De Moivre's Theorem is a cornerstone of complex analysis, enabling easy computation of complex number powers. It transforms multiplication into a simple process using polar coordinates, aiding in solving complex equations and expanding trigonometric expressions. The theorem's proof relies on mathematical induction and is vital for deriving multiple-angle formulas in trigonometry, as well as for practical applications in mathematics.
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De Moivre's Theorem is a pivotal theorem in complex analysis, attributed to Abraham de Moivre, which facilitates the computation of powers of complex numbers
Modulus and Argument
De Moivre's Theorem states that for any complex number expressed in polar form, the nth power of z is given by r^n(cos nθ + isin nθ)
Simplification of Powers
This theorem simplifies the process of raising complex numbers to integer powers, bypassing the need for cumbersome binomial expansion
De Moivre's Theorem is applied by first representing a complex number in polar form and then raising it to the desired power, making it advantageous for higher powers where manual multiplication is impractical
The proof of De Moivre's Theorem employs mathematical induction, verifying the theorem for n = 1 and demonstrating its validity for k + 1 by multiplying z^k by z and using trigonometric identities
The theorem is also valid for negative integer powers, with the proof extending from the positive case using mathematical induction and the properties of complex conjugates
De Moivre's Theorem is crucial for solving complex equations of the form x^n = a + ib, where a + ib is in polar form, by determining the magnitude and argument of the solution
By raising (cos θ + isin θ)^n to the power of n and applying the binomial theorem, De Moivre's Theorem aids in expanding trigonometric expressions and generating multiple-angle formulas
De Moivre's Theorem has practical applications in various complex number problems, such as expanding complex numbers to a given power and deriving identities for complex numbers in polar form
De Moivre's Theorem is a fundamental aspect of complex number theory, streamlining the calculation of integer powers of complex numbers and aiding in the resolution of complex equations
The theorem's utility spans finding roots of complex numbers, facilitating trigonometric expansions, and deriving multiple-angle formulas
De Moivre's Theorem's importance in mathematical education and practice renders it an indispensable tool for both students and professionals
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The theorem is essential for computing roots of complex numbers and establishing ______-angle formulas.
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