Integral involving the log gamma function












3














I have used the Kummer representation series of loggamma function but does not look promissing to tackle this integral. Any idea to calculate this integral in closed-form ?



$$int_{0}^{1}ln(x)lnGamma(x)dx$$






definite-integrals fourier-series gamma-function







share|cite|improve this question



























    3














    I have used the Kummer representation series of loggamma function but does not look promissing to tackle this integral. Any idea to calculate this integral in closed-form ?



    $$int_{0}^{1}ln(x)lnGamma(x)dx$$






    definite-integrals fourier-series gamma-function







    share|cite|improve this question

























      3












      3








      3


      2





      I have used the Kummer representation series of loggamma function but does not look promissing to tackle this integral. Any idea to calculate this integral in closed-form ?



      $$int_{0}^{1}ln(x)lnGamma(x)dx$$






      definite-integrals fourier-series gamma-function







      share|cite|improve this question













      I have used the Kummer representation series of loggamma function but does not look promissing to tackle this integral. Any idea to calculate this integral in closed-form ?



      $$int_{0}^{1}ln(x)lnGamma(x)dx$$






      definite-integrals fourier-series gamma-function




      definite-integrals fourier-series gamma-function






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Nov 25 at 19:02









      Kays Tomy

      1997




      1997






















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          I am quite skeptical about a possible closed form of this integral.



          For an approximation, I should use the expansion
          $$log (Gamma (x))=-log (x)-gamma x+frac{pi ^2 x^2}{12}+frac{x^3 psi ^{(2)}(1)}{6}+frac{pi ^4
          x^4}{360}+frac{x^5 psi ^{(4)}(1)}{120}+frac{pi ^6
          x^6}{5670}+Oleft(x^7right)$$           and integrate termwise to end with
          $$int_0^1 log(x)log (Gamma (x))=-2+frac{gamma }{4}-frac{pi ^2}{108}-frac{pi ^4}{9000}-frac{pi
          ^6}{277830}-frac{psi ^{(2)}(1)}{96}-frac{psi ^{(4)}(1)}{4320}$$           which is $approx -1.93056$ while numerical integration leads to $approx -1.92922$.



          Expanding $log (Gamma (x))$ to $Oleft(x^{10}right)$ would lead to
          $$int_0^1 log(x)log (Gamma (x))=-2+frac{gamma }{4}-frac{pi ^2}{108}-frac{pi ^4}{9000}-frac{pi
          ^6}{277830}-frac{pi ^8}{6123600}-frac{pi ^{10}}{113201550}-frac{psi
          ^{(2)}(1)}{96}-frac{psi ^{(4)}(1)}{4320}-frac{psi
          ^{(6)}(1)}{322560}-frac{psi ^{(8)}(1)}{36288000}$$           which is $approx -1.92922$.






          share|cite|improve this answer





















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            active

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            0














            I am quite skeptical about a possible closed form of this integral.



            For an approximation, I should use the expansion
            $$log (Gamma (x))=-log (x)-gamma x+frac{pi ^2 x^2}{12}+frac{x^3 psi ^{(2)}(1)}{6}+frac{pi ^4
            x^4}{360}+frac{x^5 psi ^{(4)}(1)}{120}+frac{pi ^6
            x^6}{5670}+Oleft(x^7right)$$             and integrate termwise to end with
            $$int_0^1 log(x)log (Gamma (x))=-2+frac{gamma }{4}-frac{pi ^2}{108}-frac{pi ^4}{9000}-frac{pi
            ^6}{277830}-frac{psi ^{(2)}(1)}{96}-frac{psi ^{(4)}(1)}{4320}$$             which is $approx -1.93056$ while numerical integration leads to $approx -1.92922$.



            Expanding $log (Gamma (x))$ to $Oleft(x^{10}right)$ would lead to
            $$int_0^1 log(x)log (Gamma (x))=-2+frac{gamma }{4}-frac{pi ^2}{108}-frac{pi ^4}{9000}-frac{pi
            ^6}{277830}-frac{pi ^8}{6123600}-frac{pi ^{10}}{113201550}-frac{psi
            ^{(2)}(1)}{96}-frac{psi ^{(4)}(1)}{4320}-frac{psi
            ^{(6)}(1)}{322560}-frac{psi ^{(8)}(1)}{36288000}$$             which is $approx -1.92922$.






            share|cite|improve this answer


























              0














              I am quite skeptical about a possible closed form of this integral.



              For an approximation, I should use the expansion
              $$log (Gamma (x))=-log (x)-gamma x+frac{pi ^2 x^2}{12}+frac{x^3 psi ^{(2)}(1)}{6}+frac{pi ^4
              x^4}{360}+frac{x^5 psi ^{(4)}(1)}{120}+frac{pi ^6
              x^6}{5670}+Oleft(x^7right)$$               and integrate termwise to end with
              $$int_0^1 log(x)log (Gamma (x))=-2+frac{gamma }{4}-frac{pi ^2}{108}-frac{pi ^4}{9000}-frac{pi
              ^6}{277830}-frac{psi ^{(2)}(1)}{96}-frac{psi ^{(4)}(1)}{4320}$$               which is $approx -1.93056$ while numerical integration leads to $approx -1.92922$.



              Expanding $log (Gamma (x))$ to $Oleft(x^{10}right)$ would lead to
              $$int_0^1 log(x)log (Gamma (x))=-2+frac{gamma }{4}-frac{pi ^2}{108}-frac{pi ^4}{9000}-frac{pi
              ^6}{277830}-frac{pi ^8}{6123600}-frac{pi ^{10}}{113201550}-frac{psi
              ^{(2)}(1)}{96}-frac{psi ^{(4)}(1)}{4320}-frac{psi
              ^{(6)}(1)}{322560}-frac{psi ^{(8)}(1)}{36288000}$$               which is $approx -1.92922$.






              share|cite|improve this answer
























                0












                0








                0






                I am quite skeptical about a possible closed form of this integral.



                For an approximation, I should use the expansion
                $$log (Gamma (x))=-log (x)-gamma x+frac{pi ^2 x^2}{12}+frac{x^3 psi ^{(2)}(1)}{6}+frac{pi ^4
                x^4}{360}+frac{x^5 psi ^{(4)}(1)}{120}+frac{pi ^6
                x^6}{5670}+Oleft(x^7right)$$                 and integrate termwise to end with
                $$int_0^1 log(x)log (Gamma (x))=-2+frac{gamma }{4}-frac{pi ^2}{108}-frac{pi ^4}{9000}-frac{pi
                ^6}{277830}-frac{psi ^{(2)}(1)}{96}-frac{psi ^{(4)}(1)}{4320}$$                 which is $approx -1.93056$ while numerical integration leads to $approx -1.92922$.



                Expanding $log (Gamma (x))$ to $Oleft(x^{10}right)$ would lead to
                $$int_0^1 log(x)log (Gamma (x))=-2+frac{gamma }{4}-frac{pi ^2}{108}-frac{pi ^4}{9000}-frac{pi
                ^6}{277830}-frac{pi ^8}{6123600}-frac{pi ^{10}}{113201550}-frac{psi
                ^{(2)}(1)}{96}-frac{psi ^{(4)}(1)}{4320}-frac{psi
                ^{(6)}(1)}{322560}-frac{psi ^{(8)}(1)}{36288000}$$                 which is $approx -1.92922$.






                share|cite|improve this answer












                I am quite skeptical about a possible closed form of this integral.



                For an approximation, I should use the expansion
                $$log (Gamma (x))=-log (x)-gamma x+frac{pi ^2 x^2}{12}+frac{x^3 psi ^{(2)}(1)}{6}+frac{pi ^4
                x^4}{360}+frac{x^5 psi ^{(4)}(1)}{120}+frac{pi ^6
                x^6}{5670}+Oleft(x^7right)$$                 and integrate termwise to end with
                $$int_0^1 log(x)log (Gamma (x))=-2+frac{gamma }{4}-frac{pi ^2}{108}-frac{pi ^4}{9000}-frac{pi
                ^6}{277830}-frac{psi ^{(2)}(1)}{96}-frac{psi ^{(4)}(1)}{4320}$$                 which is $approx -1.93056$ while numerical integration leads to $approx -1.92922$.



                Expanding $log (Gamma (x))$ to $Oleft(x^{10}right)$ would lead to
                $$int_0^1 log(x)log (Gamma (x))=-2+frac{gamma }{4}-frac{pi ^2}{108}-frac{pi ^4}{9000}-frac{pi
                ^6}{277830}-frac{pi ^8}{6123600}-frac{pi ^{10}}{113201550}-frac{psi
                ^{(2)}(1)}{96}-frac{psi ^{(4)}(1)}{4320}-frac{psi
                ^{(6)}(1)}{322560}-frac{psi ^{(8)}(1)}{36288000}$$                 which is $approx -1.92922$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 26 at 6:24









                Claude Leibovici

                118k1157132




                118k1157132






























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