Tangent in terms of Hyperbolic Tangent

Theorem

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

Then:

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

where:

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


Proof

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

$\blacksquare$


Also see


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.82$: 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: $5$
  • 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
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.