## Faqts : Science : Mathematics : Number : Complex

##### + Search    Did You Find This Entry Useful? 10 of 11 people (91%) answered YesRecently 9 of 10 people (90%) answered Yes

### Entry

##### Math: Number: Complex: e^( x . i ) = cos( x ) + sin( x ) . i : Proof: Can you give a proof?

Aug 27th, 2009 00:37
techi gity, Knud van Eeden,

```----------------------------------------------------------------------
--- Knud van Eeden --- 13 April 2005 - 06:56 pm ----------------------
Math: Number: Complex: e^( x . i ) = cos( x ) + sin( x ) . i : Proof:
Can you give a proof?
---
A possible algebraic proof:
---
1. The definition of a complex number (in a rectangular coordinate
system) is:
z = (real number) + (imaginary number)                  
2. If you choose z to be a unit vector turning around
in a unit circle, you have in this special case
z = cos( x ) + sin( x ) . i                             
2. Differentating this  to x gives
dz
---- = -sin( x ) + cos( x ) . i                         
dx
3. Getting another i in this equation 
1. The definition of i is
i = SquareRoot( -1 )                              
2. Squaring  gives
2                        2
i   = ( SquareRoot( -1 ) )  = -1                  
3. Writing  in this way
dz
---- = - 1. sin( x ) + cos( x ) . i                     
dx
4. Putting  in  gives
dz       2
---- = - i . sin( x ) + cos( x ) . i                    
dx
4. Separating the common factor i in  gives
dz
---- = i . ( cos( x ) + sin( x ) . i )                  
dx
5. Now using  and putting this in , you can thus write
dz
---- = i . z                                            
dx
6. This is a differential equation which you can solve
by separation of the variables
dz
---- = i . dx                                           
z
7. Integrating both sides
dz
INTEGRAL( ---- ) = INTEGRAL( i . dx )                   
z
8. Because i is constant, you can take it out of the
integral
ln( z ) = INTEGRAL( dx ) . i                            
9. Thus you get
ln( z ) = x . i + constant                              
10. Find the constant
1. Now z is a unit vector turned over an angle
x in a unit circle.
When you choose the angle x equal to e.g. zero,
then the unit vector falls together with
the horizontal axis, thus the imaginary
component is zero, and only the real component
remains. Which has a value of 1, then
length of the unit vector.
2. Thus solve the equation
ln( 1 ) = 0 . i + constant                        
3. Which becomes
0       = 0 + constant                            
4. Or thus
constant = 0                                      
11. Putting the constant back in the equation,
by putting  in  gives
ln( z ) = x . i + 0                                  
or thus
ln( z ) = x . i                                       
12. Now introducing e
1. Let both sides of  be a power of e
ln( z )     x . i
e        =  e                                        
or after working out
x . i
z = e                                                
13. Replacing z on the left side  gives
x . i
e      = cos( x ) + sin( x ) . i                     
14. Interpretation of this result:
x . i
e
is thus nothing else but a presentation
of a unit vector turned over an angle
x (in a unit circle)
x . i
figure: e      is thus a unit vector in unit circle
i
|
-     /
s|    /
imaginary  ^i|   /
axis       |n|  /  unit vector
| | /   with length 1
x|/ x
-----------+-----|----- 1
|cos x
|  --> real axis
|
|
|
|
|
|
15. Applying this expression , by choosing a few
angles x.
1. Choosing the angle x equal to 360 degrees,
or thus 2 PI radians                               
Putting  in  gives
(2 . PI) . i
e              = cos( 2 . PI ) + sin( 2 . PI ) . i   
(2 . PI) . i
e              = 1 + 0 . i                           
or thus
(2 . PI) . i
e             = 1
---
2. Choosing the angle x equal to 180 degrees,
Putting  in  gives
( PI ) . i
e              = cos( PI ) + sin( PI ) . i           
or thus
( PI ) . i
e              = -1 + 0 . i                          
or thus
( PI ) . i
e              = -1                                  
---
3. Choosing the angle x equal to 90 degrees,
or thus PI / 2 radians                             
Putting  in  gives
( PI / 2 ) . i
e              = cos( PI / 2 ) + sin( PI / 2 ) . i   
or thus
( PI / 2 ) . i
e              = 0 + 1 . i                           
or thus
( PI / 2 ) . i
e              = i                                   
---
---
---
Math: Number: Complex: Link: Overview: Can you give an overview of
http://www.faqts.com/knowledge_base/view.phtml/aid/35633/fid/1793
----------------------------------------------------------------------
http://blogtact.com/
http://uddannelsepainternet.blogtact.com/
http://anxietygk.blogtact.com/
http://insuranceru.blogtact.com/
http://ajandekotlet.blogtact.com/
http://reisenenglandschottland.blogtact.com/
http://voyageangleterreecosse.blogtact.com/
http://travelengscoru.blogtact.com/
http://voyageafrique.blogtact.com/
http://automobilecommentaires.blogtact.com/
http://geschenkeidee.blogtact.com/
http://itavaltahotellit.blogtact.com/
http://petsru.blogtact.com/
http://insurancebg.blogtact.com/
http://haziallatellatas.blogtact.com/
http://landerderwelt.blogtact.com/
http://europetravelru.blogtact.com/
http://flyvninger.blogtact.com/
http://geschenkideeen.blogtact.com/```