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

Euler's Formula 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:

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

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

Also see

• Euler's Identity
• Sum of Hyperbolic Sine and Cosine equals Exponential

Source of Name

This entry was named for Leonhard Paul Euler.

Historical Note

Leonhard Paul Euler famously published what is now known as Euler's Formula 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): Appendix $6$: Basic Algebra