Complex Numbers and Functions. Complex Differentiation

Chp13 Complex Numbers and Functions.
Complex Differentiation
Prepared By
Hesham Ali
Marwa Ghaith

COMPLEX NUMBERS AND
THEIR GEOMETRIC
REPRESENTATION

Introduction
 Consider the quadratic equation;
𝑥2
+ 1 = 0
 It has no solutions in the real number system since
𝑥2
= -1
 Similarly 𝑥2
+ 16 = 0
1x 12
i
ix 416 

Introduction
 • Power of "i"
1,,,,,,1 210
 iiii
1)1()(
1
25254100
45
224
23




ii
iiii
iii
iiii

Introduction
 A complex numbers is a number consisting a Real
and Imaginary part. z = (x, y).
 It can be written in the form(Cartesian form) :
Z = x + yi
Real Imaginary

Introduction
 Then x is known as the real part of z and y as the
imaginary part. We write x = Re z and y = Im z.
 Note that real numbers are complex – a real number
is simply a complex number with zero imaginary part.

Introduction
 By definition, two complex numbers are equal if
and only if their real parts are equal and their
imaginary parts are equal.
 (0, 1) is called the imaginary unit and is denoted
by i, ).1,0(i

Algebra of complex numbers
 Addition of complex numbers
If a + bi and c + di are two complex numbers then
addition of complex numbers are ,
(a + bi) + (c + di) = (a + c) + (b + d)i
 Example:
(2 + 4i) + (5 + 3i) = (2 + 5) + (4 + 3)i = 7 + 7i

 Subtraction of Complex numbers
 If a + bi and c + di are two complex numbers then
subtraction of complex numbers are ,
(a + bi) - (c + di) = (a - c) + (b - d)i
 Example:
(3 + 2i) - (2 + 3i) = (3 - 2) + (2 - 3)i = 1 - 1i

 Multiplication of Complex numbers
multiplication of complex numbers is,
(a + bi)(c + di) = (ac -bd) + ( ad + bc)i
Example:
 (2 + 3i)(4 + 5i)=(2x4- 3x5)+(2x5+ 3x4)i =-7+22i

 Division of Complex numbers
Division of complex numbers is,

Complex Plane (Argand diagram)
We choose two perpendicular
coordinate axes, the horizontal
x-axis, called the real axis,
and the vertical y-axis, called
the imaginary axis.

Addition can be represented
graphically on the complex
plane.

Subtraction can be represented
graphically on the complex
plane.

Complex Conjugate Numbers
 The complex conjugate of
complex number Z = x + yi,
is
 It is obtained geometrically
by reflecting the point z in
the real axis. Figure shows this
for z = 5 + 2i and its
conjugate = 5 - 2i.
yixz 
z
z

 By addition and subtraction,
bibbaabiabiazz
aibbaabiabiazz
2)()()()(
2)()()()(


),(
2
1
Re zzxz  )(
2
1
Im zz
i
yz 

 The complex conjugate is important because it permits
us to switch from complex to real. Indeed, by
multiplication the complex number with it’s conjugate.
Let z = a+bi, then :
2222
)())(( babiabiabiazz 
2
zzz 

 So when you need to divide one complex number by
another, you multiply the numerator and denominator
of the problem by the conjugate of the denominator.
 Example : Divide 10 + 5i by 4 – 3i.
i21

2121 )( zzzz  2121 )( zzzz 
2121 )( zzzz 
2
1
2
1
)(
z
z
z
z


POLAR FORM OF COMPLEX
NUMBERS.
POWERS AND ROOTS

Polar form
 Polar coordinates will help
us understand complex
numbers geometrically
,cosrx  sinry 

Polar form
 z = x + yi takes the so-called polar form,
 It could be written as
 r is called the absolute value or modulus of z and
is denoted by
)sin(cos  irz 
z
zzyxrz  22
 rz

Polar form
 θ is called the argument of z and is denoted by arg z.
Thus θ = arg z

 Geometrically, is the directed angle from the positive x-
axis to OP. Here, as in calculus, all angles are measured
in radians and positive in the counterclockwise sense.
x
y
tan

Polar form
 The Principal Argument is between -π and π
 The unique value of θ such that –π < θ < π is
called principle value of the argument.
 the other values of θ are θ= θ+2nπ n= 1, 2,… 

Polar form
 Pr6: Represent in polar form,
 Ans :
i
i
53
2
1
103


 cos2)sin(cos2 i

Triangle Inequality
2121 zzzz 
The generalized triangle
inequality,
nn zzzzzz  .......... 2121

Multiplication and Division in Polar Form
 Let,
 Then ,
 the absolute value of a product equals the product of
the absolute values of the factors,
),sin(cos 1111  irz  )sin(cos 2222  irz 
)]sin()[cos( 21212121   irrzz
2121 zzzz 

 the argument of a product equals the sum of the
 arguments of the factors,
 Division.
2121 argarg)arg( zzzz 
)]sin()[cos( 2121
2
1
2
1
  i
r
r
z
z

 the argument of a division equals the subtraction of
the arguments of the factors,
21
2
1
argarg)arg( zz
z
z

2
1
2
1
z
z
z
z


Integer Powers of z
 De Moivre’s Formula
If n is an integer,
Then,
),sin(cos  ninrz nn

,)]sin(cos[ nn
irz  

Roots of z
 If then ,
 Let and
 The absolute values on both sides must be equal;
,
The argument , k : integer 0,1,…,n-1
),sin(cos  irz 
n
wz 
n
zw 
),sin(cos  iRw 
)sin(cos)sin(cos  irzninRw nn

n
rR 
n
k

2


Roots of z

 These n values lie on a circle of radius with center at
the origin and constitute the vertices of a regular
polygon of n sides.
 The principal value of w when k=0
,
2
sin
2
cos 




 



n
k
i
n
k
rz nn 

Roots of z
 Taking z=1, we have r=1 , θ=0
 These n values are called the nth roots of unity.
,
2
sin
2
cos1 






n
k
i
n
kn 

Roots of z
 They lie on the circle of radius 1 and center 0, briefly
called the unit circle

Roots of z
 Pr 22) Find and graph all roots .
3
43 i

Roots of z
)
3
49.0
sin
3
49.0
(cos5
),
3
29.0
sin
3
29.0
(cos5
)
3
9.0
sin
3
9.0
(cos5
),
2
sin
2
(cos
9.0)
3
4
(tan,543
,43
3
1
2
3
1
1
3
1
0
3
1
122
3




















iw
iw
iw
n
k
i
n
k
rw
r
zwiz
k

Circles and Disks. Half-Planes
 open circular disk :The set of
all points z which satisfy the
inequality |z – a|<, where 
is a positive real number is
called an open disk or
neighborhood of a .

 Close circular disk :The set of
all points z which satisfy the
inequality |z – a|≤, where 
is a positive real number.

 Open annulus This is the set of
all z whose distance |z – a|
from a is greater than 1 but
less than 2 . Similarly, the
closed annulus 1≤|z – a|≤2

 Half-Planes. By the (open)
upper half-plane we mean the
set of all points z=x+yi such
that y>0 . Similarly, the
condition y<0 defines the lower
half-plane, x>0 the right half-
plane, and x<0 the left half-
plane.

Concepts on Sets in the Complex Plane
 point set in the complex plane we mean any sort of
collection of finitely many or infinitely many points.
 Interior Point A point is called an interior point
of S if and only if there exists at least one
neighborhood of z0 which is completely contained in
S.
Sz 0

 Open Set If every point of a set S is an interior point
of S, we say that S is an open set.
 Closed Set if S contains all of its boundary points,
then it is called a closed set.
 Sets may be neither open nor closed. Neither
ClosedOpen

 Connected An open set S is said to be connected if
every pair of points z1 and z2 in S can be joined by a
polygonal line that lies entirely in S. .
S
z
1
z
2

Complex Function
 Complex function of a complex variable. A function f defined
on S is a rule which assigns to each z  S a complex number w.
The number w is called a value of f at z and is denoted by
f(z), i.e.,
w = f(z).
The set S is called the domain of definition of f. Although the
domain of definition is often a domain, it need not be.
 Ex , zzzfw 3)( 2


Complex Function
 w is complex, and we write w=u+iv where u and v
are the real and imaginary parts
 Hence u becomes a real function of x and y, and so
does v. We may thus write
),(),()( yxivyxuzfw 

Complex Function
 Properties of a real-valued function of a real
variable are often exhibited by the graph of the
function. But when w = f(z), where z and w are
complex, no such convenient graphical representation
is available because each of the numbers z and w is
located in a plane rather than a line.

Complex Function
 Graph of Complex Function
x u
y v
z-plane w-planedomain of
definition
range
w = f(z)

Limit, Continuity
 A function w = f(z) is said to have the limit l as z
approaches a point z0 if for given small positive
number we can find positive number such that for
all in a disk we have
 z may approach z0 from any direction in the complex
plane.

0zz   0zz lzf )(

Limit, Continuity
 We call f(z) continuous at z0 if:
 F is defined in a neighborhood of z0,
 The limit exists, and

 A function f is said to be continuous on a set S if it is
continuous at each point of S. If a function is not
continuous at a point, then it is said to be singular at
the point.
)()(lim 0
0
zfzf
zz



Limit, Continuity
 One can show that f(z) approaches a limit precisely
when its real and imaginary parts approach limits,
and the continuity of f(z) is equivalent to the
continuity of its real and imaginary parts.

Derivatives
 Differentiation of complex-valued functions is completely
analogous to the real case:
 Definition. Derivative. Let f(z) be a complex-valued function
defined in a neighborhood of z0. Then the derivative of f(z)
at z0 is given by
Provided this limit exists. F(z) is said to be differentiable at
z0.
z
zfzzf
zf
z 



)()(
lim)( 00
0
0

Derivatives
 The function is differentiable for all z and has the
derivative because
zzz
z
zzzzz
zf
z
zzzf
zf
zz
z
2)2(lim
)(2
lim)(
)(
lim)(
0
222
0
0
2
0
0









2
)( zzf 
zzf 2)( 

Derivatives
 not Differentiable
yixzzzf  ,)(
z
iyx
iyx
z
z
z
zzz
z
zzzf










 )()(

Derivatives
 Properties of Derivatives
       
     
           
         
  
 
        Rule.Chain''
.0if,
''
'
'''
.constantanyfor''
'''
000
02
0
0000
0
00000
00
000
zgzgfzgf
dz
d
zg
zg
zgzfzfzg
z
g
f
zgzfzgzfzfg
czcfzcf
zgzfzgf













Analytic. (Holomorphic).
 Definition. A complex-valued function f (z) is said to be
analytic, or equivalently, holomorphic, on an open set  if it
has a derivative at every point of . (The term “regular” is
also used.)
 If f (z) is analytic on the whole complex plane, then it is said to
be an entire function.

Analytic. (Holomorphic).
 Rational Function.
 Definition. If f and g are polynomials in z, then h (z) = f
(z)/g(z), g(z)  0 is called a rational function.
 Remarks.
 All polynomial functions of z are entire.
 A rational function of z is analytic at every point for which
its denominator is nonzero.
 If a function can be reduced to a polynomial function which
does not involve z , then it is analytic.

CAUCHY–RIEMANN
EQUATIONS.
LAPLACE’S EQUATION

Cauchy-Riemann Equations
 If the function f (z) = u(x,y) + iv(x,y) is differentiable at z0 =
x0 + iy0, then the limit
 can be evaluated by allowing z to approach zero from any
direction in the complex plane.
z
zfzzf
zf
z 



)()(
lim)( 00
0
0

 If it approaches along the x-axis, then z = x, and we obtain
But the limits of the bracketed expression are just the first partial
derivatives of u and v with respect to x, so that:
   
x
yxivyxuyxxivyxxu
zf
x 



),(),(),(),(
lim)(' 00000000
0
0













 x
yxvyxxv
i
x
yxuyxxu
zf
xx
),(),(
lim
),(),(
lim)(' 0000
0
0000
0
0
).,(),()(' 00000 yx
x
v
iyx
x
u
zf







 If it approaches along the y-axis, then z =iy, and we obtain
 And, therefore









 yi
yxuyyxu
zf
y
),(),(
lim)(' 0000
0
0 








 yi
yxvyyxv
i
y
),(),(
lim 0000
0
).,(),()(' 00000 yx
y
v
yx
y
u
izf







 By definition, a limit exists only if it is unique. Therefore, these
two expressions must be equivalent. Equating real and
imaginary parts, we have that
x
v
y
u
y
v
x
u










and

 We mention that, if we use the polar form
and set , then the Cauchy–Riemann
equations are


u
r
v
v
r
u
r
r
1
1
,


)sin(cos  irz 
),(),()(  rivruzf 

 Ex, Prove that f (z) is analytic and find its derivative.
 The first partials are continuous and satisfy the Cauchy-
Riemann equations at every point.
ye
x
v
ye
y
u
ye
y
v
ye
x
u
yieyezf
xxxx
xx
sin,sin,cos,cos
:Solution
sincos)(













.sincos)(' yieye
x
v
i
x
u
zf xx








Laplace’s Equation. Harmonic Functions

 Harmonic Conjugate;
 Given a function u(x,y) harmonic in, say, an open disk, then we
can find another harmonic function v(x,y) so that u + iv is an
analytic function of z in the disk. Such a function v is called a
harmonic conjugate of u.

 Ex, Construct an analytic function whose real part is:
 Solution: First verify that this function is harmonic.
.3),( 23
yxyxyxu 
.066
6and16
6and33
2
2
2
2
2
2
2
2
22


















xx
y
u
x
u
and
x
y
u
xy
y
u
x
x
u
yx
x
u

 Integrate (1) with respect to y:
16)2(
33)1( 22












xy
y
u
x
v
andyx
x
u
y
v
 
 
)(3),()3(
33
33
32
22
22
xhyyxyxv
yyxv
yyxv





 Now take the derivative of v(x,y) with respect to x:
 According to equation (2), this equals 6xy – 1. Thus,
).('6 xhxy
x
v



.3),(
.)(and,)(So
.1ly,Equivalent.1)('and
16)('6
32
CxyyxyxvAnd
Cxxhxxh
x
h
xh
xyxhxy







 

 The desired analytic function f (z) = u + iv is:
   Cxyyxiyxyxzf  3223
33)(

Exponential Function
 The complex exponential function is one of the
most important analytic functions
 If z = 3 + 4i then

 For real z = x, imaginary part y = 0
 is analytic for all z
1 0

 The derivative of the exponential function is:

 General rule of the exponential functions that
ea × eb = e(a +b)

 Since z = x + iy
ez = e(x +iy)= ex eiy
 For pure imaginary complex number where z = iy
Euler Formula
 The polar form of a complex number, z = r (cosӨ + i sinӨ(
Can be written:

 Substitution of in
 Substitution of will yield
01

For pure imaginary exponent the exponential function has
absolute value of 1
Q:What is the absolute value of exponential function if x
doesn’t equal to zero ?

 Periodicity of with period
1

 Example 1:
In the polar form :

 Solve :

 It is obvious that many properties of exp z are
the same as the properties of exp x with an
exception in the periodicity of exp z with

TRIGONOMETRIC AND
HYPERBOLIC FUNCTIONS.
EULER’S FORMULA

Trigonometric and Hyperbolic Functions.
Euler’s Formula
Real trigonometric
function
Complex
trigonometric
function

Euler’s Formula
By addition and subtraction we obtain

Euler’s Formula
 Substitute ( z= x+iy) instead of x, we obtain
 functions in this formula are unrelated in real

Euler’s Formula
 Example 1: Prove that :

Euler’s Formula
Example 2:
Solve cos z = 5 (which has no real solution)

Euler’s Formula
 General formula for trigonometric functions:

Euler’s Formula
 The complex hyperbolic cosine and sine are defined
by the formulas:
 Complex Trigonometric and Hyperbolic Functions
are related:

LOGARITHM. GENERAL
POWER. PRINCIPAL VALUE

Logarithm. General power. Principal
value
 The natural logarithm of z = x+ iy is donated by ln
z or log z

value
 Where
 the complex natural logarithm is infinitely many-
valued. The value of ln z corresponding to the
principle value Arg z is denoted by Ln z and it is
called the principle value of ln z, given by

value
 Ln z is a single- valued since the other values of arg z differ
by integer multiples of . the other values of ln z given by:
 All the values of ln z have the same real part but their imaginary parts differ by integer
multiples of .If z is a positive real then Arg z= 0 and Ln z becomes same as the real
natural logarithm in calculus but if it is a negative number then Arg z = and

value

value
 From and for positive real r we
obtain
 Since arg ( ) is multivalued so:

value
 The familiar relations of natural logarithm are still
applicable for complex value:

value
 Analyticity of the Logarithm

value
 General Powers:
The general power of a complex number z= x +iy
are defined by the formula:
Where c complex and

value
 If c = 1/n then,
=

Complex Numbers and Functions. Complex Differentiation

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