Exponential of Zero/Proof 5
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Theorem
- $\exp 0 = 1$
Proof
| \(\ds \map \exp {z + \paren {-z} }\) | \(=\) | \(\ds \exp z \, \map \exp {-z}\) | Exponential of Sum | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \exp {z - z}\) | \(=\) | \(\ds \dfrac {\exp z} {\exp z}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \exp 0\) | \(=\) | \(\ds 1\) |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $\text{(i)}$