Definition:Sine/Complex Function
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Definition
The complex function $\sin: \C \to \C$ is defined as:
\(\ds \forall z \in \C: \, \) | \(\ds \sin z\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1 } } {\paren {2 n + 1}!}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds z - \frac {z^3} {3!} + \frac {z^5} {5!} - \frac {z^7} {7!} + \cdots + \paren {-1}^n \frac {z^{2 n + 1 } } {\paren {2 n + 1}!} + \cdots\) |
Also see
- Radius of Convergence of Power Series Expansion for Sine Function: this power series converges for all values of $z \in \C$.
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.1$. Introduction: $(4.2)$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): sine
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): sine series: 1.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): sine series: 1.