Difference between revisions of "De Moivre's Theorem"
(→See Also) |
(Tag: Undo) |
||
Line 36: | Line 36: | ||
==Generalization== | ==Generalization== | ||
− | + | ==See Also== | |
[[Category:Theorems]] | [[Category:Theorems]] | ||
[[Category:Complex numbers]] | [[Category:Complex numbers]] |
Latest revision as of 01:42, 11 January 2024
DeMoivre'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
, for
, the case is obviously true.
- Assume true for the case
. Now, the case of
:
- Therefore, the result is true for all positive integers
.
- If
, the formula holds true because
. Since
, the equation holds true.
- If
, one must consider
when
is a positive integer.
And thus, the formula proves true for all integral values of .
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
.