Complex Numbers and Functions.pdf

(83 KB) Pobierz
Complex Numbers and Functions
Complex Numbers and Functions
______________________________________________________________________________________________
Natural is the most fertile source of Mathematical Discoveries
- Jean Baptiste Joseph Fourier
The Complex Number System
Definition:
A complex number z is a number of the form zab
= +
, where the symbol i
= −
1
is called imaginary unit and ab R
,
.
a is called the real part and b the imaginary
part of z , written
a
Re and b
z
Im .
z
With this notation, we have z
Re
z i z
Im .
The set of all complex numbers is denoted by
{
CaibabR
,
.
If b
0, then zai
= + =
0, is a real number. Also if a
a
0, then z
= + =
0
ib ib
, is
a imaginary number; in this case, z is called pure imaginary number .
Let aib
and cid
be complex numbers, with abcd R
,,,
.
1. Equality
aib cid
+ = +
if and only if ac bd
and
.
Note:
In particular, we have zab
= + =
0 if and only if a
0
and
b
0
.
2. Fundamental Algebraic Properties of Complex Numbers
(i). Addition
(
aib cid ac ibd
) (
= + +
) (
) (
).
(ii). Subtraction
(
aib cid ac ibd
) (
= − +
) (
) (
).
(iii). Multiplication
(
a ib c id
)(
) (
ac bd i ad bc
) (
).
Remark
(a). By using the multiplication formula, one defines the nonnegative integral
power of a complex number z as
zzz zzzz
1
,
2
,
3
2
,
3
,
z zz
n
1
.
Further for z
0, we define the zero power of z is 1; that is, z 0
1
.
(b). By definition, we have
i 2
= −
1
, i
3
= −
i
, i 4
1
(iv). Division
If cid
+ ≠
0 , then
aib
cid
ac bd
cd
2
2
+
i
bc ad
cd
2 .
+ +
− +
n
.
2
139172035.007.png 139172035.008.png 139172035.009.png
Remark
(a). Observe that if aib
+ =
1, then we have
cid
c
cd
+
i
d
cd
.
(b). For any nonzero complex number z , we define
z
1
1 ,
z
where z − 1 is called the reciprocal of z .
(c). For any nonzero complex number z , we now define the negative integral
power of a complex number z as
z
1
1 ,
z
2
− −
z z
1 1
,
z
3
z z
2
1
,
3
,
z
− + −
n
z z
1 1
.
z
1
(d). i
1
= = −
i
, i −
2
= −
1, i
3
i
, i −
4
1.
i
and
12 3
(i). Commutative Laws of Addition and Multiplication :
, ,
z
,
zz z z
zz z z
1
+ = +
=
2
2
1
;
.
(ii) Associative Laws of Addition and Multiplication :
12
21
z
1
(
z z
2
3
) (
z z
1
2
)
z
3
;
zzz zz z
123
(
) (
12 3
) .
(iii). Distributive Law :
zz z zz zz
12 3
(
)
12 13
.
(iv). Additive and Multiplicative identities :
z
+ = + =
⋅ = ⋅ =
00
11
z z
;
z
z z
.
(v). z
+ − = − + =
()()
z
z z
0
Complex Conjugate and Their Properties
Definition:
Let
a
ib
C
,
a
,
b
R
.
The complex conjugate , or briefly conjugate , of z is
defined by
zaib
= −
.
For any complex numbers zz z C
, ,
12
,
we have the following algebraic properties of
the conjugate operation:
,
(ii). zz zz
1
+ = +
2
1
2
1
− = −
2
1
2
,
(iii). zz z z
12
= ⋅
1 2
,
n
3. More Properties of Addition and Multiplication
For any complex numbers zz z
z
(i). zz zz
139172035.001.png 139172035.002.png
 
(iv).
z
z
1
=
z
z
1
, provided z 2 0
,
2
2
(v). zz
,
(vi).
z
n
( )
z
n
, for all nZ
,
(vii). zz
if and only if Im
z
0
if and only if Re 0,
(ix). zz
= −
z
z
2Re ,
(x). zz i z
+ =
z
− =
(Im),
2
(xi).
zz
Re
z
) ( )
2
Im .
z
2
Modulus and Their Properties
Definition:
The modulus or absolute value of a complex number za b
= +
, ab R
,
is defined as
2 2 .
That is the positive square root of the sums of the squares of its real and imaginary
parts.
z
ab
For any complex numbers zz z C
, ,
12
,
we have the following algebraic properties of
modulus:
(i). z
0
;
and z
0
if and only if
z
0
,
(ii). zz
12
z z
1 2
,
(iii).
z
z
1
z
1
, provided z 2
0
,
z
2
2
(iv). zz z
= = −
,
(v). z
zz
,
(vi). z
Re
z
Re ,
z
Im Im ,
(viii). zz z z
z
z
1
+ ≤
2
1
2
, ( triangle inequality )
(ix).
z
1
z
2
≤ +
z z
1
2
.
The Geometric Representation of Complex Numbers
In analytic geometry, any complex number za babR
= +
,,
can be represented by
(,) in xy- plane or Cartesian plane. When the xy- plane is used in this
way to plot or represent complex numbers, it is called the Argand plane 1 or the
complex plane . Under these circumstances, the x- or horizontal axis is called the axis
of real number or simply, real axis whereas the y- or vertical axis is called the axis of
imaginary numbers or simply, imaginary axis .
1 The plane is named for Jean Robert Argand, a Swiss mathematician who proposed the representation of complex numbers in
1806.
(viii). z
(vii). z
a point zPab
139172035.003.png 139172035.004.png
Furthermor e, a nother possible representation of the complex number z in this plane is
as a vector OP . We display zab
= +
as a directed line that begins at the origin and
terminates at the point Pab
( , ). Hence the modulus of z , that is z , is the distance of
(,) from the origin. However, there are simple geometrical relationships
between the vectors for zab
= +
, the negative of z ;
z and the conjugate of z ; z
z is vector for z reflected through the origin,
whereas z is the vector z reflected about the real axis.
The addition and subtraction of complex numbers can be interpreted as vector
addition which is given by the parallelogram law . The ‘ triangle inequality ’ is
derivable from this geometric complex plane. The length of the vector zz
1
2
is
zz
1
2
, which must be less than or equal to the combined lengths z
1
z
2
. Thus
zz z z
1
+ ≤
2
1
2
.
Polar Representation of Complex Numbers
Frequently, points in the complex plane, which represent complex numbers, are
defined by means of polar coordinates. The complex number zxiy
= +
can be
xy it
follows that there is a corresponding way to write complex number in polar form.
r
instead of its rectangular coordinates ( , ),
We see that r is identical to the modulus of z ; whereas
is the directed angle from
the positive x- axis to the point P . Thus we have
xr
cos
and yr
sin
,
where
rz x y
y
x
= =
2
2 ,
tan
.
will be expressed
in radians and is regarded as positive when measured in the counterclockwise
direction and negative when measured clockwise . The distance r is never negative.
For a point at the origin; z
the argument of z and write
θ =
arg .
z The angle
is undefined since a ray like
that cannot be constructed. Consequently, we now defined the polar for m of a
complex number zxiy
0, r becomes zero. Here
= +
as
zr
(cos
i
sin )
(1)
is that it is multivalued , which means for a
nonzero complex number z , it has an infinite number of distinct arguments (since
sin(
= θ
2
k
) sin , cos(
2
k
) cos ,
k Z
) . Any two distinct arguments of z
differ each other by an integral multiple of 2
, thus two nonzero complex number
zr
1
(cos
1
i
sin )
and zr
2
2
(cos
2
i
sin )
2
are equal if and only if
k ,
where k is some integer. Consequently, in order to specify a unique value of arg ,
rr
1
2
and
θ θ
1
= +
2 2
z
we
zPab
in the Ar ga nd plane. The vector
located as polar coordinate (, )
We called
Clearly, an important feature of arg z
1
1
139172035.005.png
may restrict its value to some interval of length. For this, we introduce the concept of
principle value of the argument (or principle argument ) of a nonzero complex number
z , denoted as Arg z , is defined to be the unique value that satisfies
π
Arg .
Hence, the relation between arg z and Arg z is given by
arg
z
z
Arg
2
k
,
k Z
.
Multiplication and Division in Polar From
The polar description is particularly useful in the multiplication and division of
complex number. Consider zr
1
1
(cos
1
i
sin )
1
and zr
2
2
(cos
2
i
sin ).
1. Multiplication
Multiplying z 1 and z 2 we have
θ θ θ θ
When two nonzero complex are multiplied together, the resulting product has a
modulus equal to the product of the modulus of the two factors and an argument
equal to the sum of the arguments of the two factors; that is,
zz rr z z
zz rr
12
12
cos(
1
2
) sin(
i
1
2
) .
12
12
1 2
,
arg(
zz
12
)
= + =
θ θ
2
arg( ) arg( ).
z
1
z
1. Division
Similarly, dividing z 1 by z 2 we obtain
z
z
1
r
r
1
cos(
θ θ
) sin(
i
θ θ
) .
1
2
1
2
2
2
The modulus of the quotient of two complex numbers is the quotient of their
modulus, and the argument of the quotient is the argument of the numerator less
the argument of the denominator, thus
z
z
1
= =
r
r
1
z
z
1
,
2
2
2
arg
z
z
1
= − =
θ θ
arg( ) arg( ).
z
z
1
2
1
2
2
Euler ’s Formula and Exponential Form of Complex Numbers
For any real
, we could recall that we have the familiar Taylor series representation
of sin
, cos
and e θ :
3
5
sin
θ θ
35
!
!
3
,
− ∞ < < ∞
,
2
4
cos
= −
1
24
!
!
3
,
− ∞ < < ∞
,
2
3
e
= + +
1
23
!
− ∞ < < ∞
3
,
,
Thus, it seems reasonable to define
z
2
1
2
= −
!
139172035.006.png

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin