Hyperbolic Tangent in terms of Tangent

Theorem

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

Then:

$i \tanh z = \map \tan {i z}$

where:

$\tan$ denotes the tangent function
$\tanh$ denotes the hyperbolic tangent
$i$ is the imaginary unit: $i^2 = -1$.


Proof

\(\ds \map \tan {i z}\) \(=\) \(\ds \frac {\map \sin {i z} } {\map \cos {i z} }\) Definition of Complex Tangent Function
\(\ds \) \(=\) \(\ds \frac {i \sinh z} {\map \cos {i z} }\) Hyperbolic Sine in terms of Sine
\(\ds \) \(=\) \(\ds \frac {i \sinh z} {\cosh z}\) Hyperbolic Cosine in terms of Cosine
\(\ds \) \(=\) \(\ds i \tanh z\) Definition of Hyperbolic Tangent

$\blacksquare$


Also see


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.76$: Relationship between Hyperbolic and Trigonometric Functions
  • 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
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