Euler's Formula/Proof
< Euler's Formula(Redirected from Euler's Formula/Proof 3)
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Theorem
Let $z \in \C$ be a complex number.
Then:
- $e^{i z} = \cos z + i \sin z$
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 |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $\text{(ii)}$: $(4.14)$
- 2001: Christian Berg: Kompleks funktionsteori: $\S 1.5$
- For a video presentation of the contents of this page, visit the Khan Academy.