### Complex numbers

**Chapter 3 ****Complex functions****Chapter 3.1 Introduction**You know addition, subtraction, multiplication and division. But what about

**complex**functions?

There are used in electricity, automatics,,,

-rational

-exponential

**complex**functions mainly

The other

**complex**functions are used rarely and we will not work on them. The

**complex**functions diagrams are more complicated than “normal”-

**real number functions**diagrams. Every

**2-d**

**z**point is assigned with the

**complex**

**2-d**function point. We step in

**4-d**space.

It’s difficult to imagine for everyman!

**Chapter 3.2 Rational complex functions**This function is calcualated when addition, subtraction, multiplication and division are used only.

**Fig. 3-1**

Rational Complex Function

**Chapter 3.3 Complex exponential function exp(z)**

**Chapter 3.3.1 Definition**

**Fig. 3-2**

Complex exponential function

**Chapter 3.3.2 Number e=2,7182818…**Irrational number

**e=2,71828..**is the

**fourth**you should to know. The first are

**0,1**and

**π.**Everybody knows

**0**and

**1**origin*.

**π**number is a “

**circumference/diameter**ratio”. But

**e**number? Is any so simple defintion. There is , but not so obvious. Additionally, it came from the world of banks.

*

**0**and

**1.**Where are they from? It looks like an existential question.

**Fig. 3-3e=2,7182818… **number and

**bank interests**

**a.**I established

**1$**as a deposit in the very good bank. The inerests are

**100%**per year! Note that

**1=100%**

**b.**It should be

**2$**after a

**1**year

**c.**I agreed with a bank, that the interests is the same-

**100% per year**, but I will receive the

**1.5$**cash after half a year.

**Capitalization**is after after half a year in

**banker**language.

**d**, The

**capitalization**after after a

**1**year but after

**2**capitalisations

**e.**The

**capitalization**after a

**1 year but after 3 capitalisations**

**f.**The

**capitalization**after a

**1 year but after 10 capitalisations**

**g.**The same as

**f.**but in decimal formula

**h.**The

**capitalization**after a

**1 year but after 100 capitalisations**

**i.**The

**capitalization**after a

**1 year but after 1 000 000 capitalisations**

You can test the calculations by calculator

The more is more is

**capitalizations**in a year –> we are near to the so called

**continuous**

**capitalization**

**Conclusion:**

**e**is a

**continuous**

**capitalization**:

-after

**one**year

-when

**100%**interests

-when we invested

**1$**

**Chapter 3.3.3 Rational exponential function exp(x)**There’s no problems with

**exp(x)**calculations, when x=0,1,2…n

**Fig. 3-4**

exp(x)exponencial function when

exp(x)

**x**is a

**natural number.**

But what about any

**real**number for example

**x=1.234**?. We do likewise all the scientific calculators. We exchange

**exp(x)**into

**power series.**

**Maclaurin**series here:

**Fig. 3-5**

**Exponential**function

**exp(x)**as a Maclaurin power series.

It’s possible to calculate any

**exp(x)**when

**4**fundamental mathematical operation are used only.

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

**exp(z)**into

**Maclaurin**power series here:

**Fig. 3-6**

**Exponential**function

**exp(z)**as a Maclaurin power series.

The

**complex**number

**z**is here, and not a

**real**number as in the

**Fig. 3-5.**

**Chapter 3.3.5 exp(jωt) as a exp(z) special case**

**Fig. 3-6**formula is a

**exp(z)**function definition. We can calculate any

**exp(z)**with the aid of

**4**fundamental arithmetic operations. It isn’t difficult but very exhausting job! What is more! The diagram is hard to imagine too.–>see

**chapter. 3.1**. Hence, we will study a special case when

**z=jωt**for

**t=0…∞**. This is

**exp(jωt)**function ideal to analize the phenomenons from the electricity, automatics, acoustics and others.

**Fig. 3-7**

**Exp(jt)**exponential function i.e.

**exp(z)**for

**z=jωt**when

**ω=1/sec**here, is so called pulsation.

The upper

**Im z**

**green 0…∞**semi-axis is a

**exp(z)**function domain. The

**red**circle is a

**exp(jωt)**independent function variable. The

**red**circle points are calculated when

**green**points

**j0**,

**jπ/6,**

**jπ/2**,

**j3π/2**i

**j2π**are inserted into formula

**Fig. 3-6**.

These

**red**circle points are:

exp(

**j0**)=+1

exp(

**jπ/6**)=0.866..+j0.5

exp(

**jπ/2**)=+j

exp(

**jπ**)=-1

exp(

**j3π/2**)=-j

exp(j

**2π**)=+1

There are other intermediate and not labelled

**green**points on the

**green 0…∞**semi-axis. The appropriate

**red**circle points are calculated similarly. It means that

**red**circle is a

**0…j2π**semi-axis

**exp(jt**) function value. The point

**z=1+j0**did a

**1**full rotation. The next

**rotations**will be made for

**j2π…j4π**,

**j4π…j6π…**The

**complex**

**exp(jωt)**is a

**periodical**function! Note that the

**real**

**exp(x)**function is a contradiction of the

**periodical**function.

**Chapter 3.3.6 How did red Fig. 3-7 circle arise?**Other words. How did we calculate

**exp(jt)**?

We can’t use Euler formula

**exp(jt)=cos(t)+jsin(t)**

because we don’ know it yet.

Let’s try other possible methods.

**exp(j0).**This is the easiest problem.

**exp(j0)= exp(0)=+1**because

**j0=0**is real number.

**exp(jπ/6)**It isn’t so nice now! We have to use

**Fig. 3-6**formula with

**4**principal mathematical operations only.

This job is very exhausting so I propose

**WolframAlfa-**the brilliant mathematical tool.

Call www.wolframalpha.com and follow the picture instructions.

Note that program uses symbol

**i**instead of “electrical”

**j**.

Write into dialog window first

**6**Maclaurin power series components.

The will calculate an approximation of the

**e(jπ/6).**

*****see

**chapter 2.3.2**in the article

**“Fourier Series”**

**Fig. 3-8**

exp(jπ/6)as first

exp(jπ/6)

**6**Maclaurin power series components–>see

**Fig. 3-6**

There were

**6**Maclaurin power series components only, but the calculated value

**exp(jπ/6)≈0.866…+j0.5**is very near to the ideal value!

It’s better seen when polar coordinates are used. Compare

Calculated

**and**

r≈1.00003

r≈1.00003

**θ≈29.9993º**

Ideal (theoretical)

**r=1**and

**θ=30º**

Conclusion:

**exp(jπ/6)**point lies on the circle

**r=1**and

**θ=30º**

**.**

The next **red **circle points will be calculated by the specialized **exp(z) **function.

This function:**1**. has more compnents than **62**. uses the knowledge that is a periodical function

Let’s calculate!

**exp(jπ/2)**

**Fig. 3-9**exp(jπ/2)

Ideal!

**exp(jπ)**

**Fig. 3-10**exp(jπ)

This is reportedly the most beautiful mathematical formula!

**exp(jπ)=-1**or another words

**exp(jπ)+1=0**

**exp(j3π/2)**

**Rys. 3-11**exp(j3π/4)

**exp(j2π)**

**Rys. 3-12**exp(j2π)

Note, that

**exp(j2π)= exp(j0)=+1**–>periodical function

**Chapter 3.3.6 Euler formula****exp(jt)=cos(t)+jsin(t)**or more generally

**exp(jωt)=cos(ωt)+jsin(ωt)**

when

**ω**jest is so called

**angular velocity**or

**pulsation**in

**radian/sec**

It’s more known as

**angular**version when

**α=ωt**

**exp(jα)=cos(α)+jsin(jα)**

This is **XVIII **century formula and has an easy geometrical** interpretation**

**Fig. 3-13z=exp(jα)=cos(α)+jsin(jα)**Test it for

**α=0,**

**π/6, π/2, π,3π/2**and

**2π**. The effect is the same as

**chapter 3.3.6**.

This is

**z**complex number interpretation as a

**vector**with length=

**|z|**(module z) and an angular

**α**. see

**chapter 2.3 multiplication**

So any complex number is

**z=|z|exp(jα)=|z|[cos(α)+jsin(jα)]**

Note

**The**

**|z|=1**in the

**Fig. 3-13**.