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Prove two Integrals formulae in two dimension

Evaluating Complex Line IntegralsA closed form for $int_0^1_2F_1left(-frac14,frac54;,1;,fracx2right)^2dx$How can I prove that these integrals do not converge?Integral Representation of Terminating Hypergeometric functionEvaluating a certain integral which generalizes the $_3F_2$ hypergeometric functionHow to prove that $_2F_1(1,-k;1-k;-z) = 1 + fraczk1-k~_2F_1(1,1-k;2-k,-z).$Help needed finding a closed form or approximation for an integral(hypergeometric function)How to prove these two integrals are equal?Closed form expression for a Gamma-like integral involving a powered Appell hypergeometric functionHow to prove those Integrals formulae

0

$begingroup$

I come across the following two integral formulae

1. The first integral formula is
   beginequation
   int_Bbb C d^2z |z|^2a|z-x|^2c|z-1|^2b
   =
   fracS(a)S(c)S(a+c)|I_0x|^2+fracS(b)S(a+b+c)S(a+c)|I_1infty|^2
   endequation
   with $S(a)=sinpi a$ and
   beginequation
   I_0x=x^1+a+cfracGamma(a+1)Gamma(c+1)Gamma(a+c+2) ~_2F_1(-b,a+1,a+c+2,x)
   endequation
   beginequation
   I_1infty=fracGamma(-a-b-c-1)Gamma(b+1)Gamma(-a-c)
   ~ _2F_1(-a-b-c-1,-c,-a-c,x)
   endequation
   where $_2F_1$ is the hypergeometric function.
   Here the integral is performed on complex plane $Bbb C$ with coordinate $z=x+iy$.


2. The second similar integral is
   beginequation
   int_Bbb R^d d^d zfrac1^2b=fracpi^d/2fracGamma(d/2-a)Gamma(d/2-b)Gamma(a+b-d/2)Gamma(a)Gamma(b)Gamma(d-a-b)
   endequation
   Here $z$ is coordinate in $d$ dimensional Euclidean space $Bbb R^d$.

It seems that those integrals are similar with Feymann integrals in the calculation of Feymann diagram. When $d=2$, those integrals may occured in the calculation of 2D CFT. I tried to prove those formulae, but I failed.
So my question is how to prove the above formulas.

integration mathematical-physics conformal-field-theory

share|cite|improve this question

edited Mar 22 at 14:38

Andrews

1,2812423

asked Mar 1 at 2:43

phys_studentphys_student

112

$endgroup$

migrated from physics.stackexchange.com Mar 1 at 11:08

This question came from our site for active researchers, academics and students of physics.

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  I voted to migrate this to Math.SE.
  $endgroup$
  – AccidentalFourierTransform
  Mar 1 at 2:47
  

  
  
  
  

add a comment | 

0

$begingroup$

I come across the following two integral formulae

1. The first integral formula is
   beginequation
   int_Bbb C d^2z |z|^2a|z-x|^2c|z-1|^2b
   =
   fracS(a)S(c)S(a+c)|I_0x|^2+fracS(b)S(a+b+c)S(a+c)|I_1infty|^2
   endequation
   with $S(a)=sinpi a$ and
   beginequation
   I_0x=x^1+a+cfracGamma(a+1)Gamma(c+1)Gamma(a+c+2) ~_2F_1(-b,a+1,a+c+2,x)
   endequation
   beginequation
   I_1infty=fracGamma(-a-b-c-1)Gamma(b+1)Gamma(-a-c)
   ~ _2F_1(-a-b-c-1,-c,-a-c,x)
   endequation
   where $_2F_1$ is the hypergeometric function.
   Here the integral is performed on complex plane $Bbb C$ with coordinate $z=x+iy$.


2. The second similar integral is
   beginequation
   int_Bbb R^d d^d zfrac1^2b=fracpi^d/2fracGamma(d/2-a)Gamma(d/2-b)Gamma(a+b-d/2)Gamma(a)Gamma(b)Gamma(d-a-b)
   endequation
   Here $z$ is coordinate in $d$ dimensional Euclidean space $Bbb R^d$.

It seems that those integrals are similar with Feymann integrals in the calculation of Feymann diagram. When $d=2$, those integrals may occured in the calculation of 2D CFT. I tried to prove those formulae, but I failed.
So my question is how to prove the above formulas.

integration mathematical-physics conformal-field-theory

share|cite|improve this question

edited Mar 22 at 14:38

Andrews

1,2812423

asked Mar 1 at 2:43

phys_studentphys_student

112

$endgroup$

migrated from physics.stackexchange.com Mar 1 at 11:08

This question came from our site for active researchers, academics and students of physics.

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  I voted to migrate this to Math.SE.
  $endgroup$
  – AccidentalFourierTransform
  Mar 1 at 2:47
  

  
  
  
  

add a comment | 

$begingroup$

I come across the following two integral formulae

1. The first integral formula is
   beginequation
   int_Bbb C d^2z |z|^2a|z-x|^2c|z-1|^2b
   =
   fracS(a)S(c)S(a+c)|I_0x|^2+fracS(b)S(a+b+c)S(a+c)|I_1infty|^2
   endequation
   with $S(a)=sinpi a$ and
   beginequation
   I_0x=x^1+a+cfracGamma(a+1)Gamma(c+1)Gamma(a+c+2) ~_2F_1(-b,a+1,a+c+2,x)
   endequation
   beginequation
   I_1infty=fracGamma(-a-b-c-1)Gamma(b+1)Gamma(-a-c)
   ~ _2F_1(-a-b-c-1,-c,-a-c,x)
   endequation
   where $_2F_1$ is the hypergeometric function.
   Here the integral is performed on complex plane $Bbb C$ with coordinate $z=x+iy$.


2. The second similar integral is
   beginequation
   int_Bbb R^d d^d zfrac1^2b=fracpi^d/2fracGamma(d/2-a)Gamma(d/2-b)Gamma(a+b-d/2)Gamma(a)Gamma(b)Gamma(d-a-b)
   endequation
   Here $z$ is coordinate in $d$ dimensional Euclidean space $Bbb R^d$.

It seems that those integrals are similar with Feymann integrals in the calculation of Feymann diagram. When $d=2$, those integrals may occured in the calculation of 2D CFT. I tried to prove those formulae, but I failed.
So my question is how to prove the above formulas.

integration mathematical-physics conformal-field-theory

share|cite|improve this question

edited Mar 22 at 14:38

Andrews

1,2812423

asked Mar 1 at 2:43

phys_studentphys_student

112

$endgroup$

share|cite|improve this question

edited Mar 22 at 14:38

Andrews

1,2812423

asked Mar 1 at 2:43

phys_studentphys_student

112

share|cite|improve this question

edited Mar 22 at 14:38

Andrews

1,2812423

asked Mar 1 at 2:43

phys_studentphys_student

112

edited Mar 22 at 14:38

Andrews

1,2812423

edited Mar 22 at 14:38

Andrews

1,2812423

asked Mar 1 at 2:43

phys_studentphys_student

112

asked Mar 1 at 2:43

phys_studentphys_student

112