advanced engg math module 1

7/25/2019 Advanced Engg Math Module 1

1/19

Module 1Complex Numbers

Engr. Gerard AngSchool of EECE


2/19

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


3/19

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 & = " +


4/19

"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


5/19

$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# %%


6/19

*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 -#


7/19


#. 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 + ) = ;;;


8/19


%. Multiplication and Di0ision of Complex Numberslet andthen

)

and

!llustrationIf , and < = ' , find


9/19


'. 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 ( ' )


10/19


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

%# # #


11/19

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(% ' ) #


12/19

$rigonometric and 5 perbolic"unctions of Complex Numbers

1. Euler4s "ormula

#. elationships bet+een $rigonometric "unctions and 5 perbolic"unctions


13/19


%. $rigonometric "unctions of Complex Numbers

'. 5 perbolic "unctions


14/19

Sample &roblems

4.aluate the following:%# sin ( ' )

# os (*@ ' )


15/19


. elationships 6et+een 5 perbolic and $rigonometric"unctions

3. 5 perbolic "unctions of Complex Numbers


16/19

Sample &roblems

4.aluate the following:%# sinh (& + )

# osh ( ' )


17/19

!n0erse $rigonometric and 5 perbolic"unctions of Complex Numbers

1. !n0erse $rigonometric "unctions of Complex Numbers

#. !n0erse 5 perbolic "unctions of Complex Numbers


18/19

Sample &roblems

4.aluate the following:%# sinh"% ( ' )


19/19

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&

advanced engg math module 1

Documents