Hyperbolic Cosine in terms of Cosine

Theorem

Let $z \in \C$ be a complex number.

Then:

$\cosh z = \map \cos {i z}$

where:

$\cos$ denotes the complex cosine

$\cosh$ denotes the hyperbolic cosine

$i$ is the imaginary unit: $i^2 = -1$.

$\ds \map \cos {i z}$

$=$

$\ds \frac {e^{i \paren {i z} } + e^{-i \paren {i z} } } 2$

$\ds $

$=$

$\ds \frac {e^{-z} + e^z} 2$

$i^2 = -1$

$\ds $

$=$

$\ds \cosh z$

Definition of Hyperbolic Cosine

$\blacksquare$

1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $(4.22)$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.75$: Relationship between Hyperbolic and Trigonometric Functions
1969: J.C. Anderson, D.M. Hum, B.G. Neal and J.H. Whitelaw: Data and Formulae for Engineering Students (2nd ed.) ... (previous) ... (next): $4.$ Mathematics: $4.3$ Trigonometric identities and hyperbolic functions
1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): cosh or ch
1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): hyperbolic function