- De Moivre's formula
De Moivre's formula, named after
Abraham de Moivre , states that for anycomplex number (and, in particular, for anyreal number ) "x" and anyinteger "n" it holds that:left(cos x+isin x ight)^n=cosleft(nx ight)+isinleft(nx ight).,
The formula is important because it connects
complex numbers ("i" stands for theimaginary unit ) and trigonometry. The expression "cos "x" + "i" sin "x" is sometimes abbreviated to "cis "x".By expanding the left hand side and then comparing the real and imaginary parts under the assumption that "x" is real, it is possible to derive useful expressions for cos("nx") and sin("nx") in terms of cos("x") and sin("x"). Furthermore, one can use this formula to find explicit expressions for the "n"-th roots of unity, that is, complex numbers "z" such that "zn" = 1.
Derivation
Although historically proved earlier, de Moivre's formula can easily be derived from Euler's formula
:e^{ix} = cos x + isin x,
and the exponential law
:left( e^{ix} ight)^n = e^{inx} .,
Then, by
Euler's formula ,:e^{i(nx)} = cos(nx) + isin(nx).,
Proof by induction
We consider three cases.
For "n" > 0, we proceed by
mathematical induction . When "n" = 1, the result is clearly true. For our hypothesis, we assume the result is true for some positive integer "k". That is, we assume:left(cos x + i sin x ight)^k = cosleft(kx ight) + i sinleft(kx ight). ,
Now, considering the case "n" = "k" + 1:
:egin{alignat}{2} left(cos x+isin x ight)^{k+1} & = left(cos x+isin x ight)^{k} left(cos x+isin x ight)\ & = left [cosleft(kx ight) + isinleft(kx ight) ight] left(cos x+isin x ight) &&qquad mbox{by the induction hypothesis}\ & = cos left(kx ight) cos x - sin left(kx ight) sin x + i left [cos left(kx ight) sin x + sin left(kx ight) cos x ight] \ & = cos left [ left(k+1 ight) x ight] + isin left [ left(k+1 ight) x ight] &&qquad mbox{by the trigonometric identities}end{alignat}
We deduce that the result is true for "n" = "k" + 1 when it is true for "n" = "k". By the principle of mathematical induction it follows that the result is true for all positive integers "n"≥1.
When "n" = 0 the formula is true since cos (0x) + isin (0x) = 1 + i0 = 1, and (by convention) z^0 = 1.
When "n" < 0, we consider a positive integer "m" such that "n" = −"m". So:egin{align} left(cos x + isin x ight)^{n} & = left(cos x + isin x ight)^{-m}\ & = frac{1}{left(cos x + isin x ight)^{m\ & = frac{1}{left(cos mx + isin mx ight)}\ & = cosleft(mx ight) - isinleft(mx ight)\ & = cosleft(-mx ight) + isinleft(-mx ight)\ & = cosleft(nx ight) + isinleft(nx ight).end{align}
Hence, the theorem is true for all integer values of "n".
Formulas for cosine and sine individually
Being an equality of
complex number s, one necessarily has equality both of thereal part s and of theimaginary part s of both members of the equation. If "x", and therefore also cos x and sin x, arereal number s, then the identity of these parts can be written (interchanging sides) as:egin{alignat}2 cos(nx)&=sum_{k=0}^{lfloor n/2 floor}{ binom{n}{2k(-1)^k(cos{x})^{n-2k}(sin{x})^{2k}& &=sum_{k=0}^{lfloor n/2 floor}{ binom{n}{2k(cos{x})^{n-2k}((cos{x})^2-1)^k\ sin(nx)&=sum_{k=0}^{lfloor (n-1)/2 floor}{ binom{n}{2k+1(-1)^k(cos{x})^{n-2k-1}(sin{x})^{2k+1}& &=(sin{x})sum_{k=0}^{lfloor(n-1)/2 floor}{ binom{n}{2k+1(cos{x})^{n-2k-1}((cos{x})^2-1)^k.\end{alignat}These equations are in fact even valid for complex values of "x", because both sides areholomorphic function s of "x", and two such functions that coincide on the real axis necessarily coincide on the wholecomplex plane . Here are the concretre instances of these equations for n=2 and n=3::egin{alignat}2 cos(2x) &= (cos{x})^2 +((cos{x})^2-1) &&= 2(cos{x})^2-1\ sin(2x) &= 2(sin{x})(cos{x})\ cos(3x) &= (cos{x})^3 +3cos{x}((cos{x})^2-1) &&= 4(cos{x})^3-3cos{x}\ sin(3x) &= 3(cos{x})^2(sin{x})-(sin{x})^3 &&= 3sin{x}-4(sin{x})^3.\end{alignat}The right hand side of the formula for cos(nx) is in fact the value T_n(cos x) of theChebyshev polynomial T_n at cos x.Generalization
The formula is actually true in a more general setting than stated above: if "z" and "w" are complex numbers, then
:left(cos z + isin z ight)^w
is a
multivalued function while:cos (wz) + i sin (wz),
is not. Therefore one can state that
:cos (wz) + i sin (wz) , is one value of left(cos z + isin z ight)^w.,
Applications
This formula can be used to find the n^{th} roots of a complex number. If z is a complex number, written in polar form as
: z=rleft(cos x+isin x ight),,
then
: z^}^{frac{1}{n}= left [ rleft( cos x+isin x ight) ight] ^ }^{frac{1}{n}= r^}^{frac{1}{n} left [ cos left( frac{x+2kpi}{n} ight) + isin left( frac{x+2kpi}{n} ight) ight]
where k is an integer, to get the n different roots of z one only needs to consider values of k from 0 to n-1.
ee also
*
Euler's formula
*Root of unity References
* Milton Abramowitz and Irene A. Stegun, "
Handbook of Mathematical Functions ", (1964) Dover Publications, New York. ISBN 0-486-61272-4. "(p. 74)".External links
* [http://demonstrations.wolfram.com/DeMoivresTheoremForTrigIdentities/ De Moivre's Theorem for Trig Identities] by Michael Croucher,
The Wolfram Demonstrations Project .
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