Beta and Gamma Functions
                                                    Dr. Kamlesh Jangid
                                              Department of HEAS (Mathematics)
                                     Rajasthan Technical University, Kota-324010, India
                                                E-mail: kjangid@rtu.ac.in
        Dr. Kamlesh Jangid (RTU Kota)                      Improper Integrals                                                  1 / 20
                                    Outline
  Outline
     1  Improper Integrals
           TheGammaFunction
           TheBetaFunction
           Properties
     Dr. Kamlesh Jangid (RTU Kota)  Improper Integrals                       2 / 20
                             Improper Integrals
  Improper Integrals
          Improper integrals are definite integrals that cover an unbounded
          area. For example, Z ∞ 1 dx
                              1   x2
          Animproperintegralisadefinite integral that has either one or
          both limits infinite or an integrand that approaches infinity at one or
          morepoints in the range of integration.
          Henceimproper integrals are of two type.
     Dr. Kamlesh Jangid (RTU Kota)  Improper Integrals                      3 / 20
                                             Improper Integrals
           1   Type-1: (Infinite limits of integration): In this kind of integrals one
               or both limits of integration are infinity. For example, Z ∞ 1 dx is
                                                                                                 1    x2
               an improper integral.                             Z
                                                                     b 1
               It can be viewed as the limit lim                         2dx.
                                                         b→∞ 1 x
           2   Type-2: (Discontinuous integrand or integrands with vertical
               asymptotes): In this type of improper integrals the endpoints are
               finite, but the integrand function is unbounded at one (or two) of the
                                                    Z 1 1
               endpoints. For example,                    √ dx.
                                                      0      x Z
                                                                     1 1
               It can be viewed as the limit lim                        √ dx.
                                                               +           x
                                                         a→0        a
        Dr. Kamlesh Jangid (RTU Kota)                 Improper Integrals                                            4 / 20