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In general practice, the functions are classified in two major classes, 232, 102 ENGINEERING MATHEMATHH, A function, consisting of, powers of, independent, variables is called sufficient to understand and evaluate various definite integrals in totakt, as algebraic., 9.1, Chapter Objectives, i.e. algebraic and transcendental Functions. But these functions are, The object of this chapter is to learn the concept of new class of, functions called special functions. Gamma and beta functions, discussed with their applications., Gamma and beta functions are very elementary special functions, and find application in the evaluations of definite integrals and in, describing various physical phenomena., The functions, such as, trigonometric, exponential,, logarithmic etc., are called as, transcendental., 9.2, Gamma Function, An integral is, called as, The gamma function, denoted by I is the extended form of factorial, function i.e. factorial of non integral values. It is defined by an, integral :, improper, improper if, either its one of, the limits is oo or, |(x) = e*pr-1 dt, x > 0. AD, T(x) = et*-' dt, x > 0., .(1), %3D, the integrand, becomes o at, Here x is called as parameter of the integral or argument of the, gamma function., Integral (1) is also named as Euler's Integral of second kind defined, over positive real line. An integral of same kind defined over truncated, (punctured) real line at 'a' is called as incomplete gamma function and, is expressed as, any of the limits, of integration., y(a, x) = " et*-1 dt, x > 0., .(2), %3D, Recurrence Relation :, Rewriting (1) as, I(x + 1) = e* dt., Ja, Euler, Integrating by parts we write, A relation, between similar, r(x + 1) = (-e t"); + x " ep-1 dt, functions for, .(3), finding, successive values, is called as, r(x + 1) = x I(x)., Particularly, when x =, 1, the value of T(1) is given by, r(1) = e* dt = (-e+) =1,, reccurence., Using (3), we derive, %3D, Scanned with CamScanner

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Gamma and Beta Funchions;, by, Gumma function The Gamma function, deno ted, r is the extended form f factorial functHon, i.e. factorial of, non integral values. It is, defined by an improper integral:, In, n >o, Here n is called, parameter of the integral, as, ar argument d the gamma function, This, integral is aso named as Eule's Integia of, Second Kind depined oves positive greal line ., Re currence Relation:, Se-, Inti, (n +1) -1, Integrating, by past we write, Inti, (-e-n ", Sen dn, -2, + n, n in, Scanned with CamScanner