# Definition:Sine/Complex Function

< Definition:Sine(Redirected from Definition:Complex Sine Function)

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## Definition

The complex function $\sin: \C \to \C$ is defined as:

\(\displaystyle \sin z\) | \(=\) | \(\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1 } } {\paren {2 n + 1}!}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle z - \frac {z^3} {3!} + \frac {z^5} {5!} - \frac {z^7} {7!} + \cdots + \paren {-1}^n \frac {z^{2 n + 1 } } {\paren {2 n + 1}!} + \cdots\) |

## Also see

- Radius of Convergence of Power Series over Factorial: this power series converges for all values of $z \in \C$.

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 4.1$. Introduction: $(4.2)$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**sine series**:**1.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**sine series**:**1.**