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