Difference between revisions of "De Moivre's Theorem"
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This is one proof of de Moivre's theorem by [[induction]]. | This is one proof of de Moivre's theorem by [[induction]]. | ||
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:If <math>n=0</math>, the formula holds true because <math>\cos(0x)+i\sin(0x)=1+i0=1=z^0.</math> | :If <math>n=0</math>, the formula holds true because <math>\cos(0x)+i\sin(0x)=1+i0=1=z^0.</math> |
Revision as of 09:22, 31 August 2024
De Moivre's Theorem is a very useful theorem in the mathematical fields of complex numbers. It allows complex numbers in polar form to be easily raised to certain powers. It states that for and , .
Proof
This is one proof of de Moivre's theorem by induction.
- If :
- If , the formula holds true because
- Assume the formula is true for . Now, consider :
- Therefore, the result is true for all nonnegative integers .
- If , one must consider when is a positive integer.
And thus, the formula proves true for all integral values of .
Generalization
Note that from the functional equation where , we see that behaves like an exponential function. Indeed, Euler's identity states that . This extends de Moivre's theorem to all .