De Moivre's Formula

Theorem

Let $z \in \C$ be a complex number expressed in complex form:

$z = r \paren {\cos x + i \sin x}$

Then:

$\forall \omega \in \C: \paren {r \paren {\cos x + i \sin x} }^\omega = r^\omega \paren {\map \cos {\omega x} + i \, \map \sin {\omega x} }$

Exponential Form

can also be expressed thus in exponential form:

$\forall \omega \in \C: \paren {r e^{i \theta} }^\omega = r^\omega e^{i \omega \theta}$

Integer Index

This result is often given for integer index only:

Let $z \in \C$ be a complex number expressed in complex form:

$z = r \paren {\cos x + i \sin x}$

Then:

$\ds \forall n \in \Z: \, $

$\ds \paren {r \paren {\cos x + i \sin x} }^n$

$=$

$\ds r^n \paren {\map \cos {n x} + i \map \sin {n x} }$

$\ds $

$=$

$\ds r^n \cos n x + i r^n \sin n x$

Rational Index

Some sources give it for rational index:

Let $z \in \C$ be a complex number expressed in complex form:

$z = r \paren {\cos x + i \sin x}$

Then:

$\forall p \in \Q: \paren {r \paren {\cos x + i \sin x} }^p = r^p \paren {\map \cos {p x} + i \, \map \sin {p x} }$

Proof 1

$\ds \paren {r \paren {\cos x + i \sin x} }^\omega$

$=$

$\ds \paren {r e^{i x} }^\omega$

Euler's Formula

$\ds $

$=$

$\ds r^\omega e^{i \omega x}$

Power of Power

$\ds $

$=$

$\ds r^\omega \paren {\map \cos {\omega x} + i \map \sin {\omega x} }$

Euler's Formula

$\blacksquare$

Also presented as

is also often presented in the simpler form:

$\forall \omega \in \C: \paren {\cos x + i \sin x}^\omega = \map \cos {\omega x} + i \, \map \sin {\omega x}$

Also known as

is also known as De Moivre's Theorem.

Source of Name

This entry was named for Abraham de Moivre.

Historical Note

was discovered by Abraham de Moivre around $1707$.

Sources

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 6$: Complex Numbers: De Moivre's Theorem: $6.9$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): de Moivre's theorem
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): de Moivre's theorem
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 4$: Complex Numbers: De Moivre's Theorem: $4.10.$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): De Moivre's Theorem
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $6$: Basic Algebra