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. 3-3
e=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 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(j2π)=+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 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
r≈1.00003 and 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 6
2. 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. 3-13
z=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.