Entry
Math: Number: Complex: e^( x . i ) = cos( x ) + sin( x ) . i : Proof: Can you give a proof?
Apr 16th, 2005 07:10
Knud van Eeden,
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--- 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?
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A possible algebraic proof:
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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
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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]
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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]
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Internet: see also:
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Math: Number: Complex: Link: Overview: Can you give an overview of
links?
http://www.faqts.com/knowledge_base/view.phtml/aid/35633/fid/1793
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