Properties of Complex Numbers

Properties of Complex Numbers

Properties of Complex Numbers
Swapnil Sunil Jain
December 26, 2006

Properties of Complex Numbers

Conjugate Properties

(

z

1

+

z

2

)

*

=

z

1

*

+

z

2

*

(

z

1

⁢

z

2

)

*

=

z

1

*

⁢

z

2

*

(

z

1

z

2

)

*

=

z

1

*

z

2

*

(

z

n

)

*

=

(

z

*

)

n

f

⁢

(

z

*

)

=

f

*

⁢

(

z

)

z

⁢

z

*

=

|

z

|

2

Re() and Im() Properties

z

=

R

⁢

e

⁢

(

z

)

=

j

⁢

I

⁢

m

⁢

(

z

)

R

⁢

e

⁢

(

z

)

=

z

+

z

*

2

I

⁢

m

⁢

(

z

)

=

z

-

z

*

2

⁢

j

R

⁢

e

⁢

(

z

*

)

=

R

⁢

e

⁢

(

z

)

I

⁢

m

⁢

(

z

*

)

=

-

I

⁢

m

⁢

(

z

)

R

⁢

e

⁢

(

z

)

≤

|

z

|

I

⁢

m

⁢

(

z

)

≤

|

z

|

R

⁢

e

⁢

(

z

1

+

z

2

)

=

R

⁢

e

⁢

(

z

1

)

+

R

⁢

e

⁢

(

z

2

)

I

⁢

m

⁢

(

z

1

+

z

2

)

=

I

⁢

m

⁢

(

z

1

)

+

I

⁢

m

⁢

(

z

2

)

R

⁢

e

⁢

(

z

)

=

I

⁢

m

⁢

(

j

⁢

z

)

I

⁢

m

⁢

(

z

)

=

R

⁢

e

⁢

(

-

j

⁢

z

)

Abs() and Arg() Properties

|

z

|

≡

[

R

⁢

e

⁢

(

z

)

2

+

I

⁢

m

⁢

(

z

)

2

]

1

2

|

z

1

⁢

z

2

|

=

|

z

1

|

⁢

|

z

2

|

|

z

1

z

2

|

=

|

z

1

|

|

z

2

|

|

z

*

|

=

|

z

|

|

z

1

+

z

2

|

≤

|

z

1

|

+

|

z

2

|

arg

⁢

(

z

)

≡

{

arctan

⁡

(

I

⁢

m

⁢

(

z

)

R

⁢

e

⁢

(

z

)

)

,

x

>

0

arctan

⁡

(

I

⁢

m

⁢

(

z

)

R

⁢

e

⁢

(

z

)

)

+

π

,

x

<

0

π

2

,

x

=

0

,

y

>

0

-

π

2

,

x

=

0

,

y

<

0

arg

⁢

(

z

*

)

=

-

arg

⁢

(

z

)

arg

⁢

(

z

1

⁢

z

2

)

=

arg

⁢

(

z

1

)

+

arg

⁢

(

z

2

)

arg

⁢

(

z

1

z

2

)

=

arg

⁢

(

z

1

)

-

arg

⁢

(

z

2

)

Some Tips

1

j

=

-

j

-

j

2

=

1

a

l

⁢

o

⁢

g

⁢

(

b

)

=

b

l

⁢

o

⁢

g

⁢

(

a

)

a

=

e

l

⁢

n

⁢

(

a

)

Power Properties

For

z

=

r

⁢

(

cos

⁡

(

θ

)

+

j

⁢

sin

⁡

(

θ

)

)

,

z

n

=

r

n

⁢

(

cos

⁡

(

n

⁢

θ

)

+

j

⁢

sin

⁡

(

n

⁢

θ

)

)

z

1

n

=

r

1

n

⁢

[

cos

⁡

(

θ

+

2

⁢

k

⁢

π

n

)

+

j

⁢

sin

⁡

(

θ

+

2

⁢

k

⁢

π

n

)

]

for

⁢

k

=

0

,

1

,

…

,

n

-

1

Trigonometric and Logarithmic Properties

ln

⁡

(

z

)

=

ln

⁡

(

|

z

|

⁢

e

j

⁢

arg

⁢

(

z

)

)

≡

ln

⁡

(

|

z

|

)

+

j

⁢

arg

⁢

(

z

)

=

ln

⁡

(

|

z

|

)

+

j

⁢

(

θ

+

2

⁢

k

⁢

π

)

exp

⁡

(

z

)

=

exp

⁡

(

R

⁢

e

⁢

(

z

)

)

⁢

[

cos

⁡

(

I

⁢

m

⁢

(

z

)

)

+

j

⁢

sin

⁡

(

I

⁢

m

⁢

(

z

)

)

]

cos

⁡

(

z

)

=

1

2

⁢

(

e

j

⁢

z

+

e

-

j

⁢

z

)

sin

⁡

(

z

)

=

1

2

⁢

j

⁢

(

e

j

⁢

z

-

e

-

j

⁢

z

)

arccos

⁡

(

z

)

=

-

j

⁢

ln

⁡

(

z

+

j

⁢

(

1

-

|

z

|

2

)

1

2

)

arcsin

⁡

(

z

)

=

-

j

⁢

ln

⁡

(

j

⁢

z

+

(

1

-

|

z

|

2

)

1

2

)

cosh

⁡

(

z

)

=

1

2

⁢

(

e

z

+

e

-

z

)

sinh

⁡

(

z

)

=

1

2

⁢

(

e

z

-

e

-

z

)

arccosh

⁢

(

z

)

=

-

j

⁢

ln

⁡

(

z

+

j

⁢

(

|

z

|

2

-

1

)

1

2

)

arcsinh

⁢

(

z

)

=

-

j

⁢

ln

⁡

(

z

+

(

|

z

|

2

+

1

)

1

2

)

Title

Properties of Complex Numbers

Canonical

name

PropertiesOfComplexNumbers1

Date of creation

2013-03-11 19:29:52

Last modified on

2013-03-11 19:29:52

Owner

swapnizzle (13346)

Last modified by

(0)

Numerical id

1

Author

swapnizzle (0)

Entry type

Definition

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