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complex analysis the first topic for the subject engineering mathematics. it is from a reference material
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17.1 Complex Numbers
17.2 Powers and Roots
17.3 Sets in the Complex Plane
17.4 Functions of a Complex Variable
17.5 Cauchy-Riemann Equations
17.6 Exponential and Logarithmic Functions
17.7 Trigonometric and Hyperbolic Functions
17.8 Inverse Trigonometric and Hyperbolic Functions
z = x + iy , the real number x is called the real part and
y is called the imaginary part:
Re( z ) = x, Im( z ) = y
A complex number is any number of the z = a + ib
where a and b are real numbers and i is the imaginary
units.
DEFINITION 17.
Complex Number
2
2
2
2
1 2 1 2
2
2
2
2
1 2 1 2
2
1
1 2 1 2 1 2 1 2 1 2
1 2 1 2 1 2
1 2 1 2 1 2
1 1 1 2 2 2
Suppose ,
x y
y x x y
i
x y
x x y y
z
z
z z x x y y i y x x y
z z x x i y y
z z x x i y y
z x iy z x iy
Addition :
Subtraction :
Multiplication :
Division :
1 2 1 2
1 2 1 2
1 2 1 2
1 1
(^2 )
Conjugate If z is a complex number, then the number obtained
by changing the sign of its imaginary part is called the complex
conjugate or, simply, the conjugate of z.
Fig 17.1 is called the complex plane and a complex
number z is considered as a position vector.
The modulus or absolute value of z = x + iy , denoted
by │ z │, is the real number
DEFINITION 17.
Modulus or Absolute Values
z x y z z
2 2
Polar Form
Referring to Fig 17.3, we have
, it is called the principal argument, denoted by
Arg (z ).
Solution
See Fig 17.4 that the point lies in the fourth quarter.
Express 1 3 i in polar form.
sin
2 cos
, arg( )
tan
z i
z
r z i
) sin(
2 cos(
z i
Then
for z 2
(cos sin )
Suppose (cos sin )
2 2 2 2
1 1 1 1
z r i
z r i
(sin cos cos sin )]
[(cos cos sin sin )
1 2 1 2
1 2 1 2 1 2 1 2
i
z z r r
(sin cos cos sin )]
[(cos cos sin sin )
1 2 1 2
1 2 1 2
2
1
2
1
i
r
r
z
z
From the addition formulas from trigonometry,
Thus we can show
[cos( ) sin( )]
1 2 1 2 1 2 1 2
1 2 1 2
2
1
2
1
2
1
2
1
1 2 1 2
1 2
2
1
1 2 1 2
When z = cosθ + i sinθ, we have | z |= r = 1 and so (8)
yields
This last result is known as DeMoivre’s formula and
is useful in deriving certain trigonometric identities.
i n i n
n
A number w is an nth root of a nonzero number z if
w
n
The root corresponds to k=0 called the principal nth
root.
cos cos , sin sin
(cos sin ) (cos sin )
1 /
k n
n
k
n n
r r
n i n r i
n n
n
