Euler's Formula/Corollary

Corollary to Euler's Formula

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.

Corollary

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

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

Proof

\(\ds e^{-i z}\)

\(=\)

\(\ds \map \cos {-z} + i \map \sin {-z}\)

Euler's Formula

\(\ds \)

\(=\)

\(\ds \cos z + i \map \sin {-z}\)

Cosine Function is Even

\(\ds \)

\(=\)

\(\ds \cos z - i \sin z\)

Sine Function is Odd

$\blacksquare$

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.

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $\text{(ii)}$: $(4.16)$