Difference between revisions of "Diamond integral"
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| + | The $\Diamond$ integral was introduced to fix the issues with the [[Diamond alpha derivative | $\Diamond_{\alpha}$ integral]]. | ||
| + | |||
=Properties= | =Properties= | ||
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
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<strong>Theorem (Constant Multiple):</strong> The following formula holds: | <strong>Theorem (Constant Multiple):</strong> The following formula holds: | ||
$$\int_a^b \alpha f(t) \Diamond t= \alpha \int_a^b f(t) \Diamond t.$$ | $$\int_a^b \alpha f(t) \Diamond t= \alpha \int_a^b f(t) \Diamond t.$$ | ||
| + | <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"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
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: █
