Euler's Formula

Theorem

Let $z \in \C$ be a complex number.

Then:

$e^{i z} = \cos z + i \sin z$

where:

$e^{i z}$ denotes the complex exponential function

$\cos z$ denotes the complex cosine function

$\sin z$ denotes complex sine function

$i$ denotes the imaginary unit.

Real Domain

This result is often presented and proved separately for arguments in the real domain:

Let $\theta \in \R$ be a real number.

Then:

$e^{i \theta} = \cos \theta + i \sin \theta$

Corollary

$e^{-i z} = \cos z - i \sin z$

Proof

As Complex Sine Function is Absolutely Convergent and Complex Cosine Function is Absolutely Convergent, we have:

$\ds \cos z + i \sin z$

$=$

$\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \dfrac {z^{2 n} } {\paren {2 n}!} + i \sum_{n \mathop = 0}^\infty \paren {-1}^n \dfrac {z^{2 n + 1} } {\paren {2 n + 1}!}$

Definition of Complex Cosine Function and Definition of Complex Sine Function

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty \paren {\paren {-1}^n \dfrac {z^{2 n} } {\paren {2 n}!} + i \paren {-1}^n \dfrac {z^{2 n + 1} } {\paren {2 n + 1}!} }$

Sum of Absolutely Convergent Series

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty \paren {\dfrac {\paren {i z}^{2 n} } {\paren {2 n}!} + \dfrac {\paren {i z}^{2 n + 1} } {\paren {2 n + 1}!} }$

Definition of Imaginary Unit

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty \dfrac {\paren {i z}^n} {n!}$

$\ds $

$=$

$\ds e^{i z}$

Definition of Complex Exponential Function

$\blacksquare$

Examples

Example: $e^{i \pi / 4}$

$e^{i \pi / 4} = \dfrac {1 + i} {\sqrt 2}$

Example: $e^{i \pi / 2}$

$e^{i \pi / 2} = i$

Example: $e^{-i \pi / 2}$

$e^{-i \pi / 2} = -i$

Example: $e^{i \pi}$

$e^{i \pi} = -1$

Example: $e^{2 i \pi}$

$e^{2 i \pi} = 1$

Example: $e^{2 k i \pi}$

$\forall k \in \Z: e^{2 k i \pi} = 1$

Also known as

in this and its corollary form, along with Euler's sine identity and Euler's cosine identity are also found referred to as Euler's identities.

However, this allows for confusion with Euler's identity:

$e^{i \pi} + 1 = 0$

To compound the confusion, is also itself often referred to as Euler's identity.

It is wise when referring to it by name, therefore, to ensure that the equation itself is also specified.

Also see

Source of Name

This entry was named for Leonhard Paul Euler.

Historical Note

Leonhard Paul Euler famously published what is now known as in $1748$.

However, it needs to be noted that Roger Cotes first introduced it in $1714$, in the form:

$\map \ln {\cos \theta + i \sin \theta} = i \theta$

Sources

1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Euler's formula
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Euler's identities
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Euler's formula
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Euler's identities
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Euler's formula
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): complex exponential
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $6$: Basic Algebra