Complex Cosine Function is Entire
Theorem
Let $\cos: \C \to \C$ be the complex cosine function.
Then $\cos$ is entire.
Proof 1
By the definition of the complex cosine function, $\cos$ admits a power series expansion about $0$:
- $\ds \cos z = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}$
By Complex Function is Entire iff it has Everywhere Convergent Power Series, to show that $\cos$ is entire it suffices to show that this series is everywhere convergent.
From Radius of Convergence from Limit of Sequence: Complex Case, it is sufficient to show that:
- $\ds \lim_{n \mathop \to \infty} \size {\frac {\paren {-1}^{n + 1} } {\paren {2 n + 2}!} \times \frac {\paren {2 n}!} {\paren {-1}^n} } = 0$
We have:
\(\ds \lim_{n \mathop \to \infty} \size {\frac {\paren {-1}^{n + 1} } {\paren {2 n + 2}!} \times \frac {\paren {2 n}!} {\paren {-1}^n} }\) | \(=\) | \(\ds \size {-1} \lim_{n \mathop \to \infty} \size {\frac {\paren {2 n}!} {\paren {2 n + 2} \paren {2 n + 1} \paren {2 n}!} }\) | Definition of Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \paren {\frac 1 {\paren {2 n + 2} \paren {2 n + 1} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
Hence the result.
$\blacksquare$
Proof 2
Let:
\(\ds \map f z\) | \(=\) | \(\ds \exp z\) | ||||||||||||
\(\ds \map g z\) | \(=\) | \(\ds i z\) | ||||||||||||
\(\ds \map h z\) | \(=\) | \(\ds -i z\) |
for all $z \in \C$.
By Complex Exponential Function is Entire, we have that $f$ is entire.
By Polynomial is Entire, we have that $g$ and $h$ are entire.
Therefore, by Composition of Entire Functions is Entire, we have that both $f \circ g$ and $f \circ h$ are entire.
By Linear Combination of Entire Functions is Entire, we then have that:
- $\dfrac 1 2 \paren {f \circ g + f \circ h}$
is entire.
Then:
\(\ds \frac 1 2 \paren {\map {\paren {f \circ g} } z + \map {\paren {f \circ h} } z}\) | \(=\) | \(\ds \frac 1 2 \paren {\map \exp {i z} + \map \exp {-i z} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cos z\) | Euler's Cosine Identity |
Therefore $\cos$ is an entire function.
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): entire function
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): entire function (integral function)