How to prove De Moivres theorem using eulers formula
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(69 votes)
Published: Dec 03, 2003 12:00 a.m.
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The proof of De Moivres is almost always done by mathematical induction and this is the way that it is usually asked for in exams. The method of proving using Eulers formula is also valid, knowing alternate methods is good as it expands mathematical knowledge.
First i will go through eulers formula
im using x where it would usually be theta
if f(x)= cosx + isinx
then f’(x) = -sinx + icosx (simple differentiation)
= i(cosx + isinx) (taking i as factor nb i^2 =-1)
= i.f(x) (i times f(x) for those unfamiliar with the notation)
from here we can conclude that f’(x) / f(x) = i
and the integral of f’(x)/f(x)dx [I{f’(x)/f(x)]dx =
the integral of i dx,
i.e I{f’(x)/f(x)}dx = I{i}dx
integrating both sides gives,
ln (cosx + isinx) = ix+c
sub x=0, =>> ln1 = 0+c , ==>> c=0
take exponential both sides,
cosx + isinx = e^ix
and e^-ix = cosx - isinx (using -x, and cos(-x) = cosx
Now for the de moivres proof,
RTP:(cosx + isinx)^n = cosnx + isinnx {n>=1, nEJ)
by eulers formula, cosx+isinx = e^ix
so, lhs= (e^ix)^n = e^inx
writing it back as the original
lhs = cosnx + isinnx = rhs,
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 | Dec 16, 2004 12:32 am -
I tried this.It really works! |
 | Dec 16, 2004 4:19 am -
lol i really truely hope that dusnt werk otherwise im talkin to albert freakin einstein lol
dude that is soooo complicated for a dumass oldshagger like myself |
 | Dec 16, 2004 5:42 pm -
it does work, the idea of the proof is to show that it does. |
 | Mar 31, 2005 4:51 pm -
nice it is good to see some eggs on mathmatics |
 | Apr 13, 2005 8:25 am -
I derived your MOTHERS maxwells equations in 5 steps using tensors! |
 | Mar 26, 2006 9:30 am -
i already knew this and its common sense really so 2 stars |
| Mar 26, 2006 9:30 am -
5stars btw |
 | Sep 09, 2006 4:29 pm -
You know what i am wondering?
I am wondering if this is maths or physics.
But nyways thanks to you,im gonna fail in both. |
 | Jan 18, 2007 10:06 pm -
I think I’m going to cum. |
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