Difference between revisions of "Diamond integral"
(Created page with "=References= [http://www.emis.de/journals/BMMSS/pdf/acceptedpapers/2013-02-023-R1.pdf The Diamond Integral on Time Scales]") |
|||
| (2 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
| + | The $\Diamond$ integral was introduced to fix the issues with the [[Diamond alpha derivative | $\Diamond_{\alpha}$ integral]]. | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$\int_a^a f(t) \Diamond t=0.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$\int_a^b f(t) \Diamond t= \int_a^c f(t) \Diamond t + \int_c^b f(t) \Diamond t.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$\int_a^b f(t) \Diamond t= -\int_b^a f(t) \Diamond t.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem (Sum Rule):</strong> The following formula holds: | ||
| + | $$\int_a^b (f+g)(t) \Diamond t= \int_a^b f(t) \Diamond t + \int_a^b g(t) \Diamond t.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem (Constant Multiple):</strong> The following formula holds: | ||
| + | $$\int_a^b \alpha f(t) \Diamond t= \alpha \int_a^b f(t) \Diamond t.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem (Mean Value Theorem):</strong> Let $f,g \colon \mathbb{T} \rightarrow \mathbb{R}$ be bounded and $\Diamond$-integrable on $[a,b] \cap \mathbb{T}$, and let $g$ be nonnegative or nonpositive on $[a,b] \cap \mathbb{T}$. Let $m$ and $M$ be the infimum and supremum respectively of $f$. Then there exists a real number $K$ satisfying $m \leq K \leq M$ such that | ||
| + | $$\displaystyle\int_a^b (fg)(t) \Diamond t = K \displaystyle\int_a^b g(t) \Diamond t.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
=References= | =References= | ||
| − | [http:// | + | [http://arxiv.org/pdf/1306.0988.pdf The Diamond Integral on Time Scales] |
Latest revision as of 08:05, 10 March 2015
The $\Diamond$ integral was introduced to fix the issues with the $\Diamond_{\alpha}$ integral.
Properties
Theorem: The following formula holds: $$\int_a^a f(t) \Diamond t=0.$$
Proof: █
Theorem: The following formula holds: $$\int_a^b f(t) \Diamond t= \int_a^c f(t) \Diamond t + \int_c^b f(t) \Diamond t.$$
Proof: █
Theorem: The following formula holds: $$\int_a^b f(t) \Diamond t= -\int_b^a f(t) \Diamond t.$$
Proof: █
Theorem (Sum Rule): The following formula holds: $$\int_a^b (f+g)(t) \Diamond t= \int_a^b f(t) \Diamond t + \int_a^b g(t) \Diamond t.$$
Proof: █
Theorem (Constant Multiple): The following formula holds: $$\int_a^b \alpha f(t) \Diamond t= \alpha \int_a^b f(t) \Diamond t.$$
Proof: █
Theorem (Mean Value Theorem): Let $f,g \colon \mathbb{T} \rightarrow \mathbb{R}$ be bounded and $\Diamond$-integrable on $[a,b] \cap \mathbb{T}$, and let $g$ be nonnegative or nonpositive on $[a,b] \cap \mathbb{T}$. Let $m$ and $M$ be the infimum and supremum respectively of $f$. Then there exists a real number $K$ satisfying $m \leq K \leq M$ such that $$\displaystyle\int_a^b (fg)(t) \Diamond t = K \displaystyle\int_a^b g(t) \Diamond t.$$
Proof: █
