Formula:KLS:01.14:09

<< Formula:KLS:01.14:08

formula in Some q-analogues of special functions

Formula:KLS:01.14:10 >>

${\displaystyle{\displaystyle{\displaystyle\mathrm{e}_{q}\!\left(\mathrm{i}z\right)=\mathrm{cos}_{q}\!\left(z\right)+\mathrm{i}\mathrm{sin}_{q}\!\left(z\right)\quad\textrm{and}\quad\mathrm{E}_{q}\!\left(\mathrm{i}z\right)=\mathrm{Cos}_{q}\!\left(z\right)+\mathrm{i}\>\mathrm{Sin}_{q}\!\left(z\right)}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\mathrm{e}_{q}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-analogue of the exponential function ${\displaystyle{\displaystyle{\displaystyle\mathrm{e}_{q}}}}$ used in KLS : http://drmf.wmflabs.org/wiki/Definition:qexpKLS
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}_{q}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-analogue of the cosine function ${\displaystyle{\displaystyle{\displaystyle\cos_{q}}}}$ used in KLS : http://drmf.wmflabs.org/wiki/Definition:qcosKLS
${\displaystyle{\displaystyle{\displaystyle\mathrm{E}_{q}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-analogue of the exponential function ${\displaystyle{\displaystyle{\displaystyle\mathrm{E}_{q}}}}$ used in KLS : http://drmf.wmflabs.org/wiki/Definition:qExpKLS
${\displaystyle{\displaystyle{\displaystyle\mathrm{Cos}_{q}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-analogue of the cosine function ${\displaystyle{\displaystyle{\displaystyle\mathrm{Cos}_{q}}}}$ used in KLS : http://drmf.wmflabs.org/wiki/Definition:qCosKLS
${\displaystyle{\displaystyle{\displaystyle\mathrm{Sin}_{q}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-analogue of the sine function ${\displaystyle{\displaystyle{\displaystyle\mathrm{Sin}_{q}}}}$ used in KLS : http://drmf.wmflabs.org/wiki/Definition:qSinKLS

Bibliography

Equation in Section 1.14 of KLS.

URL links

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<< Formula:KLS:01.14:08

formula in Some q-analogues of special functions

Formula:KLS:01.14:10 >>