Complex Number Power Formula Derivation
DeMoivre’s Theorem can be derived with the help of Mathematical Induction as follows:
P(n): (cos x + i sin x)n = cos(nx) + isin(nx) ⇢ (1)
For n = 1, we have
P(1) = (cos x + i sin x)1
P(1) = cos(1x) + i sin(1x)
P(1) = cos(x) + i sin(x)
That is true and thus, P(1) is true.
Assuming P(k) is true, i.e.
P(k) = (cos x + i sin x)k = cos(kx) + i sin(kx) ⇢ (2)
Now, we just have to prove that the P(k+1) is also true.
P(k+1) = (cos x + i sin x)k+1
⇒ P(k+1) = (cos x + i sin x)k (cos x + i sin x)
⇒ P(k+1) = (cos (kx) + i sin (kx)) (cos x + i sin x) [Using (i)]
⇒ P(k+1) = cos (kx) cos x − sin(kx) sinx + i (sin(kx) cosx + cos(kx) sinx)
⇒ P(k+1) = cos {(k + 1)x} + i sin {(k + 1)x}
⇒ P(k+1) = (cos x + i sin x)k+1 = cos {(k + 1)x} + i sin {(k + 1)x}
Thus, P(k+1) is also true, thus by the principal of mathematical induction, P(n) is true.
Hence the result is proved.
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Complex Number Power Formula
Complex Numbers are numbers that can be written as a + ib, where a and b are real numbers and i (iota) is the imaginary component and its value is √(-1), and are often represented in rectangle or standard form. 10 + 5i, for example, is a complex number in which 10 represents the real component and 5i represents the imaginary part. Depending on the values of a and b, they might be wholly real or purely fictitious. When a = 0 in a + ib, ib is a totally imaginary number, and when b = 0, we get a, which is a strictly real number.
In this article, we will learn about, complex number power formulas, their examples, and others in detail.
Table of Content
- Complex Number Definition
- Complex Number Power Formula
- Complex Number Power Formula Derivation
- Solved Example on Complex Number Power Formula
- FAQs