COMPLEX NUMBERS COMPLEX NUMBERS
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COMPLEX NUMBERSCOMPLEX NUMBERS
In this unit we will discuss ……
� Introduction and basic definition of Complex numbers.
� Algebraic properties of Complex numbers.
� De Moivre’s theorem and its expansion.
� Exponential form of Complex numbers.
� Logarithm of a Complex numbers.
� Hyperbolic and Inverse hyperbolic functions.
DEFINITION OF COMPLEX NUMBERS
i=−1
Complex number Z = a + bi is defined as an
ordered pair (a, b), where a & b are real numbers
and . a = Re (z) b = im(z))
� Two complex numbers are equal iff their real as well as
imaginary parts are equal
� Complex conjugate to z = a + ib is z = a - ib
� (0, 1) is called imaginary unit i = (0, 1).
ALGEBRA OF COMPLEX NUMBERS
� Addition and subtraction of complex numbers is defined asidbcadicbia )()()()( ±+±=+±+
� Multiplication of complex numbers is defined as
iadbcbdacdicbia )()())(( ++−=++
� Division of complex numbers is defined as� Division of complex numbers is defined as
idc
adbc
dc
bdac
dic
bia2222
)(
)(
+
−++
+=+
+
� Relation between z and z
(((( )))) )Z(Z
Z
Z
Z;ZZZZ
,zzz;zz;z)z(
i
zzzIm,
zzzRe
0
22
2
2
1
2
12121
2
≠≠≠≠====
====
============
−−−−====
++++====
GEOMETRICAL REPRESENTATION OF COMPLEX NUMBERS
If z = a + ib, is a complex
number than in cartesian form
it is as good as (a, b)
For polar form, let us take
a = r cos θ and b = r sin θ
z = rcos θ + i rsin θ
= r(cos θ + i sin θ),
= r cis θ
πθπb
tanθ
π
bar
≤≤≤≤<<<<========
±±±±±±±±====++++====
====++++====
−−−− - , a
Arg(z)
...2.........1,0,K k,2Arg(z)arg(z)
, z
1
22
Geometrically, IzI is distance of point z from origin.
� θ is directed angle from positive X – axis to (0, 0) – (a, b)
� θ between - π < θ < π is called principal argument and
denoted by Arg (z)
The absolute value or modulus o the number z = a + bi is
denoted by |z| given by 22baz +=
2121 )inequalitytriangular(zzzz ++++≤≤≤≤++++
ABSOLUTE VALUE & DISTANCE
Distance between the points z1 = a1+b1i and z2 = a2+b2i is
denoted by 2
21
2
2121 ) ()( bbaazz −+−=−
1212 zzzz −−−−≤≤≤≤−−−−
An important interpretation regarding multiplication
given by polar form of complex number
z1 = r1 (cos θ1 + i sin θ1 )
z2 = r2 (cos θ2 + i sin θ2 )
z1z2= r1 r2 (cos θ1 + i sin θ1 ) (cos θ2 + i sin θ2)
=r1r2(cos θ1cos θ2 - sin θ1sin θ2)+i(sin θ1cos θ2+cos θ1sin θ2)
= r1r2 [cos(θ1 + θ2)+i sin (θ1 + θ2)] = r1r2 cis (θ1 + θ2)
� The modulus of the product is product of the moduli
� The argument of the product is sum of the argument
|z1z2|=|z1 || z2|
arg (z1z2)= arg z1 + arg z2
z1
z2
θ1 + θ2
θ2
θ1
z1 z2
EXAMPLES
Q. Find the complex conjugate of i
i
−−−−
++++
1
23
Q. Determine Region in z – plane represented by
)z
z(argand)zz(arg
,izandizIf
2
121
21 32231
++++====++++−−−−====Q.
1<|z-2|<3
Q. Express the
complex number
in polar form
and find the
principle argument.
i++++−−−− 3
Q. Express the
complex number
in polar form
and find the
principle argument.
31 i++++
De Moivre’s Theorem
If n is a rational number than the value or one of the
values of (cos θ + i sin θ)n is cos nθ + i sin nθ.
In particular, (cos θ + i sin θ)n = cos nθ + i sin nθ
for n = 0, ±1, ±2 ………….
For any complex number z = r e i θ
and n = 0, ±1, ±2 …………., we have zn = rn e i nθ
Q. 90903131 )i()i(Evaluate −−−−++++++++
θsiniθcos
)θsiniθ(cos)θsiniθ(cos
)θsiniθ(cos)θsiniθ(costhatovePr 77
5533
22
31
2
232
++++====
−−−−−−−−
−−−−++++Q.
Examples - De Moivre’s Theorem
)θsiniθ(cos)θsiniθ(cos 5533 −−−−−−−−
Q. 4311
311
58
46i
)i()i(
)i()i(thatovePr ====
++++−−−−
−−−−++++
Q.
−−−−
−−−−====−−−−++++++++++++++++
++++
2424211
1 θnπncos
θπ)θcosiθsin()θcosiθsin(
nnn Cos
n
Roots of a complex number
n
θsini
n
θcos)θsiniθ(cos n ++++====++++
1
If n is a positive integer than is one of the root of
that is
n
θsini
n
θcos ++++
n)θsiniθ(cos1
++++
nn
++++++++
++++====
++++++++++++====++++
n
θπksini
n
θπk[cos
)]θπksin(i)θπk[cos()θsiniθ(cosn
n
22
22
11
Remaining roots can be obtained by periodic nature of sine and cosine
It gives all roots of for K = 0, 1, 2, 3, …(n – 1) n)θsiniθ(cos1
++++
Examples:
Q. Solve Z4 + 1 = 0
)i(),i(),i(),i( −−−−−−−−−−−−++++−−−−++++ 12
11
2
11
2
11
2
1
Q. Find fifth root of i++++−−−− 3Q. Find fifth root of i++++−−−− 3
++++
++++
++++
++++
++++
30
53
30
532
30
41
30
412
30
29
30
292
30
17
30
172
662
51
5151
5151
πsini
πcos
,π
siniπ
cos,π
siniπ
cos
,π
siniπ
cos,π
siniπ
cos
Q. Solve the equation x 4 – x3 + x2 – x +1 = 0 using De
Moivre’s theorem.
(((( ))))
++++++++
++++
++++
7722
5
3
5
32
552
5151
5151
,π
siniπ
cos,πsiniπcos
,π
siniπ
cos,π
siniπ
cos
(((( ))))
++++
++++++++
5
9
5
92
5
7
5
722
51
5151
πsini
πcos
,π
siniπ
cos,πsiniπcos
θsiniz
z,θcosz
z 21
21
====−−−−====++++
Expansion of De Moivre’s Theorem
θsin)i(z
z,θcosz
znnnn
n
21
21
====
−−−−====
++++
θnsiniz
z,θncosz
zn
n
n
n2
12
1====
−−−−====
++++
zz
Examples:
Q. Express Cos6 θ in terms of cosines of multiples of θ.
Let z is a complex number, then ez is called
exponential function
ez = e x + iy = e x e iy
For each y ∈ R , complex number e iy is defined as
Known as Euler’s formulayiyeiy
sincos +=
EXPONENTIAL FORM OF COMPLEX NUMBER
Known as Euler’s formulayiyeiy
sincos +=
)sin(cos , yiyeeeeeiyxzForxiyxiyxz
+===+=+
(((( )))) (((( )))) ysineeIm,ycoseeRexzxz
========
)zRe(ee),z(imy)earg(xzz
================
LOGARITHMIC FORM OF COMPLEX NUMBER
zLogwze,Cw,zIf ew
====⇒⇒⇒⇒====∈∈∈∈
w)z(Log
Ik,kiπw)z(Log
ze,Now
e
iπkw
====
∈∈∈∈++++====
====++++
2
2
iπk)iyxlog()iyx(Log
reiyxzAsθi
2++++++++====++++
====++++====
iπk)iyxlog()iyx(Log 2++++++++====++++
iπktani)yxlog(
iπkθiyxlog
iπk)elog()rlog(
iπk)relog(
θi
θi
22
1
2
2
2
122
22
++++++++++++====
++++++++++++====
++++++++====
++++====
−−−−
x
y
x
ym
1222
2
1 −−−−++++====++++++++====++++ tanπk)]iyx(Log[I),yxlog()]iyx(LogRe[
Examples:
Q. Prove that 22
2
ba
ab
iba
ibalogitan
−−−−====
++++
−−−−
Q. Find general value of log (-3) and log (- i).
Q. Separate real and imaginary parts of
1) log (1+i)
2) log (4+3i)
Circular functions of complex number
i
eexsin,
eexcos
ixixixix
22
−−−−−−−−−−−−
====++++
====
Hyperbolic functionsHyperbolic functions
xx
xxxxxx
ee
eextanh,
eexsinh,
eexcosh
−−−−
−−−−−−−−−−−−
++++
−−−−====
−−−−====
++++====
22
HYPERBOLIC AND CIRCULAR FUNCTIONS
sin h (ix) = i sin x
cos h (ix) = cos x
tan h (ix) = i tan x
cosec h (ix) = -i cosec x
sec h (ix) = sec x
cot h (ix) = -i cot x
HYPERBOLIC IDENTITIES
1
1
1
22
22
22
====−−−−
====++++
====−−−−
zheccoszhcot
zhtanzhsec
zhsinzhcos
1====−−−− zheccoszhcot
INVERSE HYPERBOLIC FUNCTIONS
++++
====
−−−−++++====
++++++++====
−−−−
−−−−
−−−−
x1 an
os
ln)x(ht
)xx(ln)x(hc
)xx(ln)x(hsin
1
1
1
1
21
21
====−−−−
x-1 an ln)x(ht
2
1
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