De Moivre's Theorem for Rational Exponents

In summary, De Moivre's Theorem for Rational Exponents is a mathematical theorem used to raise complex numbers to rational powers. It is an extension of De Moivre's Theorem and has a formula of (cosθ + isinθ)^n = cos(nθ) + isin(nθ), where n is a rational number and θ is the angle of the complex number. To simplify complex numbers, the rational exponent must first be converted to a fraction and then the formula (cosθ + isinθ)^b = cos(bθ) + isin(bθ) is used. There are restrictions for using this theorem, including the rational exponent being in the form of a/b and the complex number being in polar form.
  • #1
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##cosθ + isinθ = cos(θ + 2kπ) + isin(θ + 2kπ)## for ##k ∈ ℤ##
##[cosθ + isinθ]^n = [cos(θ + 2kπ) + isin(θ + 2kπ)]^n##
##cos(nθ) + isin(nθ) = cos(nθ + 2nkπ) + isin(nθ + 2nkπ)##
##cos(nθ + 2mπ) + isin(nθ + 2mπ) = cos(nθ + 2nkπ) + isin(nθ + 2nkπ)## for ##m ∈ ℤ##

Now consider the special case ##n = 1/p## for ##p ∈ ℤ##

##cos(\frac{θ}{p} + 2mπ) + isin(\frac{θ}{p} + 2mπ) = cos(\frac{θ}{p} + \frac{2kπ}{p}) + isin(\frac{θ}{p} + \frac{2kπ}{p})##

Is this a contradiction?
 
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  • #2
Where do you see a contradiction?
 
  • #3
MohammedRady97 said:
##cosθ + isinθ = cos(θ + 2kπ) + isin(θ + 2kπ)## for ##k ∈ ℤ##
##[cosθ + isinθ]^n = [cos(θ + 2kπ) + isin(θ + 2kπ)]^n##
##cos(nθ) + isin(nθ) = cos(nθ + 2nkπ) + isin(nθ + 2nkπ)##
##cos(nθ + 2mπ) + isin(nθ + 2mπ) = cos(nθ + 2nkπ) + isin(nθ + 2nkπ)## for ##m ∈ ℤ##

Now consider the special case ##n = 1/p## for ##p ∈ ℤ##

##cos(\frac{θ}{p} + 2mπ) + isin(\frac{θ}{p} + 2mπ) = cos(\frac{θ}{p} + \frac{2kπ}{p}) + isin(\frac{θ}{p} + \frac{2kπ}{p})##

Is this a contradiction?
No. The left side and the right side are two different things. The left side simply expresses the fact [itex]e^{2in\pi}=1[/itex] while the right side simply says there are p pth roots of [itex]e^{i\theta}[/itex]
 
  • #4
mathman said:
No. The left side and the right side are two different things. The left side simply expresses the fact [itex]e^{2in\pi}=1[/itex] while the right side simply says there are p pth roots of [itex]e^{i\theta}[/itex]

The very fact that the RHS and LHS seem to represent two different things is the reason I'm confused as to whether the equality holds.
 
  • #5
mathman said:
No. The left side and the right side are two different things. The left side simply expresses the fact [itex]e^{2in\pi}=1[/itex] while the right side simply says there are p pth roots of [itex]e^{i\theta}[/itex]

Also, could you please elaborate on what the RHS represents?
 
  • #6
[itex]e^{i(\theta+2k\pi)}[/itex] is the same number for all integer values of k. However [itex]e^{\frac{i(\theta+2k\pi)}{p}}[/itex] will have p possible values for k =0,1,...,p-1. These are the pth roots.
 
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  • #7
mathman said:
[itex]e^{i(\theta+2k\pi)}[/itex] is the same number for all integer values of k. However [itex]e^{\frac{i(\theta+2k\pi)}{p}}[/itex] will have p possible values for k =0,1,...,p-1. These are the pth roots.

But this means that, technically, they're not equal. Right? They're only equal in special cases where k/p is an integer.
 
  • #8
MohammedRady97 said:
But this means that, technically, they're not equal. Right? They're only equal in special cases where k/p is an integer.
I hate to keep repeating myself. The left side of the expression represents one particular pth root. The right side is any pth root, depending on k, which is different, unless k/p is an integer.
 

What is De Moivre's Theorem for Rational Exponents?

De Moivre's Theorem for Rational Exponents is a mathematical theorem that allows us to raise complex numbers to rational powers. It is an extension of De Moivre's Theorem, which deals with raising complex numbers to integer powers.

What is the formula for De Moivre's Theorem for Rational Exponents?

The formula for De Moivre's Theorem for Rational Exponents is (cosθ + isinθ)^n = cos(nθ) + isin(nθ), where n is a rational number and θ is the angle of the complex number.

How do you use De Moivre's Theorem for Rational Exponents to simplify complex numbers?

To simplify a complex number using De Moivre's Theorem for Rational Exponents, we first convert the rational exponent to a fraction in the form of a/b. Then, we use the formula (cosθ + isinθ)^b = cos(bθ) + isin(bθ) to raise the complex number to the power of b. Finally, we use the formula (a+bi)^a = a^a + (a^(a-1)b)i to raise the result to the power of a.

What are the restrictions for using De Moivre's Theorem for Rational Exponents?

The restrictions for using De Moivre's Theorem for Rational Exponents are that the rational exponent must be in the form of a/b, where a and b are integers, and b cannot be equal to 0. Additionally, the complex number must be in polar form, with the angle θ in radians.

What are the applications of De Moivre's Theorem for Rational Exponents?

De Moivre's Theorem for Rational Exponents has various applications in mathematics and physics, particularly in the fields of complex analysis and differential equations. It is also used in engineering, specifically in the analysis of electrical circuits and signal processing.

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