Complex Cosine Function is Unbounded/Proof 1

Theorem

The complex cosine function is unbounded.

Proof

Let $K \in \R_{>0}$ be an arbitrary real number.

Let $p = \ln {2 K}$.

Let $z = i p$, where $i$ denotes the imaginary unit.

Then:

\(\ds \cos z\)

\(=\)

\(\ds \dfrac {\map \exp {i \paren {i p} } + \map \exp {-i \paren {i p} } } 2\)

Euler's Cosine Identity

\(\ds \)

\(=\)

\(\ds \dfrac {\exp p + \map \exp {-p} } 2\)

simplifying: $i^2 = -1$

\(\ds \)

\(>\)

\(\ds \dfrac {\exp p} 2\)

as $\map \exp {-p} > 0$

\(\ds \)

\(=\)

\(\ds \dfrac {\map \exp {\ln {2 K} } } 2\)

\(\ds \)

\(=\)

\(\ds K\)

\(\ds \leadsto \ \ \)

\(\ds \cmod {\map \cos {i p} }\)

\(>\)

\(\ds K\)

Thus for any $K$ we can find $z \in \C$ such that $\cmod {\cos z} > K$.

Hence the result by definition of unbounded.

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$