Post on 26-Feb-2018
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Module 1Complex Numbers
Engr. Gerard AngSchool of EECE
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Definition of a Complex Number
A complex number z is an ordered pair (x,y) of realnumbers x and y written as
z = (x,y)Where:
x = the real part of z written as x = Re zy = the imaginary part of z written as y = Im z
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Geometrical epresentationof a Complex Number
A omplex number z an be plot asa point (x,y) in the xy plane, now
alled the complex plane or
sometimes alled as the Arganddiagram named after !ean"Robert Argand#
!llustration
$lot the following:z% = & ' z = * +
z = " ' z & = " +
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"orms of Complex Numbers1. ectangular "orm
z = x ' yWhere:
x = real part y = imaginary part = "operator #. $rigonometric "orm
z = r( os- ' sin-) z = r is-%. &olar "orm
Where:r is the absolute .alue, amplitude or modulus (mod) of z
- is the argument (arg) or phase of z'.Exponential "orm
Where: - = argument in radians
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$he ()*perator
/he ()operator is an operator used to indi ate the ounter" lo 0wiserotation of a .e tor through 123#
&o+ers of ( & = ( ) = %
= "%*
= (&
) = = ( ) = "
!llustration,4.aluate the following:
%# 1 %*
# 2 *# &
# 2 5# %%
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*perations on Complex Numbers
1. E-ualit of $+o Complex Numbe rs6et: z % = x % ' y %
z = x ' y
/hen z % = zIf x% = x and y % = y
!llustrationIf , find R and -#
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*perations on Complex Numbers
#. Addition/Subtraction of Complex Numberslet z % = x % ' y % and z = x ' y
thenz% 7 z = (x % ' y %) 7 (x ' y )
z% 7 z = (x % 'x ) 7 (y% ' y )8ote:
Add9subtra t real part to real part and imaginarypart to imaginary part#
!llustration(* ' ) ' ( + &) + (5 + ) = ;;;
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*perations on Complex Numbers
%. Multiplication and Di0ision of Complex Numberslet andthen
)
and
!llustrationIf , and < = ' , find
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*perations on Complex Numbers
'. ationalization of a Complex Number let z = x 7 ythenwhere: is the omplex on ugate of z
!llustration,
Rationalize
. 2ogarithm of a Complex Number let
then ln z = ln r ' -
!llustration, 4.aluate ln ( ' )
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*perations on Complex Numbers
3. &o+ers and oots of Complex Numberslet z = r( os- ' sin-) andthen z n = r n( os n- ' sin n-) De Moi0re4s "ormula
alsowhere: 0 = 2, %, , , n + %
!llustrationind the roots of the following
%# # #
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Sample &roblems
%# >implify:a# (* ' &)( ' )( + ) b#
# If , find the real and imaginary parts of the omplexnumber #
# If (a ' b) ' (a + b) = ( ' *) ' ( + ), find the .alues ofa and b
If x and y are real, sol.e the e?uation, #*# ind the modulus and argument of ( + )(* ' % )9(% ' ) #
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$rigonometric and 5 perbolic"unctions of Complex Numbers
1. Euler4s "ormula
#. elationships bet+een $rigonometric "unctions and 5 perbolic"unctions
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$rigonometric and 5 perbolic"unctions of Complex Numbers
%. $rigonometric "unctions of Complex Numbers
'. 5 perbolic "unctions
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Sample &roblems
4.aluate the following:%# sin ( ' )
# os (*@ ' )
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$rigonometric and 5 perbolic"unctions of Complex Numbers
. elationships 6et+een 5 perbolic and $rigonometric"unctions
3. 5 perbolic "unctions of Complex Numbers
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Sample &roblems
4.aluate the following:%# sinh (& + )
# osh ( ' )
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!n0erse $rigonometric and 5 perbolic"unctions of Complex Numbers
1. !n0erse $rigonometric "unctions of Complex Numbers
#. !n0erse 5 perbolic "unctions of Complex Numbers
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Sample &roblems
4.aluate the following:%# sinh"% ( ' )
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2oci &roblems
We are sometimes re?uired to find the lo us of a point whi h mo.es inthe Argand diagram a ording to some stated ondition#
Sample &roblemsIf z = x ' y, find the e?uation of the lo us defined by the following:
a# mod z = *#b# arg z = @9
#d# arg (z ) = " 9&