Difference between revisions of "Incomplete beta function"

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The incomplete beta function is defined by
 
The incomplete beta function is defined by
$$B_x(a,b)=\displaystyle\int_0^x t^{a-1}(1-t)^{b-1} dt.$$
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$$B_x(a,b)=\displaystyle\int_0^x t^{a-1}(1-t)^{b-1} \mathrm{d}t.$$
  
 
=Properties=
 
=Properties=
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[[Relationship between incomplete beta and hypergeometric 2F1]]<br />
<strong>Theorem:</strong> The following formula holds:
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$$B_x(a,b)=\dfrac{x^a}{a} {}_2F_1(a,1-b;a+1;x),$$
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=References=
where $B_x$ denotes the [[incomplete beta function]] and ${}_2F_1$ denotes the [[hypergeometric pFq]].
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[http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_263.htm]
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<strong>Proof:</strong>  █
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[[Category:SpecialFunction]]
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Latest revision as of 01:56, 23 December 2016

The incomplete beta function is defined by $$B_x(a,b)=\displaystyle\int_0^x t^{a-1}(1-t)^{b-1} \mathrm{d}t.$$

Properties

Relationship between incomplete beta and hypergeometric 2F1

References

[1]