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→Geometry: 1848 ref via Biodiversity Heritage |
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<math display="block">\exp(j\theta) = \cosh(\theta) + j\sinh(\theta).</math>
This formula can be derived from a [[power series]] expansion using the fact that [[hyperbolic cosine|cosh]] has only even powers while that for [[hyperbolic sine|sinh]] has odd powers.<ref>James Cockle (1848) [https://www.biodiversitylibrary.org/item/20157#page/452/mode/1up On a New Imaginary in Algebra], ''Philosophical Magazine'' 33:438</ref> For all real values of the [[hyperbolic angle]] {{mvar|θ}} the split-complex number {{math|1=''λ'' = exp(''jθ'')}} has norm 1 and lies on the right branch of the unit hyperbola. Numbers such as {{mvar|λ}} have been called [[versor#Hyperbolic versor|hyperbolic versors]].
Since {{mvar|λ}} has modulus 1, multiplying any split-complex number {{mvar|z}} by {{mvar|λ}} preserves the modulus of {{mvar|z}} and represents a ''hyperbolic rotation'' (also called a [[Lorentz boost]] or a [[squeeze mapping]]). Multiplying by {{mvar|λ}} preserves the geometric structure, taking hyperbolas to themselves and the null cone to itself.
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