Faqts : Science : Mathematics : Number : Complex

+ Search
Add Entry AlertManage Folder Edit Entry Add page to http://del.icio.us/
Did You Find This Entry Useful?

10 of 11 people (91%) answered Yes
Recently 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)                  [1]
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]
2. Differentating this [1] to x gives
   dz
  ---- = -sin( x ) + cos( x ) . i                         [3]
   dx
3. Getting another i in this equation [3]
   1. The definition of i is
        i = SquareRoot( -1 )                              [4]
   2. Squaring [4] gives
         2                        2
        i   = ( SquareRoot( -1 ) )  = -1                  [5]
   3. Writing [3] in this way
   dz
  ---- = - 1. sin( x ) + cos( x ) . i                     [6]
   dx
   4. Putting [5] in [6] gives
   dz       2
  ---- = - i . sin( x ) + cos( x ) . i                    [7]
   dx
4. Separating the common factor i in [7] gives
   dz
  ---- = i . ( cos( x ) + sin( x ) . i )                  [8]
   dx
5. Now using [2] and putting this in [8], you can thus write
   dz
  ---- = i . z                                            [9]
   dx
6. This is a differential equation which you can solve
   by separation of the variables
   dz
  ---- = i . dx                                           [10]
    z
7. Integrating both sides
             dz
  INTEGRAL( ---- ) = INTEGRAL( i . dx )                   [11]
              z
8. Because i is constant, you can take it out of the
   integral
  ln( z ) = INTEGRAL( dx ) . i                            [12]
9. Thus you get
  ln( z ) = x . i + constant                              [13]
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                        [14]
    3. Which becomes
        0       = 0 + constant                            [15]
    4. Or thus
        constant = 0                                      [16]
11. Putting the constant back in the equation,
    by putting [16] in [13] gives
     ln( z ) = x . i + 0                                  [17]
    or thus
    ln( z ) = x . i                                       [18]
12. Now introducing e
    1. Let both sides of [18] be a power of e
      ln( z )     x . i
     e        =  e                                        [19]
       or after working out
          x . i
     z = e                                                [20]
13. Replacing z on the left side [2] gives
      x . i
     e      = cos( x ) + sin( x ) . i                     [21]
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 [21], by choosing a few
    angles x.
    1. Choosing the angle x equal to 360 degrees,
       or thus 2 PI radians                               [22]
       Putting [22] in [21] gives
      (2 . PI) . i
     e              = cos( 2 . PI ) + sin( 2 . PI ) . i   [23]
      (2 . PI) . i
     e              = 1 + 0 . i                           [24]
     or thus
      (2 . PI) . i
     e             = 1
    ---
    2. Choosing the angle x equal to 180 degrees,
       or thus PI radians                                 [25]
       Putting [25] in [21] gives
      ( PI ) . i
     e              = cos( PI ) + sin( PI ) . i           [26]
     or thus
      ( PI ) . i
     e              = -1 + 0 . i                          [27]
     or thus
      ( PI ) . i
     e              = -1                                  [28]
    ---
    3. Choosing the angle x equal to 90 degrees,
       or thus PI / 2 radians                             [29]
       Putting [29] in [21] gives
      ( PI / 2 ) . i
     e              = cos( PI / 2 ) + sin( PI / 2 ) . i   [30]
     or thus
      ( PI / 2 ) . i
     e              = 0 + 1 . i                           [31]
     or thus
      ( PI / 2 ) . i
     e              = i                                   [32]
---
---
Internet: see also:
---
Math: Number: Complex: Link: Overview: Can you give an overview of 
links?
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/