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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 z = a + ib , where the symbol i = − 1 is called imaginary unit and a b , ∈ R. a is called the real part and b the imaginary part of z , written a = Re z and b = Im. z With this notation, we have z = Re z + i Im. z
The set of all complex numbers is denoted by
C = { a + ib a b , ∈ R }.
If b = 0, then z = a + i 0 = a , is a real number. Also if a = 0, then z = 0 + ib = ib , is a imaginary number; in this case, z is called pure imaginary number.
Let a + ib and c + id be complex numbers, with a b c d , , , ∈ R.
- Equality a + ib = c + id if and only if a = c and b = d. Note: In particular, we have z = a + ib = 0 if and only if a = 0 and b = 0.
- (^) Fundamental Algebraic Properties of Complex Numbers (i). Addition ( a + ib ) + ( c + id ) = ( a + c ) + i b ( + d ). (ii). Subtraction ( a + ib ) − ( c + id ) = ( a − c ) + i b ( − d ). (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 z^1 = z , z^2 = zz , z^3 = z z^2 , !, z n^ = z n −^1 z. 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 c + id ≠ 0 , then a ib c id
ac bd c d i^
bc ad c d
+ =^
^
+^
^
Remark (a). Observe that if a + ib = 1, then we have 1 c id^2 2 2
c c d
i
d
^
+^
^
(b). For any nonzero complex number z , we define z^ −^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 z
− (^1) = 1 , z − (^2) = z − (^1) z (^) − (^1) , z − (^3) = z − (^2) z − (^1) , !, z − n (^) = z − n + (^1) z − (^1).
(d). i −^1 = (^1) i^ = − i , i −^2 = −1, i −^3 = i , i −^4 =1.
- More Properties of Addition and Multiplication For any complex numbers z z , (^) 1 , z (^) 2 and z 3 , (i). Commutative Laws of Addition and Multiplication : z z z z z z z z
1 2 2 1 1 2 2 1
(ii) Associative Laws of Addition and Multiplication : z z z z z z z z z z z z
1 2 3 1 2 3 1 2 3 1 2 3
(iii). Distributive Law : z (^) 1 ( z (^) 2 + z (^) 3 ) = z z 1 (^) 2 + z z 1 3. (iv). Additive and Multiplicative identities : z z z z z z
(v). z + −( z ) = ( − z ) + z = 0.
Complex Conjugate and Their Properties
Definition: Let z = a + ib ∈ C , a , b ∈ R. The complex conjugate , or briefly conjugate , of z is defined by z = a − ib.
For any complex numbers (^) z z , (^) 1 , z (^) 2 ∈ C ,we have the following algebraic properties of the conjugate operation:
(i). (^) z (^) 1 + z (^) 2 = z (^) 1 + z 2 ,
(ii). z (^) 1 − z (^) 2 = z (^) 1 − z 2 ,
(iii). z z 1 (^) 2 = z (^) 1 ⋅ z 2 ,
Furthermore, another possible representation of the complex number z in this plane is as a vector OP. We display z = a + ib as a directed line that begins at the origin and terminates at the point P a b ( , ). Hence the modulus of z , that is z , is the distance of z = P a b ( , ) from the origin. However, there are simple geometrical relationships between the vectors for z = a + ib , the negative of z ; − z and the conjugate of z ; z in the Argand plane. The vector (^) − 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 z (^) 1 + z 2 is
z (^) 1 + z 2 , which must be less than or equal to the combined lengths z (^) 1 + z 2. Thus z (^) 1 + z (^) 2 ≤ z (^) 1 + z 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 z = x + iy can be located as polar coordinate ( , r θ )instead of its rectangular coordinates ( , x y ), it follows that there is a corresponding way to write complex number in polar form.
We see that r is identical to the modulus of z ; whereas θ is the directed angle from the positive x- axis to the point P. Thus we have x = r cos θ and y = r sin θ , where
r z x y y x
tan θ.
We called θ the argument of z and write θ = arg. z The angle θ 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 = 0, r becomes zero. Here θ is undefined since a ray like that cannot be constructed. Consequently, we now defined the polar for m of a complex number z = x + iy as z = r (cos θ + i sin θ ) (1) Clearly, an important feature of arg z =θ 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 π ) = sin θ , cos(θ + 2 k π ) = cos θ, k ∈ Z ). Any two distinct arguments of z differ each other by an integral multiple of 2^ π^ , thus two nonzero complex number z (^) 1 = r 1 (^) (cos θ 1 + i sin θ 1 )and z (^) 2 = r 2 (^) (cos θ 2 + i sin θ 2 )are equal if and only if r 1 (^) = r 2 and θ 1 = θ 2 + 2 k π , where k is some integer. Consequently, in order to specify a unique value of arg , z we
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 z <^ π. Hence, the relation between arg z and Arg z is given by arg z = Arg z + 2 k π, k ∈ Z.
Multiplication and Division in Polar From
The polar description is particularly useful in the multiplication and division of complex number. Consider z (^) 1 = r 1 (^) (cos θ 1 + i sin θ 1 )and z (^) 2 = r 2 (^) (cos θ 2 + i sin θ 2 ).
- Multiplication Multiplying z 1 and z 2 we have z z 1 (^) 2 = r r 1 2 (^) ( cos( θ 1 + θ 2 ) + i sin(θ 1 +θ 2 ) .) 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, z z r r z z z z z z
1 2 1 2 1 2 1 2 1 2 1 2
arg( ) θ^ θ arg( ) arg( ).
- Division Similarly, dividing z 1 by z 2 we obtain ( )
z z
r r i
1 2
1 2 1 2 1 2
= cos( θ − θ ) + sin( θ −θ). 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
r r
z z z z
z z
1 2
1 2
1 2 1 2
1 2 1 2
arg θ θ arg( ) arg( ).
Euler ’s Formula and Exponential Form of Complex Numbers
For any real θ, we could recall that we have the familiar Taylor series representation of sin θ , cos θ and e^ θ^ :
sin (^)!! , ,
cos !!
!! ,^ ,
θ θ θ^ θ^ θ
θ θ^ θ^ θ
θ
θ θ θ θ
3 5
2 4
2 3
e!
Thus, it seems reasonable to define
for any positive integer n. By the same argument, it can be shown that (3) is also true for any nonpositive integer n. Which is known as de Moivre’s^3 formula , and more precisely, we have the following theorem:
Theorem: ( de Moivre ’s Theorem ) For any^ θ^ and for any integer n ,
( cos θ + i sin θ ) n^ = cos n θ + i sin n θ.
In term of exponential form, it essentially reduces to
( e i^^ θ) n^ = ein θ.
Roots of Complex Numbers
Definition: Let n be a positive integer ≥ 2, and let z be nonzero complex number. Then any complex number w that satisfies w n^ = z is called the n-th root of z , written as w (^) = nz.
Theorem: Given any nonzero complex number z = re i θ^ , the equation w n^ = z has precisely n solutions given by
w r
k n i^
k k n = n + ^
+^
^
^
cos sin (^) ,
θ 2 π θ 2 π k = 0 1, , ", n − 1 ,
or
w (^) k n^ r e
i (^) nk
+ ^
θ 2 π , k = 0 1, , ", n − 1 ,
where n^ r denotes the positive real n -th root of r = z and θ = Arg z.
Elementary Complex Functions
Let A and B be sets. A function f from A to B , denoted by f : A → B is a rule which assigns to each element a ∈ A one and only one element b ∈ B , we write b = f ( a ) and call b the image of a under f. The set A is the domain-set of f , and the set B is the codomain or target-set of f. The set of all images
f ( A )= { f ( a ): a ∈ A }
is called the range or image-set of f. It must be emphasized that both a domain-set and a rule are needed in order for a function to be well defined. When the domain-set is not mentioned, we agree that the largest possible set is to be taken.
The Polynomial and Rational Functions
(^3) This useful formula was discovered by a French mathematician, Abraham de Moivre (1667 - 1754).
- Complex Polynomial Functions are defined by P z ( ) = a (^) 0 + a z 1 + !+ a (^) n − 1 z n −^1 + a zn n , where a (^) 0 , a (^) 1 , " , a (^) n ∈ C and n ∈ N. The integer n is called the degree of polynomial P z ( ), provided that a (^) n ≠ 0. The polynomial p z ( ) = az + b is called a linear function.
- Complex Rational Functions are defined by the quotient of two polynomial functions; that is, R z
P z ( ) (^) Q z
where P z ( ) and Q z ( )are polynomials defined for all z ∈ C for which Q z ( ) ≠ 0. In particular, the ratio of two linear functions: f z
az b ( ) = (^) cz d
- with^ ad^ −^ bc ≠^ 0, which is called a linear fractional function or Mobius ##^ transformation.
The Exponential Function
In defining complex exponential function, we seek a function which agrees with the exponential function of calculus when the complex variable z = x + iy is real; that is we must require that f ( x + i 0 ) = e x for all real numbers x , and which has, by analogy, the following properties: e z^^1 e z^^2 = ez^^1 + z^2 , e z^1^^ e z^^2 = ez^^1 − z^2 for all complex numbers z (^) 1 , z 2. Further, in the previous section we know that by
Euler ’s identity, we get e iy^ = cos y + i sin y , y ∈ R. Consequently, combining this
we adopt the following definition:
Definition: Let z = x + iy be complex number. The complex exponential function e z^ is defined to be the complex number e z^ = e x^ + iy^^ = e x (cos y + i sin y ).
Immediately from the definition, we have the following properties: For any complex numbers z (^) 1 , z (^) 2 , z = x + iy , x y , ∈ R ,we have (i). e z^^1 e z^^2 = ez^^1 + z^2 , (ii). e z^^1 e z^^2 = ez^1^^ −^ z^2 ,
(iii). e iy^ = 1 forallreal y ,
(iv). e z^ = ex ,
(v). e z^ = ez , (vi). arg( e z^ ) = y + 2 k π, k ∈ Z , (vii). e z^ ≠ 0,
cot
cos z (^) sin ,
z = (^) z csc z = (^) sin z ,
where z ≠ k π , k ∈ Z.
As in the case of the exponential function, a large number of the properties of the real trigonometric functions carry over to the complex trigonometric functions. Following is a list of such properties.
For any complex numbers w z , ∈ C ,we have (i). sin 2 z + cos 2 z = 1 , 1 + tan 2 z =sec 2 z , 1 + cot 2 z =csc 2 z ; (ii). sin( w ± z ) = sin w cos z ±cos w sin , z cos( w ± z ) = cos w cos z $sin w sin , z tan( ) tan^ tan tan tan
w z w^ z ; w z
(iii). sin( − z ) = −sin , z tan( − z ) = −tan , z csc( − z ) = −csc , z cot( − z ) = −cot z , cos( − z ) =cos , z sec( − z ) =sec ; z (iv). For any k ∈ Z , sin( z + 2 k π ) =sin , z cos( z + 2 k π) =cos , z sec( z + 2 k π ) =sec , z csc( z + 2 k π) =csc , z tan( z + k π ) =tan , z cot( z + k π) =cot z ,
(v). sin z = sin z , cos z = cos z , tan z =tan z , sec z = sec z , csc z = csc z , cot z =cot z ;
Hyperbolic Functions
The complex hyperbolic functions are defined by a natural extension of their definitions in the real case.
Definition: For any complex number z , we define the complex hyperbolic sine and the complex hyperbolic cosine as
sinh z ,
e z^ ez
cosh z.
e z^ e z
Let z^ = x^ + iy^ , x y ,^ ∈ R. It is directly from the previous definition, we obtain the following identities: sinh sinh cos cosh sin , cosh cosh cos sinh sin ,
z x y i x y z x y i x y
sinh sinh sin , cosh sinh cos.
z x y z x y
(^2 2 ) (^2 2 )
Hence we obtain (i). sinh z = 0 if and only if z = i k ( π), k ∈ Z ,
(ii). cosh z = 0 if and only if z = i 2 + k , k ∈ Z.
π π
Now, the four remaining complex hyperbolic functions are defined by the equations
tanh
sinh z (^) cosh ,
z = (^) z sech z = (^) z
cosh ,
for z = i + k k ∈ Z
π 2 π ,^ ; coth z cothsinh ,
z = (^) z csch z = (^) z
sinh , for z = i k ( π), k ∈ Z.
Immediately from the definition, we have some of the most frequently use identities: For any complex numbers (^) w z , ∈ C , (i). cosh 2 z − sinh 2 z = 1 , 1 − tanh 2 z =sech^2 z , coth 2 z − 1 = csch^2 z ; (ii). sinh( w ± z ) = sinh w cosh z ±cosh w sinh , z cosh( w ± z ) = cosh w cosh z ±sinh w sinh , z tanh( )
tanh tanh w z (^) tanh tanh ;
w z ± = (^) w z
(iii). sinh( − z ) = −sinh , z tanh( − z ) = −tanh , z csch ( − z ) = −csch z , coth( − z ) = −coth , z cosh( − z ) =cosh , z sech ( − z ) =sech z ;
(iv). sinh z = sinh z , cosh z = cosh z , tanh z =tanh z , sech z = sech z , csch z = csch z , coth z =coth z ;
Remark (i). Complex trigonometric and hyperbolic functions are related: sin iz = i sinh , z cos iz = cosh , z tan iz = i tanh , z sinh iz = i sin , z cosh iz = cos , z tanh iz = i tan. z (i). The above discussion has emphasized the similarity between the real and their complex extensions. However, this analogy should not carried too far. For example, the real sine and cosine functions are bounded by 1, i.e., sin x ≤ 1 and cos x ≤ 1 for all x ∈ R , but sin iy = sinh y and cos iy =cosh hy which become arbitrary large as y → ∞.
The Logarithm
(v). log( z n^ )= n log z for any integer positive n.
Complex Exponents
Definition: For any fixed complex number c , the complex exponent c of a nonzero complex number z is defined to be
z c^ = ec^ log^ z for all z ∈ C \ { } 0.
Observe that we evaluate e c^ log^ z by using the complex exponential function, but since the logarithm of z is multivalued. For this reason, depending on the value of c , z c may has more than one numerical value.
The principle value of complex exponential c , z c^ occurs when log z is replaced by principle logarithm function, Log z in the previous definition. That is,
z c^ = ec Log^^ z for all z ∈ C \ { } 0 ,
where − π ≤ Arg z ≤π If z = re i θ^ with θ = Arg z , then we get z c^ = e c^ (log^ r^ + i^^ θ)^ = e c^ log^ r^ ei θ.
Inverse Trigonometric and Hyperbolic
In general, complex trigonometric and hyperbolic functions are infinite many-to-one functions. Thus, we define the inverse complex trigonometric and hyperbolic as multiple-valued relation.
Definition: For z ∈ C , the inverse trigonometric^ arctrig z^ or trig−^1 z^ is defined by w (^) = trig −^1 z if z = trig w. Here, ‘trig w ’ denotes any of the complex trigonometric functions such as sin w , cos w , etc.
In fact, inverses of trigonometric and hyperbolic functions can be described in terms of logarithms. For instance, to obtain the inverse sine, sin −^1 z , we write w = sin −^1 z when z (^) = sin w. That is, w (^) = sin −^1 z when
z w e^ e i
iw w = = −^
− sin.
1 2 Therefore we obtain ( e iw^ ) 2 − 2 iz e ( iw ) − 1 = 0 , that is quadratic in e iw^. Hence we find that
e iw^ = iz + ( 1 − z^2 ) ,
(^12)
where ( 1 2 )
(^12) − z is a double-valued of z , we arrive at the expression
sin log ( )
log.
(^1 2 )
2
z i iz z
i z z
π
Here, we have the five remaining inverse trigonometric, as multiple-valued relations which can be expressed in terms of natural logarithms as follows:
cos log ( ),
tan log , ,
cot log , ,
sec log , ,
csc log ,.
−
−
−
−
−
^
^ ≠ ±
^
^ ≠ ±
= ± +^ −
1 2
1
1
1 2
1
2
z i z z
z i
i z i z
z i
z i
i z i z
z i
z i
z z z
z i
z z z
π
π
The principal value of complex trigonometric functions are defined by
Arc Log
Arc Log
sin ,
cos ,
z i z z
z i z z
π 2 1 1
2
2
Arc Log
Arc Log
Arc Log
Arc Log
tan , ,
cot , ,
sec , ,
csc ,.
z (^) i
i z i z z^ i
z (^) i
i z i z z^ i
z i
z z z
z i z z
z
^
^ ≠ ±
^
^ ≠ ±
= + +^ −
2
2
π
π
Definition: For any complex number z , the inverse hyperbolic , archyp z or hyp −^1 z is defined by w = hyp −^1 z if z = hyp w. Here ‘hyp’ denotes any of the complex hyperbolic functions such as sinh , z cosh , z etc.
These relations, which are multiple-valued, can be expressed in term of natural logarithms as follows:
