Difference between revisions of "De Moivre's Theorem"
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*If <math>n>0</math>: | *If <math>n>0</math>: | ||
− | : | + | :If <math>n=0</math>, the formula holds true because <math>\cos(0x)+i\sin(0x)=1+i0=1=z^0.</math> |
− | :Assume true for | + | :Assume the formula is true for <math>n=k</math>. Now, consider <math>n=k+1</math>: |
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
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\end{align*}</cmath> | \end{align*}</cmath> | ||
− | :Therefore, the result is true for all | + | :Therefore, the result is true for all nonnegative integers <math>n</math>. |
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*If <math>n<0</math>, one must consider <math>n=-m</math> when <math>m</math> is a positive integer. | *If <math>n<0</math>, one must consider <math>n=-m</math> when <math>m</math> is a positive integer. |
Revision as of 09:20, 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 .