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Background
B1
Mathematical Preliminaries
(Chapter B)
• Complex numbers
• Sinusoids
• Sketching signals (also MATLAB)
• Cramer’s rule
• Partial fraction expansion
• Vectors and matrices
• Miscellaneous (on your own)
• MATLAB (laboratories will be 70% MATLAB)
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View Full Document Background
B2
Complex numbers
•
Algebra (aljabra) introduced from alKhwarizmi’s
book into Italy through Gerard and Fibbonaci
•
Some Italian mathematicians developed an
algebraic solution to the cubic equation.
For some
equations with real solutions, the form of the
equations contained square roots of negative
numbers.
Mathematicians initially used to obtain
correct real numbers.
•
Rene’ Descarte introduced the word “imaginary”
because in his mind the numbers did not represent
anything “real”
•
By the 17
th
century, a geometry of complex
numbers was worked out by Wallis and was used
by Newton and others.
•
Complex numbers were subsequently developed by
Euler, Gauss, Cauchy and others
John Wallis (16161703)
One of the first to provide
a geometric interpretation
for complex 1.
Leonhard Euler (17071783)
Famous Swiss physicist and
mathematician
Background
B3
• Complex numbers are an extension of
ordinary numbers
• Complex plane (Cartesian and polar coordinates)
(Algebra of) Complex numbers
Re
Im
z
aj
b
z
a
z
b
=+
=
=
cos
sin
ar
br
θ
=
=
(cos
sin )
za
j
j
=
+
Real
Imaginary
z
r
a
b
b
θ
z*
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View Full Document Background
B4
• The Euler formula
• Proof: Maclaurin series to expand the terms
Algebra of Complex numbers
cos
sin
j
ej
θ
=+
()
23456
23
45
6
246
35
1
...
2!
3!
4!
5!
6!
1
...
cos
1
...
sin
...
3!
5!
j
jjjjj
jj
j
θθθθθ
θθ
θθθ
=
+
++++++
=
=+ − −
+ +
− +
=− + − +
=−
+
−
Background
B5
Algebra of Complex numbers
cos
sin
ar
br
θ
=
=
22
1
tan
ra
b
b
a
−
=+
⎛⎞
=
⎜⎟
⎝⎠
Real
Imaginary
z
r
a
b
θ
z
r
z
=
∠
=
j
z
z
ze
∠
=
111
1
j
jz
j
ee
zr
e
r
z
−
−∠
==
=
j
z
a
j
e
=
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View Full DocumentBackground
B6
•
We define z*, the conjugate of z, as
•
To find the conjugate of any number, replace (j) by (–j) in that number
•
A few formulas:
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This note was uploaded on 09/28/2009 for the course BME 343 taught by Professor Emelianov during the Fall '09 term at University of Texas at Austin.
 Fall '09
 Emelianov

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