_1 \cos\varphi_2.$$
Note that we used the Euler formula in order to derive the identities of trigonometric function.
The point is to show you that you can use the properties of the complex exponential to quickly find the form of trigonometric formulas, which are often easily forgotten.
!!! example "Properties of the complex exponential function II"
In this example, let's see what we can learn from the derivative of the exponential function:
$$\frac{d}{d\varphi} \exp({\rm i} \varphi) = {\rm i} \exp({\rm i} \varphi) .$$
Writing out the exponential in terms of cosine and sine, we see that
$$\cos'\varphi + {\rm i} \sin'\varphi = {\rm i} \cos\varphi - \sin\varphi.$$
where the prime $'$ denotes the derivative as usual. Equating real
and imaginary parts leads to $$\cos'\varphi = - \sin\varphi;$$
where the prime $'$ denotes the derivative as usual. Equating real and imaginary parts leads to
$$\cos'\varphi = - \sin\varphi;$$
$$\sin'\varphi = \cos\varphi.$$
## Summary
## 1.5. Summary
- A complex number $z$ has the form $$z = a + b \rm i$$ where $a$ and
1. A complex number $z$ has the form $$z = a + b \rm i$$ where $a$ and
$b$ are both real, and $\rm i^2 = 1$. The real number $a$ is called
the *real part* of $z$ and $b$ is the *imaginary part*. Two complex
numbers can be added, subtracted and multiplied straightforwardly.
......@@ -248,8 +262,8 @@ function helps in re-deriving trigonometric identities.
$z_2=a_2 + \rm i b_2$ is
$$\frac{z_1}{z_2} = \frac{z_1 z_2^*}{z_2 z_2^*} = \frac{(a_1 a_2 + b_1 b_2) + (-a_1 b_2 + a_2 b_1) {{\rm i}}}{a_2^2 + b_2^2}.$$
- Complex numbers can also be characterised by their *norm*
$|z|=\sqrt{a^2+b^2}$ and *argument* $\varphi$. These coordinates
2. Complex numbers can also be characterised by their *norm*
$|z|=\sqrt{a^2+b^2}$ and *argument* $\varphi$. These parameters
correspond to polar coordinates in the complex plane. For a complex
number $z = a + b {\rm i}$, its real and imaginary parts can be
expressed as $$a = |z| \cos\varphi$$ $$b = |z| \sin\varphi$$ The
......@@ -261,9 +275,9 @@ function helps in re-deriving trigonometric identities.
The complex number itself then becomes
$$z = |z| e^{{\rm i} \varphi}$$
- The most important complex function for us is the complex exponential function, which simplifies many operations on complex numbers
3. The most important complex function for us is the complex exponential function, which simplifies many operations on complex numbers
$$\exp(z) = e^{x + {\rm i}y} = e^{x} \left( \cos y + {\rm i} \sin y\right).$$
where $y$ is defined up to $2 \pi$.
where $y$ is defined up to $2 \pi$.\\
The $\sin$ and $\cos$ can be rewritten in terms of this complex exponential as
$$\cos(x) = \frac{e^{{\rm i} x} + e^{-{\rm i} x}}{2}$$
$$\sin(x) = \frac{e^{{\rm i} x} - e^{-{\rm i} x}}{2i}$$
......@@ -271,7 +285,7 @@ function helps in re-deriving trigonometric identities.
$$\frac{d}{d\varphi} e^{{\rm i} \varphi} = e^{{\rm i} \varphi} \frac{d}{d\varphi} ({\rm i} \varphi) ={\rm i} e^{{\rm i} \varphi} .$$
$$\int_{0}^{\pi} e^{{\rm i} \varphi} = \frac{1}{{\rm i}} \left[ e^{{\rm i} \varphi} \right]_{0}^{\pi} = -{\rm i}(-1 -1) = 2 {\rm i}$$
## Problems
## 1.6. Problems
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 $b/a$.
......
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