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

De Moivre's Formula 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 defined as

This result 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

De Moivre's Theorem.

Source of Name

This entry was named for Abraham de Moivre.

Historical Note

De Moivre's Formula was discovered by Abraham de Moivre around $1707$.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $6.9$: De Moivre's Theorem
• 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
• 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