Cosine in terms of Hyperbolic Cosine

Theorem

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

Then:

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

where:

$\cos$ denotes the complex cosine
$\cosh$ denotes the hyperbolic cosine
$i$ is the imaginary unit: $i^2 = -1$.


Proof

\(\ds \map \cosh {i z}\) \(=\) \(\ds \frac {e^{i z} + e^{-i z} } 2\) Definition of Hyperbolic Cosine
\(\ds \) \(=\) \(\ds \cos z\) Euler's Cosine Identity

$\blacksquare$


Also see


Sources

  • 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: $8.81$: 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: $5$
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): hyperbolic functions
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): hyperbolic functions
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