1. Chp3 MATLAB Functions: Part1
Engr/Math/Physics 25
Chp3 MATLAB Functions: Part1
Bruce Mayer, PE
Registered Electrical & Mechanical Engineer

2. Learning Goals
Understand the difference between Built-In and User-Defined Functions
Write User Defined Functions
Describe Global and Local Variables
When to use SUBfunctions as opposed to NESTED-Functions
Import Data from an External Data-File
As generated, for example, by an Electronic Data-Acquisition System

3. Functions MATLAB Has Two Types of Functions
Built-In Functions Provided by the Softeware
e.g.; sqrt, exp, cos, sinh, etc.
User-Defined Functions are .m-files that can accept InPut Arguments and Return OutPut Values

4. Getting Help for Functions
Use the lookfor command to find functions that are relevant to your application
For example typing lookfor complex returns a list of functions that operate on complex numbers (more to come):
>> lookfor complex
ctranspose.m: %' Complex conjugate transpose.
COMPLEX Construct complex result from real and imaginary parts.
CONJ Complex conjugate.
CPLXPAIR Sort numbers into complex conjugate pairs.
IMAG Complex imaginary part.
REAL Complex real part.
CPLXMAP Plot a function of a complex variable.

5. Built-In Exponential Functions
Command
Conventional Math Function
exp(x)
Exponential; ex
sqrt(x)
Square root; x
log(x)
Natural logarithm; lnx
log10(x)
Common (base 10) logarithm; logx = log10x
Note the use of log for NATURAL Logarithms and log10 for “normal” Logarithms
This a historical Artifact from the FORTRAN Language – FORTRAN designers were concerned with confusing ln with “one-n”

6. Built-In Complex-No. Functions
Command
Conventional Math Function
abs(x)
Absolute value (Magnitude or Modulus)
angle(x)
Angle of a complex number (Argument)
conj(x)
Complex Conjugate
imag(x)
Imaginary part of a complex number
real(x)
Real part of a complex number
Useful for Analyzing Periodic Systems
e.g., Sinusoidal Steady-State Electrical Ckts

7. Built-In Rounding Functions
Command
Conventional Math Function
ceil(x)
Round to nearest integer toward +
fix(x)
Round to nearest integer toward zero
floor(x)
Round to nearest integer toward −
round(x)
Round toward nearest integer.
sign(x)
Signum function: +1 if x > 0; 0 if x = 0; −1 if x < 0.
Sort of like the FFT problem we did in lab
Graph

8. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Engr/MTH/Phys 25
Complex Numbers
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

9. Complex Numbers – Math What do We Do with? Factoring
Let’s Make-Up or IMAGINE

10. Complex No.s – Basic Concept
Discriminant D
World of REAL numbers
x = 2a
b2– 4ac b
a  x2 + b  x + c = 0
Solution(s) of a quadratic equation exist only for non-negative values of D !
World of COMPLEX numbers
Solution(s) of a quadratic equation exist also for negative values of D !
x =
± j 2a
|b2– 4ac| b
In Engineering √(-1) = j in Math √(-1) = i

11. Complex Number, z, Defined
z = x + j y
Imaginary part
y = Im(z)
Real part
x = Re(z)
Im(z)
Real
numbers
Real
numbers
Im(z) = 0
Complex numbers
Complex
numbers
Im(z)  0
REAL no.s described by a (number) LINE. Complex no.s described by a PLANE
Re(z)

12. Complex No.s – Basic Rules
Powers of j
j2 = –1
j3 = –j
j4 = +1
j –1 = –j
j4n = +1; j4n+1 = +j ; j4n+2 = –1; j4n+3 = –j for n = 0, ±1, ± 2, …
Equality
z1 = x1 + j y1
z2 = x2 + j y2
z1 = z2,  x1 = x2 AND y1 = y2
Do on board j-1 = -j
Addition
z1 + z2, = (x1 + x2) + j ( y1 + y2)

13. Complex No.s – Basic Rules cont
z1 = x1 + j y1
z2 = x2 + j y2
Multiplication
z1 z2, = (x1 x2 – y1 y2) + j (x1 y2 + x2 y1)
The complex conjugate
z = x + j y
z* = x – j y
(z + z*) = x  Re(z) 2 1
(z – z*) = j y  j Im(z) 2 1
z z* = x2 + y2
Multiply by F.O.I.L. * For division multiply by 1, where on is the conjugate of the DIVISOR divided by itself z2 z1 =
(x22 + y22)
z2 z2*
z1z2*
x1 x2 + y1 y2
+ j
y1 x2 – x1 y2
Division

14. Complex No.s – Graphically
The Argand diagram
Modulus (magnitude) of z
r = mod z = |z| =
x2 + y2 x y r 
z = x + i y
Im(z)
Re(z)
Argument (angle) of z
arctan
 = arg z = x y
+ p, if x < 0
, if x > 0
x = r cos 
Polar form of a complex number z
z = r (cos  + j sin )
y = r sin 

15. Complex No.s – Polar Form
z1 = r1 (cos 1 + j sin 1)
z2 = r2 (cos 2 + j sin 2)
Multiplication
z1 z2 = r1r2 (cos (1 + 2) + j sin(1 + 2)) x y r 
z = x + j y
Im(z)
Re(z)
|z1z2| = |z1| |z2| ; arg(z1z2) = arg(z1) + arg(z2)
Division
= (cos (1 – 2) + j sin(1 – 2)) z2 z1 r2 r1
= ; arg( ) = arg(z1) – arg(z2)
|z2|
|z1| z2 z1

16. Polar Multiplication Proof
Consider:
Then
But
Then factoring out j, & Grouping
Recall Trig IDs

17. Polar Multiplication Proof cont
Using Trig ID in the Loooong Expression
So Finally
Q.E.D.

18. De Moivre’s Formula z1 z2 = r1r2 (cos (1 + 2) + j sin(1 + 2))
z1 z2…zn = r1r2 …rn [cos (1 + 2 + …+ n) + j sin(1 + 2 + …+ n)]
z1 = z2=…= zn
zn = rn (cos (n) + j sin(n))
Named after French Mathematician Abraham de Moivre ( )
r = 1
(cos  + j sin )n = cos (n) + j sin(n)
French Mathematician Abraham de Moivre ( )

19. Complex Functions A complex function A complex conjugate function
Real function
f(x) = g(x) + j h(x)
Real function
A complex conjugate function
f*(x) = g(x) – j h(x)
f(x) f*(x) = g2(x) + h2(x)
f(z) = z2 + 2z + 1; z = x + j y
Example:
g(x,y) = (x2 – y2 + 2x + 1)
h(x,y) = 2y (x + 1)
f(z) = g(x,y) + j h(x,y)

20. Verify by MuPad
From the last Line (collect comand) collect real and imaginary parts 

21. Euler’s Formula ei e–i cos(θ) sin(θ) Im(z) Re(z) 1 –1  – 
A power series for an exponential
cos(θ)
sin(θ)
e^j(-x) = cos(-x) + jsin(-x) => use trig IDs: cos(-y) = cosy & sin(-y) = -siny
A complex conjugate is also inverse

22. Complex Numbers – Engineering
Natural No.s = COUNTING No.s = 1, 2, 3, 4, ...
WHOLE No.s = 0, 1, 2, 3, 4, ... (includes the null = ZERO)
INTEGER No.s = -3, -2, -1, 0, 1, 2, 3, ...

23. Complex Number Calcs Im(z) Consider a General Complex Numbery r 
z = x + j y
Im(z)
Re(z)
Consider a General Complex Number
This Can Be thought of as a VECTOR in the Complex Plane
This Vector Can be Expressed in Polar (exponential) Form Thru the Euler Identity
Where
Then from the Vector Plot
Need to be careful abaout the values of θ when z is in the LEFT-half of the Argand Plane => must Add 180 degrees

24. Complex Number Calcs cont
Consider Two Complex Numbers
The PRODUCT n•m
The SUM, Σ, and DIFFERENCE, , for these numbers
Complex DIVISION is Painfully Tedious
See Next Slide

25. Complex Number Division
For the Quotient n/m in Rectangular Form
Use the Complex CONJUGATE to Clear the Complex Denominator
The Generally accepted Form of a Complex Quotient Does NOT contain Complex or Imaginary DENOMINATORS
The Exponential Form is Cleaner
See Next Slide

26. Complex Number Division cont.
For the Quotient n/m in Exponential Form
Look for lots of this in ENGR43
However Must Still Calculate the Magnitudes and Angles

27. Root of a Complex Number
How to Find?
Use Euler
Note that θ is in the 2nd Quadrant
Thus
In This Case

28. Root of A Complex Number
Now use Properites of Exponents
Use Euler in Reverse
In this Case
By MATLAB
(-7+19j)^(1/3)
ans = i

29. ln of a NEGATIVE Number (1)
Im(z)
What is r
z=(-19,0)
 = π –r
Re(z)
State −19 as a complex no. –r
Find Euler Reln Quantities r, & θ

30. ln of a NEGATIVE Number (2)
Im(z)
Note that θ is 180º, NOT Zero
Thus the Polar form of −19 r
z=(-19,0)
 = π –r
Re(z) –r
Taking the ln

31. Log of a NEGATIVE Number
Im(z)
Recall complex forms for −19 r
z=(-19,0)
 = π –r
Re(z) –r
Taking the common (Base-10) log

32. sin or cos of Complex number
Recall from Euler Development
By Sum & Difference Formulas

33. sin or cos of Complex number
Thus
From Previous Slide
So Finally

34. MATLAB Complex Operations
>> a = 3+2j;
>> b = -4+5i;
>> c = -5-j*4;
>> d = i;
>> Mag_b = abs(b)
Mag_b =
6.4031
>> c_star = conj(c)
c_star = i
>> ac = a*c
ac = i

35. Complex Ops >> b_d = b/d b_d = 5.0000 + 4.0000i
>> c_b = c/b
c_b = i
>> c_a = c/a
c_a = i
>> Re_d = real(d)
Re_d =
>> Im_b = imag(b)
Im_b = 5

36. Complex Ops >> b_sq = b^2 b_sq = -9.0000 -40.0000i
>> b_cu = b^3
b_cu =
2.3600e e+002i
>> cos_a = cos(a)
cos_a = i
>> exp_c = exp(c)
exp_c = i
>> log_b = log10(b)
log_b = i

37. Complex Ops >> 180*angle(z)/pi >> r = 73; ans =
>> x = -23; y = 19;
>> z2 = complex(x,y)
z2 = i
>> r = 73;
>> theta = 2*pi/11;
>> theta_deg = 180*theta/pi
theta_deg =
>> z = r*exp(j*theta)
z = i
>> abs(z)
ans = 73

38. Caveat MATLAB Accepts But NOT >> z2 = 3+7j;
>> z3 = 5 + i*11;
>> z4 = *j;
>> z2
z2 = i
>> z3
z3 = i
>> z4
z4 = i
>> z5 = 7 + j5;
??? Undefined function or variable 'j5'.

39. Leonhard Euler (1707-1783) All Done for Today
".. indeed, far and away the most prolific writer in the history of the subject" writes Howard Eves in An Introduction to the History of Mathematics. Euler's contribution to mathematics is represented here by a few of the notations conventionalized by him or in his honor. Around the world, these are read, written, and spoken thousands of times every day:
e for the base of the natural logarithm (a.k.a. "the calculus number") a, b, c for the sidelengths of a triangle ABC f(x) for functional value R and r for the circumradius and inradius of a triangle sin x and cos x for values of the sine and cosine functions i for the imaginary unit, the "square root of -1" capital sigma for summation. capital delta for finite difference.
Euler grew up near Basel, Switzerland, and studied at an early age under Johann Bernoulli. He finished studies at the University of Basel when only 15 years old. From 1727 to 1741, Euler worked in St. Perersburg, Russia, and then moved to the Akademie in Berlin. In 1766 he returned to St. Petersburg, where he remained.

40. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Engr/Math/Physics 25
Appendix
Time For
Live Demo
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

42. Complex Integration Example
Us EULER to Faciltitate (Nasty) AntiDerivation
I =
I = Re
I = Re

43. Illuminate Previous Slide
By EULER
Using Term-by-Term Integration
As “i” is just a CONSTANT
Taking the REAL Part of the above

44. Rotation Operator Im(z) r z'(x',y') z = r (cos  + j sin ) = z(x,y) a
Re(z) r –r  a
z'(x',y')
Polar form of a complex number
z = r (cos  + j sin ) =
x'= r cos( + a) = r(cos cosa – sin sina)
y'= r sin( + a) = r(sin cosa + cos sina)
x'= x cosa – y sina
y'= x sina + y cosa
The function eja can be regarded as a representation of the rotation operator R(x,y) which transforms the coordinates (x,y) of a point z into coordinates (x',y') of the rotated point z' : R(x,y) = (x',y') .

45. Periodicity
z =
z' =
z' = z'
The function ej occurs in the natural sciences whenever periodic motion is described or when a system has periodic structure.

46. Periodicity on a Circle
or n nth roots of 1 z0
Im(z)
Re(z) 1 z1 z2 2
z0 = ej0 ; z1 = ej 2p/3 ; z2 = ej 4p/3
zk = ej 2pk/3, where k = 0,1,2
zk3 = (ej 2pk/3)3 = ej 2pk = 1  = zk
z0 = ej0 = 1
z1 = ej 2p/3 = j
z2 = ej 4p/3 = – j 2 1
3 roots of third degree of 1
zk = ej 2pk/3, where k = 0, ±1

47. Periodicity on a Circle cont
or n nth roots of 1 z0
Im(z)
Re(z) z1 z2 z3
z-2
z-1
6 sixth roots of 1
Used for description of the properties of the benzene molecule.
z0 = ej0 = 1
z±1 = e±j p/3 = ± j
z±2 = e ±j 2p/3 = ± j 2 1
n nth roots of 1
zk = ej 2pk/n for k =
0, ±1, ±2, ... , ±(n –1)/ if n is odd
0, ±1, ±2, ... , ±(n/2 –1), n/2 if n is even
Such functions are important for the description of systems with circular periodicity.

48. Periodicity on a Line f(x) = f(x + a) f(x) = f(x,y,z) = a a a a a ...
Periodic function
f(x) =
f(x) = f(x + a)
Generalization for three-dimensional periodic systems
f(x,y,z) =
Such functions are important for the description of periodic systems such as crystals.

49. Trig on complex numbers