Abraham
De Moivre (1667-1754) was born in France, but fled to England in 1688,
after being imprisoned for his religious beliefs. A brilliant
mathematician, he was unable to gain a university appointment (because
he was born in France) or escape his life of poverty, gaining only a meagre income as a private tutor. He was friends with Sir Isaac Newton
and Edmund Halley (1656 - 1742), was elected to the Royal Society in
England, and to the Academies of Paris and Berlin, yet in spite of the
support of the great Leibniz (1646 - 1716), and
of Jacques Bernoulli,
he never gained a university appointment and died in relative
poverty. In spite
of this, he made many discoveries in mathematics, some of which are
attributed to others (For instance,
Stirling's Formula for factorial
approximations was known earlier by De Moivre).
This
page deals with the proof of De Moivre's Theorem, etc. It has formula
to compute the cosine and sine directly, but these require further
algebraic manipulation, to reduce the sines in the cosine formula, and
the cosines in the sine formula. This work is extended later beginning
with the cosine, where formulae
are also derived to calculate directly the coefficients of a given
power in a given expansion for cos nx and sin nx. Before that, some
pages, mentioned below, deal with the Chebyshev method.
The
following is nowadays called De Moivre's formula, clearly known to De
Moivre, but never explicitly expressed in this form by him. [1.1] where i=√(-1)
Picking out the real parts from both sides and equating them, for cosine, we get: [1.2] Where k, a nonnegative integer, is 2·p, for the pth term, p=0, 1, 2..., and k≤n
For the sine, we find: [1.3] Where k, a nonnegative integer, is 1+2·p, for the pth term, p=0, 1, 2..., and k≤n
We
can use this formula to find multiple angles of cosine and sine by
associating the real parts of the right-hand side, expanded with the Binomial Theorem, with cosine and the imaginary parts with the sine. We give examples on a following page.
On
this page, we make a limited claim, that the formula is true for all
integers n. Actually it is true in a much wider context, for complex
numbers.
Explanation
Leonhard Euler (1707-1783), was inspired by De Moivre, to formulate: [2.1]
Because [2.2]
We can substitute ix for x (where i=√-1 ) [2.3] Replacing i^{2} by -1, and i^{4} by 1, etc: [2.4]
Grouping the real and imaginary parts: [2.5] We note the first series is the expansion of cos x, and the second i·sin x, so:
(which is what Euler claimed)
By raising e^{ix} to the power n, we note, the two forms depending on how we group the indexes:
Which, assuming the expansions above for e^{ix} , sin x, cos x, etc (Proved in calculus using Maclaurin's Theorem), we have proved De Moivre's Law:
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Proof by Induction
Nonnegative n
We wish to show by mathematical induction that [1.1, repeated] Where n is a natural number
When n=1, then: cos x+i·sin x=(cos x+i·sin x) Which is true, so the theorem is true for n=1.
For n=k, where k is a nonnegative integer, by the formula, we have cos kx+i·sin kx=(cos x+i·sin x)^{k} [3.1].
Assuming that it is true for n=k, then when n=k+1: cos (k+1)x+i·sin (k+1)x= (cos kx+i·sin kx)(cos x+i·sin x) [3.2] by multiplying Equation 3.1 by (cos x+i·sin x)
Multiplying out Equation 3.2 we find: =cos kx·cos x+i^{2}·sin kx·sin x+i·sin kx·cos x+i·cos kx·sin x =cos kx·cos x-sin kx·sin x+i·sin kx·cos x+i·cos kx·sin x [because i^{2}=-1] =cos (k+1)x+i·sin (k+1) by the compound angle formula.
So if the formula is true for n=k, it is also true for n=k+1. As it true for n=1, then it is true for all n
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Negative n
Assuming (or claiming) that the formula is true for nonnegative n, let us write n as -m, where m is a positive integer: [4.1]
So:
[4.2]
Applying De Moivre's formula to the right-hand side: [4.3]
Multiplying top and bottom by cos (mx)-i·sin(mx), noting (a−b)(a+b)=a^{2}+b^{2} : [4.4] And because sin^{2}θ+cos^{2}θ=1, we have: [4.5]
Substituting n back into the formula, that is,
substituting −n
for m: [4.6]
Taking the
minuses outside, and noting that cos(−nx)=cos(nx),
and sin (−nx)=−sin(nx), we have: [4.7]
which is De Moivre's formula for negative n, so the formula works for all integers. ■