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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
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