Analytic continuation of the incomplete beta function











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Is there a rigorous proof for the analytic continuation of the incomplete beta function $B(x;a,b)$ for all values of $a$ and $b$? The incomplete beta function normally restricts the values of $a,b$ as $a>0$, $b>0$. So I would like to extend these values to negative but I cannot find a good and comprehensive reference for this.



There is a given expression in wikipedia for the analytic continuation of beta function but I cannot see one for the incomplete one. By the way, the integral that I am working on is:



$$rho=frac{b_0}{1-q}int_0^{1-(b_0/x)^{1-q}}v^{-1/2}(1-v)^{frac{1}{q-1}-1}dv=frac{b_0}{1-q}Beta(1-(b_0/x)^{1-q};frac{1}{2},frac{1}{q-1})$$



And I am working on $-infty<q<1$. Thanks for the help.






beta-function analytic-continuation







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    up vote
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    favorite












    Is there a rigorous proof for the analytic continuation of the incomplete beta function $B(x;a,b)$ for all values of $a$ and $b$? The incomplete beta function normally restricts the values of $a,b$ as $a>0$, $b>0$. So I would like to extend these values to negative but I cannot find a good and comprehensive reference for this.



    There is a given expression in wikipedia for the analytic continuation of beta function but I cannot see one for the incomplete one. By the way, the integral that I am working on is:



    $$rho=frac{b_0}{1-q}int_0^{1-(b_0/x)^{1-q}}v^{-1/2}(1-v)^{frac{1}{q-1}-1}dv=frac{b_0}{1-q}Beta(1-(b_0/x)^{1-q};frac{1}{2},frac{1}{q-1})$$



    And I am working on $-infty<q<1$. Thanks for the help.






    beta-function analytic-continuation







    share|cite|improve this question


























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Is there a rigorous proof for the analytic continuation of the incomplete beta function $B(x;a,b)$ for all values of $a$ and $b$? The incomplete beta function normally restricts the values of $a,b$ as $a>0$, $b>0$. So I would like to extend these values to negative but I cannot find a good and comprehensive reference for this.



      There is a given expression in wikipedia for the analytic continuation of beta function but I cannot see one for the incomplete one. By the way, the integral that I am working on is:



      $$rho=frac{b_0}{1-q}int_0^{1-(b_0/x)^{1-q}}v^{-1/2}(1-v)^{frac{1}{q-1}-1}dv=frac{b_0}{1-q}Beta(1-(b_0/x)^{1-q};frac{1}{2},frac{1}{q-1})$$



      And I am working on $-infty<q<1$. Thanks for the help.






      beta-function analytic-continuation







      share|cite|improve this question















      Is there a rigorous proof for the analytic continuation of the incomplete beta function $B(x;a,b)$ for all values of $a$ and $b$? The incomplete beta function normally restricts the values of $a,b$ as $a>0$, $b>0$. So I would like to extend these values to negative but I cannot find a good and comprehensive reference for this.



      There is a given expression in wikipedia for the analytic continuation of beta function but I cannot see one for the incomplete one. By the way, the integral that I am working on is:



      $$rho=frac{b_0}{1-q}int_0^{1-(b_0/x)^{1-q}}v^{-1/2}(1-v)^{frac{1}{q-1}-1}dv=frac{b_0}{1-q}Beta(1-(b_0/x)^{1-q};frac{1}{2},frac{1}{q-1})$$



      And I am working on $-infty<q<1$. Thanks for the help.






      beta-function analytic-continuation




      beta-function analytic-continuation






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      share|cite|improve this question













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      edited Nov 21 at 14:47









      Klangen

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      asked Aug 21 at 3:40









      user583893

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          There is the general complete description with the regularized Gauss hypergeometric function ${{_2}tilde{F}_{1}}(a,b,c,x)$, see 1 and 2: $$
          B_x(a,b)= Gamma(a),x^a ;{{_2}tilde{F}_{1}}(a,1-b,a+1,x), qquad -a notin mathbb{N}
          $$



          When $a le 0$ or $b le 0$, the Gauss hypergeometric function ${{_2}F_{1}}$ function can be used: If $a neq 0$ is no negative integer, the result is (3)
          $$
          B_x(a,b)= frac{x^a}{a},{{_2}F_{1}}(a,1-b,a+1,x), qquad -a notin mathbb{N},
          $$
          else if $b neq 0$ is no negative integer, then (4):
          $$
          B_x(a,b)= B(a,b)- frac{(1-x)^b x^a}{b};{{_2}F_{1}}(1,a+b,b+1,1-x), qquad -b notin mathbb{N}.
          $$






          share|cite|improve this answer





















          • Hi. Thanks for the answer. I don't actually understand this. If the parameter $b$ takes a value of zero or any negative value, I can use the last equation? Is that right?
            – user583893
            Aug 21 at 9:44










          • No, you can use it if $b ne -1, -2, dots$ otherwise $ c= b+1$ becomes zero, and the the Gauss function is undefined. The regularized function $${{_2}tilde{F}_{1}}(a,b,c,x) = frac{1}{Gamma(c)} {{_2}F_{1}}(a,b,c,x)$$ can handle this because together with $Gamma(c)$ you get a finite limit if $c$ approaches a negative integer or zero.
            – gammatester
            Aug 21 at 9:56










          • What i need is what works for negative values of b. I already tried in the mathematica, and it gave me the correct answer.
            – user583893
            Aug 21 at 10:36










          • What i need is an extension of incomplete beta function that works for negative values of the parameter $b$. But i tried feeding up negative values of $b$ to the last equation and it gave me the right answers. Now I am confused since the last equation must only work for positive $b$'s.
            – user583893
            Aug 21 at 10:40










          • No it does work for negative non-integer $b$. If $b$ is a negative integer and $a$ is not you can use the second formula, otherwise use the regularized function. But note the hint given at functions.wolfram.com/GammaBetaErf/Beta3/02/02. Using my own implementation I get for example $B_{0.5}(-1.5, -2.5)= 6.4$ (verified on Wolfram function evaluation).
            – gammatester
            Aug 21 at 10:56











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          up vote
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          accepted










          There is the general complete description with the regularized Gauss hypergeometric function ${{_2}tilde{F}_{1}}(a,b,c,x)$, see 1 and 2: $$
          B_x(a,b)= Gamma(a),x^a ;{{_2}tilde{F}_{1}}(a,1-b,a+1,x), qquad -a notin mathbb{N}
          $$



          When $a le 0$ or $b le 0$, the Gauss hypergeometric function ${{_2}F_{1}}$ function can be used: If $a neq 0$ is no negative integer, the result is (3)
          $$
          B_x(a,b)= frac{x^a}{a},{{_2}F_{1}}(a,1-b,a+1,x), qquad -a notin mathbb{N},
          $$
          else if $b neq 0$ is no negative integer, then (4):
          $$
          B_x(a,b)= B(a,b)- frac{(1-x)^b x^a}{b};{{_2}F_{1}}(1,a+b,b+1,1-x), qquad -b notin mathbb{N}.
          $$






          share|cite|improve this answer





















          • Hi. Thanks for the answer. I don't actually understand this. If the parameter $b$ takes a value of zero or any negative value, I can use the last equation? Is that right?
            – user583893
            Aug 21 at 9:44










          • No, you can use it if $b ne -1, -2, dots$ otherwise $ c= b+1$ becomes zero, and the the Gauss function is undefined. The regularized function $${{_2}tilde{F}_{1}}(a,b,c,x) = frac{1}{Gamma(c)} {{_2}F_{1}}(a,b,c,x)$$ can handle this because together with $Gamma(c)$ you get a finite limit if $c$ approaches a negative integer or zero.
            – gammatester
            Aug 21 at 9:56










          • What i need is what works for negative values of b. I already tried in the mathematica, and it gave me the correct answer.
            – user583893
            Aug 21 at 10:36










          • What i need is an extension of incomplete beta function that works for negative values of the parameter $b$. But i tried feeding up negative values of $b$ to the last equation and it gave me the right answers. Now I am confused since the last equation must only work for positive $b$'s.
            – user583893
            Aug 21 at 10:40










          • No it does work for negative non-integer $b$. If $b$ is a negative integer and $a$ is not you can use the second formula, otherwise use the regularized function. But note the hint given at functions.wolfram.com/GammaBetaErf/Beta3/02/02. Using my own implementation I get for example $B_{0.5}(-1.5, -2.5)= 6.4$ (verified on Wolfram function evaluation).
            – gammatester
            Aug 21 at 10:56















          up vote
          1
          down vote



          accepted










          There is the general complete description with the regularized Gauss hypergeometric function ${{_2}tilde{F}_{1}}(a,b,c,x)$, see 1 and 2: $$
          B_x(a,b)= Gamma(a),x^a ;{{_2}tilde{F}_{1}}(a,1-b,a+1,x), qquad -a notin mathbb{N}
          $$



          When $a le 0$ or $b le 0$, the Gauss hypergeometric function ${{_2}F_{1}}$ function can be used: If $a neq 0$ is no negative integer, the result is (3)
          $$
          B_x(a,b)= frac{x^a}{a},{{_2}F_{1}}(a,1-b,a+1,x), qquad -a notin mathbb{N},
          $$
          else if $b neq 0$ is no negative integer, then (4):
          $$
          B_x(a,b)= B(a,b)- frac{(1-x)^b x^a}{b};{{_2}F_{1}}(1,a+b,b+1,1-x), qquad -b notin mathbb{N}.
          $$






          share|cite|improve this answer





















          • Hi. Thanks for the answer. I don't actually understand this. If the parameter $b$ takes a value of zero or any negative value, I can use the last equation? Is that right?
            – user583893
            Aug 21 at 9:44










          • No, you can use it if $b ne -1, -2, dots$ otherwise $ c= b+1$ becomes zero, and the the Gauss function is undefined. The regularized function $${{_2}tilde{F}_{1}}(a,b,c,x) = frac{1}{Gamma(c)} {{_2}F_{1}}(a,b,c,x)$$ can handle this because together with $Gamma(c)$ you get a finite limit if $c$ approaches a negative integer or zero.
            – gammatester
            Aug 21 at 9:56










          • What i need is what works for negative values of b. I already tried in the mathematica, and it gave me the correct answer.
            – user583893
            Aug 21 at 10:36










          • What i need is an extension of incomplete beta function that works for negative values of the parameter $b$. But i tried feeding up negative values of $b$ to the last equation and it gave me the right answers. Now I am confused since the last equation must only work for positive $b$'s.
            – user583893
            Aug 21 at 10:40










          • No it does work for negative non-integer $b$. If $b$ is a negative integer and $a$ is not you can use the second formula, otherwise use the regularized function. But note the hint given at functions.wolfram.com/GammaBetaErf/Beta3/02/02. Using my own implementation I get for example $B_{0.5}(-1.5, -2.5)= 6.4$ (verified on Wolfram function evaluation).
            – gammatester
            Aug 21 at 10:56













          up vote
          1
          down vote



          accepted







          up vote
          1
          down vote



          accepted






          There is the general complete description with the regularized Gauss hypergeometric function ${{_2}tilde{F}_{1}}(a,b,c,x)$, see 1 and 2: $$
          B_x(a,b)= Gamma(a),x^a ;{{_2}tilde{F}_{1}}(a,1-b,a+1,x), qquad -a notin mathbb{N}
          $$



          When $a le 0$ or $b le 0$, the Gauss hypergeometric function ${{_2}F_{1}}$ function can be used: If $a neq 0$ is no negative integer, the result is (3)
          $$
          B_x(a,b)= frac{x^a}{a},{{_2}F_{1}}(a,1-b,a+1,x), qquad -a notin mathbb{N},
          $$
          else if $b neq 0$ is no negative integer, then (4):
          $$
          B_x(a,b)= B(a,b)- frac{(1-x)^b x^a}{b};{{_2}F_{1}}(1,a+b,b+1,1-x), qquad -b notin mathbb{N}.
          $$






          share|cite|improve this answer












          There is the general complete description with the regularized Gauss hypergeometric function ${{_2}tilde{F}_{1}}(a,b,c,x)$, see 1 and 2: $$
          B_x(a,b)= Gamma(a),x^a ;{{_2}tilde{F}_{1}}(a,1-b,a+1,x), qquad -a notin mathbb{N}
          $$



          When $a le 0$ or $b le 0$, the Gauss hypergeometric function ${{_2}F_{1}}$ function can be used: If $a neq 0$ is no negative integer, the result is (3)
          $$
          B_x(a,b)= frac{x^a}{a},{{_2}F_{1}}(a,1-b,a+1,x), qquad -a notin mathbb{N},
          $$
          else if $b neq 0$ is no negative integer, then (4):
          $$
          B_x(a,b)= B(a,b)- frac{(1-x)^b x^a}{b};{{_2}F_{1}}(1,a+b,b+1,1-x), qquad -b notin mathbb{N}.
          $$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Aug 21 at 8:16









          gammatester

          16.6k21631




          16.6k21631












          • Hi. Thanks for the answer. I don't actually understand this. If the parameter $b$ takes a value of zero or any negative value, I can use the last equation? Is that right?
            – user583893
            Aug 21 at 9:44










          • No, you can use it if $b ne -1, -2, dots$ otherwise $ c= b+1$ becomes zero, and the the Gauss function is undefined. The regularized function $${{_2}tilde{F}_{1}}(a,b,c,x) = frac{1}{Gamma(c)} {{_2}F_{1}}(a,b,c,x)$$ can handle this because together with $Gamma(c)$ you get a finite limit if $c$ approaches a negative integer or zero.
            – gammatester
            Aug 21 at 9:56










          • What i need is what works for negative values of b. I already tried in the mathematica, and it gave me the correct answer.
            – user583893
            Aug 21 at 10:36










          • What i need is an extension of incomplete beta function that works for negative values of the parameter $b$. But i tried feeding up negative values of $b$ to the last equation and it gave me the right answers. Now I am confused since the last equation must only work for positive $b$'s.
            – user583893
            Aug 21 at 10:40










          • No it does work for negative non-integer $b$. If $b$ is a negative integer and $a$ is not you can use the second formula, otherwise use the regularized function. But note the hint given at functions.wolfram.com/GammaBetaErf/Beta3/02/02. Using my own implementation I get for example $B_{0.5}(-1.5, -2.5)= 6.4$ (verified on Wolfram function evaluation).
            – gammatester
            Aug 21 at 10:56


















          • Hi. Thanks for the answer. I don't actually understand this. If the parameter $b$ takes a value of zero or any negative value, I can use the last equation? Is that right?
            – user583893
            Aug 21 at 9:44










          • No, you can use it if $b ne -1, -2, dots$ otherwise $ c= b+1$ becomes zero, and the the Gauss function is undefined. The regularized function $${{_2}tilde{F}_{1}}(a,b,c,x) = frac{1}{Gamma(c)} {{_2}F_{1}}(a,b,c,x)$$ can handle this because together with $Gamma(c)$ you get a finite limit if $c$ approaches a negative integer or zero.
            – gammatester
            Aug 21 at 9:56










          • What i need is what works for negative values of b. I already tried in the mathematica, and it gave me the correct answer.
            – user583893
            Aug 21 at 10:36










          • What i need is an extension of incomplete beta function that works for negative values of the parameter $b$. But i tried feeding up negative values of $b$ to the last equation and it gave me the right answers. Now I am confused since the last equation must only work for positive $b$'s.
            – user583893
            Aug 21 at 10:40










          • No it does work for negative non-integer $b$. If $b$ is a negative integer and $a$ is not you can use the second formula, otherwise use the regularized function. But note the hint given at functions.wolfram.com/GammaBetaErf/Beta3/02/02. Using my own implementation I get for example $B_{0.5}(-1.5, -2.5)= 6.4$ (verified on Wolfram function evaluation).
            – gammatester
            Aug 21 at 10:56
















          Hi. Thanks for the answer. I don't actually understand this. If the parameter $b$ takes a value of zero or any negative value, I can use the last equation? Is that right?
          – user583893
          Aug 21 at 9:44




          Hi. Thanks for the answer. I don't actually understand this. If the parameter $b$ takes a value of zero or any negative value, I can use the last equation? Is that right?
          – user583893
          Aug 21 at 9:44












          No, you can use it if $b ne -1, -2, dots$ otherwise $ c= b+1$ becomes zero, and the the Gauss function is undefined. The regularized function $${{_2}tilde{F}_{1}}(a,b,c,x) = frac{1}{Gamma(c)} {{_2}F_{1}}(a,b,c,x)$$ can handle this because together with $Gamma(c)$ you get a finite limit if $c$ approaches a negative integer or zero.
          – gammatester
          Aug 21 at 9:56




          No, you can use it if $b ne -1, -2, dots$ otherwise $ c= b+1$ becomes zero, and the the Gauss function is undefined. The regularized function $${{_2}tilde{F}_{1}}(a,b,c,x) = frac{1}{Gamma(c)} {{_2}F_{1}}(a,b,c,x)$$ can handle this because together with $Gamma(c)$ you get a finite limit if $c$ approaches a negative integer or zero.
          – gammatester
          Aug 21 at 9:56












          What i need is what works for negative values of b. I already tried in the mathematica, and it gave me the correct answer.
          – user583893
          Aug 21 at 10:36




          What i need is what works for negative values of b. I already tried in the mathematica, and it gave me the correct answer.
          – user583893
          Aug 21 at 10:36












          What i need is an extension of incomplete beta function that works for negative values of the parameter $b$. But i tried feeding up negative values of $b$ to the last equation and it gave me the right answers. Now I am confused since the last equation must only work for positive $b$'s.
          – user583893
          Aug 21 at 10:40




          What i need is an extension of incomplete beta function that works for negative values of the parameter $b$. But i tried feeding up negative values of $b$ to the last equation and it gave me the right answers. Now I am confused since the last equation must only work for positive $b$'s.
          – user583893
          Aug 21 at 10:40












          No it does work for negative non-integer $b$. If $b$ is a negative integer and $a$ is not you can use the second formula, otherwise use the regularized function. But note the hint given at functions.wolfram.com/GammaBetaErf/Beta3/02/02. Using my own implementation I get for example $B_{0.5}(-1.5, -2.5)= 6.4$ (verified on Wolfram function evaluation).
          – gammatester
          Aug 21 at 10:56




          No it does work for negative non-integer $b$. If $b$ is a negative integer and $a$ is not you can use the second formula, otherwise use the regularized function. But note the hint given at functions.wolfram.com/GammaBetaErf/Beta3/02/02. Using my own implementation I get for example $B_{0.5}(-1.5, -2.5)= 6.4$ (verified on Wolfram function evaluation).
          – gammatester
          Aug 21 at 10:56


















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          Mont Emei

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          Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog...
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