### Complex Numbers

**Chapterł 3 Complex functionsChapter 3.1 Introduction**You can now add, subtract, multiply and divide

**complex numbers**. It’s time for

**functions**.

In

**electrica**l engineering, automatics and the like, the

**complex function**is mainly used:

**-rational**function

**-exponential**function with

**base e**

Other

**complex functions**, e.g.

**trigonometric, logarithmic**, are used less often and we will not deal with them.

Time charts of

**complex functions**are more complicated than of

**real functions**. To each

**two-dimensional**point in the

**z-plane**we assign a complex number

**f(z)**, which is also a

**two-dimensional**. We enter

**four-dimensional**space! Hard to imagine for an ordinary bread eater. We will come back to this topic.

**Chapter 3.2 Complex rational functions**

We will calculate it using only **addition, subtraction, multiplication **and** division**.

We get any **power** of **z** by successive multiplications.

Free polynomial **z**, or “sum of powers” by **exponentiation, addition **and** subtraction**

Any **rational** function, i.e. the **quotient** of** polynomials**, by means of **exponentiation, addition, subtraction** and** division**

**Fig. 3-1**

Complex rational functions

**Chapter 3.3 Complex exponential function exp(z)Chapter 3.3.1 Definition**Strictly-

**exponential**with

**base e**, otherwise

**exponential**.

**Fig. 3-2**

Complexfunction

Complex

**exp(z**) with

**base e**, i.e. an

**exponential function**

**Chapter 3.3.2 Number e=2.7182818…**

The

**irrational**real number

**e=2.71828…**is the

**fourth**one to know after

**0.1**and

**π**. Everyone knows where

**0**and

**1**came from

*****. The number

**π=3.14…**is the

**circumference**to the

**radius**. But the number

**e**? Is there any simple “circumference to radius” interpretation for it. It is, but less obvious, and it came from the banking world

*****Where exactly is it from?

**Fig. 3-3**

Number

**e**and

**interest**in the

**bank**.

**a**I put a

**1**$ in the bank with

**100%**interest after a year. By the way, good interest and the bank is great.

**b**After a year, it should be

**$**

**2**. Note that

**1**is

**100%**

**c**I made an agreement with the bank that the

**interest**will also be

**100%**but I will collect money

**$**

**1.5**after six months. Using banking vocabulary – capitalization after

**half**a year

**d**If I collected it after a

**year**, i.e. after

**2**capitalizations, it would be

**2.25 $**

**e**And if there were

**3**capitalizations, after the

**3**year

**$ 2.3703**…

**f**$

**2,593**… after after

**10**capitalizations

**g**The same, just a

**different**notation

**h**PLN 2,704… after a year after

**100**capitalizations

**j**PLN

**$**2,718…. after a year after

**1,000,000**capitalizations

The more

**capitalizations**in a year, the closer we are to the so-called

**continuous capitalization**. Then, after a year,

**1$**will become

**$e =2.7182818…**

You can easily check the calculations with a calculator.

**Note:**

The number

**e**is a

**continuous capitalization**of

**1 $**1 after

**1 year**with

**100% interest**.

**Chapter 3.3.3 The real exponential function exp(x)**

Before we get to the **complex function** version, let’s take a look at the **real version**. So an exponential function with** base e**. We can easily calculate it by multiplying the values of these functions for the **integer** exponent **0, 1, 2, 3..**.**Fig. 3-4**Exponential function values for

**integer**values

But how to calculate it for e.g. for

**x=1.234**?

We’ll do what all

**calculators**do. i.e. we decompose the function into a

**power series**, e.g. a

**Maclaurin**series.

**Fig. 3-5**

The exponential function as a power series

In this way, we will calculate the value of the function for any x using only 4 basic mathematical operations.

**Note:**

symbol ! is the so-called

**factorial**

E.g.

**3!=1*2*3=6**

**Chapter 3.3.4 Complex exponential function exp(z)**

We can now calculate the value of the exponential function **exp(x)** for any real **x**. Let’s do the same thing, but for any complex number **z**. I wonder how this thing will behave? Maybe it bites, maybe it kicks?

So let’s decompose **exp(z)** into a **complex** Maclaurin **power series**.**Fig. 3-6**The

**complex exponential**function as a

**complex Maclaurin**power function.

**Chapter 3.3.5 The special case of exp(z) or exp(jωt)****Fig. 3-6** can be a definition of a **complex exponential** function! We can easily calculate its value for any of them using only **4** basic operations that we already know. No problem, but quite tedious because **z** is a pair of **numbers**! Also the plot of **exp(z)** is hard to visualize–>see** Chapter 3.1**. Therefore, we will consider its special case when** z=jωt** for **t=0…∞** That is, when the domain is the **upper half** of the imaginary axis **Im z**. We will therefore consider the complex function **exp(jωt)**, which, as it turns out later, is ideal for analysis **sine waveforms**. And this is the basis of such fields as **electrical engineering, automatics, radio engineering, acoustics…**

**Fig. 3-7**

The exponential function

**exp(z) for z=jt**, i.e. for

**z=jωt**when

**ω=1/sec**is the so-called

**pulsation**.

The domain

**exp(z)**is the points

**0…∞**of the upper

**green imaginary**semi-axis

**Im z**. The value of the function are the points on the

**red circle**. They were determined by substituting successive numbers into the formula in

**Fig. 3-6**–

**green points**of the semi-axis

**Im z**. The

**green**points

**j0, jπ/6, jπ/2, j3π/2**and

**j2π**were marked on it. They are assigned to the appropriate

**red**points

**exp(jt)**on the

**circle**, i.e

**. +1, (0.866..+j0.5), +j, -1**and

**-j.**There are also other, but not described, intermediate

**green**points on the imaginary semi-axis

**Im z**. Corresponding

**red points**on the

**circle**are also assigned to them.

The formula in

**Fig. 3-6**shows, for example, that

**for**:

**exp(j0)=+1**

**exp(jπ/6)=0.866..+j0.5**

**exp(jπ/2)=+j**

**exp(jπ)=-1**

**exp(j3π/2)=-j**

**exp(j2π)=+1**

The

**circle**is the graph of

**exp(jt)**for

**jt=0…∞**. But we only examined the range

**jt=0…2π**, for which the image

**exp(jt)**made a full rotation. It will turn out and it can be checked that the

**second**identical rotation will be made

**for t=2π…4π**, the

**third**one

**for t=4π…6π etc…**So

**exp(jt)**is a

**periodic**function!

We were curious how this

**exp(z)**creation would behave? And who would have expected that a

**complex exponential**function could be a

**periodic**function. After all, it bites and kicks!

**Chapter 3.3.6 How did the circle in Figure 3-7 come about?**

So how did we calculate the values of the complex function **exp(jt)** for **jt=0…∞**?

We can’t use** Euler’s formula**:**exp(jt)=cos(t)+jsin(t)**

because we don’t know him yet.**exp(j0)**

The first value** exp(j0)= exp(0)=+1** is trivial because** j0=0** is a **real** number.**exp(jπ/6)**

For **jπ/6** it’s not so nice anymore. So we will use the formula **Fig. 3-6**. There are only **4** basic math operations here, but tiring work. That’s why I recommend everyone a brilliant math tool **Wolfram Alfa** *

Call up **www.wolframalpha.com** and do what the drawing says. Just remember that for **WolframAlpha**, the imaginary number is **i**, not “electrical” **j**.

After entering the first **6** components from the **Maclaurin** series **Fig. 3-6** into the dialog box and clicking the appropriate button, the program will calculate the approximation **exp(jπ/6)**.**Note**:**Fig. 3-8** is slightly different from what **WolframAlfa** showed, but the content is the same. I just moved the various elements around to make the drawing more compact. This also applies to subsequent calls to **WolframAlpha**.

*I wrote more about the** WolframAlfa** program in the **Rotating Fourier Series** course in **chapter 11.2**.

**Fig. 3-8****exp(jπ/6)** as the **first 6** components of the **Maclaurin** series from **Fig. 3-6**

Note that the approximate value of **exp(jπ/6)≈0.866…+j0.5**… was calculated by the program using only the **first** **6 components** of the power series. But he did it quite decently. To the casual observer, the result heals almost exactly on a **circle** of radius** r=1** at an angle of **π/6=30°**. while the calculated data is **r=1.00003** and** θ=29.9993°** as Polar coordinates. The computed **exp(jπ/6)** is also shown in the complex plane as **position** in the complex plane. That **Wolfram** is nice!

The remaining **exp(jt) for jπ/2, j3π/2** and **j2π** will also be calculated using **Wolfram Alfa**, but with the specialized complex function **exp(z)**.

How does** exp(z)** differ from the Maclaurin series in** Fig. 3-6**?

I think two things:

-More ingredients than **6**. How much exactly? I don’t know.

-Assumption that the function is periodic.

So let’s calculate.**exp(jπ/2) calculating**

**Fig. 3-9**

**exp(jπ/2)**

How perfectly he calculated it!

**exp(jπ) calculating**

**Rys. 3-10**

**exp(jπ)=-1**

Probably

**exp(jπ)=-1**or

**exp(jπ)+1=0**

is one of the most beautiful equations of mathematics. In this

**short**formula are used all the most important numbers

**0,1,**

**π,e**and

**j**

**.**

**exp(j3π/2) calculating**

**Fig. 3-11**

**exp(j3π/2)**

**exp(j2π) calculating**

**Fig. 3-12**

**exp(j2π)**

Notre that

**exp(j2π)= exp(j0)=+1**

** Chapter 3.7 Euler Formula**

Returning to the previously mentioned **Euler’s formula****exp(jt)=cos(t)+jsin(t)**

or more generally**exp(jωt)=cos(ωt)+jsin(ωt)**

where **ω** is the so-called angular **velocity** in **radian/sec**, it is more known in the “angular version” where instead of **ωt** there is angle **α**.**exp(jα)=cos(α)+jsin(jα)**

The **formula** has been known since the** 18th** century and has a very easy graphic interpretation

**Fig. 3-13z=exp(jα)=cos(α)+jsin(jα)**Check this formula for e.g.

**α=0, π/6, π/2, π,3π/2**and

**2π**and you will get the results you calculated with

**Wolfram Alfa**in

**Chapter 3.3.5.**

E.g.

**exp(jπ/6)=0.866..+j0.5**