Moivres Theorem

Equating the separate real and imaginary parts of each side of the equation, we obtain the two fundamental angle-sum trigonometric identities  

Dividing botli numerator and denominator by cos a cos /9 and introducing tan a and tan /9, we obtain 

31) can be applied to the square of a phasor z = cis6 of modulus 1, giving z = (cis 6) = cis(20). This can, in fact, be extended to the nth power of z, giving (cis ) = cis(n0). This is a famous result known as de Moivre s theorem, which we write out in full  

Beginning with de Moivre s theorem, more useful identities involving sines and cosines can be derived. For example, setting n = 2, 

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Use the Euler formula and the De Moivre theorem to evaluate powers of complex numbers, to determine / th roots of a complex number, and to identify real and imaginary parts of functions of a complex variable... 

After cancelling the r" factors in equation (2.29), we obtain the De Moivre theorem ... 

On the other hand, the Taylor expansion of Y gives the coefficient ak with de Moivre s theorem in the form48,49 ... 

Rule 5 must be modified slightly for higher-order systems. See de Moivre s theorem. 

De Moivre s theorem can be used to determine the nth roots of unity, namely the n complex roots of the equation... 

De Moivre s theorem, Eq. (4.37), remains valid even for noninteger values of n. Replacing n by 1/m, we can write... 

Normal approximation (De Moivre-Laplace limit theorem). It follows from the interpretation as well as from the central limit theorem that for large enough values of npq (npq > 6 suffices already) the binomial distribution can be approximated by a normal distribution with expected value p = p and variance [Pg.416]

The above interpretation is the Lindberg-Levy version of the central limit theorem. The [N np, npq)] normal distribution could also be interpreted as the limiting case of the B n, p) binomial distribution (De Moivre-Laplace theorem), but that is only a special case of the general theorem phrased for sums when the limiting distribution in general is N( /t, na ) normal. 

The solution is bifurcated and, for valnes of < 4y, the argument within the square root sign in Equations (12.71) and (12.72) become imaginary and, using De Moivre s theorem and equating the real parts. 

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