Definition:Cosine/Complex Function

Definition

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

\(\ds \cos z\)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}\)

\(\ds \)

\(=\)

\(\ds 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \frac {z^6} {6!} + \cdots + \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!} + \cdots\)

Examples

Example: $4 \cos z = 3 + i$

Let:

$4 \cos z = 3 + i$

Then:

$z = \dfrac {\paren {8 n + 1} \pi} 4 - \dfrac {i \ln 2} 2$ for $n \in \Z$

or:

$z = \dfrac {\paren {8 m - 1} i \pi} 4 + \dfrac {i \ln 2} 2$ for $m \in \Z$

Also see

• Radius of Convergence of Power Series Expansion for Cosine Function: this power series converges for all values of $z \in \C$.

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.1$. Introduction: $(4.3)$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): cosine
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cosine series: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cosine series: 1.