Complex Cosine Function is Unbounded/Proof 1

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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:

\(\ds \cos z\) \(=\) \(\ds \dfrac {\exp \paren {i \paren {i p} } + \exp \paren {-i \paren {i p} } } 2\) Cosine Exponential Formulation
\(\ds \) \(=\) \(\ds \dfrac {\exp p + \exp \paren {-p} } 2\) simplifying
\(\ds \) \(>\) \(\ds \dfrac {\exp p} 2\) as $\exp \paren {-p} > 0$
\(\ds \) \(=\) \(\ds \dfrac {\exp \paren {\ln {2 K} } } 2\)
\(\ds \) \(=\) \(\ds K\)
\(\ds \leadsto \ \ \) \(\ds \cmod {\cos \paren {i p} }\) \(>\) \(\ds K\)

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

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