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Beta gamma functions

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Engineering Maths

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Beta gamma functions

  1. 1. (1) Prepared by Mr.Zalak Patel Lecturer, Mathematics BETA AND GAMMA FUNCTIONS Introduction to topic : Weightage for university exam: 06Marks No. of lectures required to teach: 02hrs Definition of Gamma function : The gamma function is denoted and defined by the integral )0(1 0 mdxxem mx Properties of Gamma function : (1) mmm )1( (2) !)1( mm when m is a positive integer. (3) aaamamam ).........2)(1()( ,when n is a positive integer. (4) m = 2 )0(12 0 2 mdxxe mx (5) )0(1 0 mdxxe t m mtx m (6) 2 1 (7) . 20 2 dxe x Definition of Beta function : The beta function is denoted and defined by the integral )0,0(,)1(),( 1 0 11 nmdxxxnmB nm ),( nmB is also denoted by ),( nm . Properties of Beta function : (1) ),( nmB = ),( mnB (2) ),( nmB = 2 dnm 2 0 1212 cossin (3) ),( nmB = dx x x nm m 0 1 )1( (4) ),( nmB = dx x xx nm nm1 0 11 )1(
  2. 2. (2) Prepared by Mr.Zalak Patel Lecturer, Mathematics Relation between Beta and Gamma functions: Relation between Beta and Gamma functions is nm nm nmB . ),( Using above relation we can derive following results dqp 2 0 cossin = 2 1 2 2 2 2 1 . 2 1 2 1 , 2 1 qp qp qp B 2 1 Euler s formula: n nn sin 1. Legendre s formula (or Duplication formula): 12 2 )2( 2 1 . n n nn
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