Difference between revisions of "Delta cpq"
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$$c_{pq}(t,s) = \dfrac{e_{p+iq}(t,s)+e_{p-iq}(t,s)}{2},$$ | $$c_{pq}(t,s) = \dfrac{e_{p+iq}(t,s)+e_{p-iq}(t,s)}{2},$$ | ||
where $e_{p+iq}$ denotes the [[delta exponential]]. | where $e_{p+iq}$ denotes the [[delta exponential]]. | ||
| + | |||
| + | =Properties= | ||
| + | {{:Pythagorean identity for alternate delta trigonometric functions}} | ||
| + | {{:Derivative of alternative delta cosine}} | ||
| + | {{:Derivative of alternative delta sine}} | ||
=See Also= | =See Also= | ||
[[Delta spq]]<br /> | [[Delta spq]]<br /> | ||
Revision as of 23:00, 2 June 2016
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) cosine function is defined by $$c_{pq}(t,s) = \dfrac{e_{p+iq}(t,s)+e_{p-iq}(t,s)}{2},$$ where $e_{p+iq}$ denotes the delta exponential.
Contents
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.
Proof
References
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.
Proof
References
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.
