Q-Sin

The function $\mathrm{Sin}_q$ is defined by $$\mathrm{Sin}_q(z)=\dfrac{E_q(iz)-E_q(-iz)}{2i},$$ where $E_q$ denotes the $q$-exponential $E_q$.

Properties

q-Euler formula for E sub q
q-derivative of q-Sine

Theorem: The general solution of the $q$-difference equation $D_q^2 y(x) + k^2 y(x) = 0$ is $y(x)=c_1 \mathrm{Cos}_q(kz) + c_2 \mathrm{Sin}_q(kz).$

Proof: █

External links

[1]

References

• 2012: Thomas Ernst: A Comprehensive Treatment of q-Calculus ... (previous) ... (next): (6.168)