Delta spq

Let $\mathbb{T}$ be a time scale and let $p$ and $q$ be rd-continuous functions that satisfy the relation $2p(t)+\mu(t)(p(t)^2+q(t)^2)=0$. The (alternative) delta sine function is defined by $$s_{pq}(t,s) = \dfrac{e_{p+iq}(t,s)-e_{p-iq}(t,s)}{2i},$$ where $e_{p+iq}$ denotes the delta exponential.

Properties

Theorem

Let $\mathbb{T}$ be a time scale. The following formula holds: $$\mathrm{c}_{pq}^2(t,s;\mathbb{T})+\mathrm{s}_{pq}^2(t,s;\mathbb{T})=1,$$ where $\mathrm{c}_{pq}$ denotes the alternative delta cosine and $\mathrm{s}_{pq}$ denotes the alternative delta sine.

Theorem

Let $\mathbb{T}$ be a time scale. The following formula holds: $$\mathrm{c}_{pq}^{\Delta}(t,s;\mathbb{T})=p(t)\mathrm{c}_{pq}(t,s;\mathbb{T})-q(t)\mathrm{s}_{pq}(t,s;\mathbb{T}),$$ where $\mathrm{c}_{pq}$ denotes the alternative delta cosine and $\mathrm{s}_{pq}$ denotes the alternative delta sine.

Theorem

Let $\mathbb{T}$ be a time scale. The following formula holds: $$\mathrm{s}_{pq}^{\Delta}(t,s;\mathbb{T})=q(t)\mathrm{c}_{pq}(t,s;\mathbb{T})+p(t)\mathrm{s}_{pq}(t,s;\mathbb{T}),$$ where $\mathrm{s}_{pq}$ denotes the alternative delta sine and $\mathrm{c}_{pq}$ denotes the alternative delta cosine.

See Also

Delta cpq