chapter 01 the gamma and beta functions (01)

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 CHAPTER 01 THE GAMMA AND BETA FUNCTIONS 1.1 INTRODUCTION TO GAMMA FUNCTION The gamma function was first defined by the great Swiss Mathematician L. Euler (1707-1783) in 1729. An effor t was made to develop a general expression having value equal to And applicable to not only for large integral values but also for any real value of n. This definition was in the form of an infinite product. A number of equivalent forms are available to define the gamma function but it is more con venient to u se its definite integral form. This form of gamma function is widely used and given by  The above definite integral is absolutely convergent for positive values of n. The above notation  and the name ‘gamma function’ were first introduce d by A. L egendre (17 52-1833) in 1814. Eq.(1.1) is also referred as Euler’s integral of second kind. The above in tegral can be transformed into the form     by the substitution  . It is evident by direct integration by parts of the following integral that           

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