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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/