Gamma Function Notes

Gamma Function Notes - Gamma Function Notes The gamma...

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Gamma Function Notes The gamma function is widely used in applied mathematics and probability. An discussion of this function provides a review of calculus and an introduction to numerical methods. The gamma function, denoted by () x Γ , is defined for 1 x > and is related to its argument in the following manner. 1 0 xt e d t −− Γ = ( 1 ) There is no analytical expression for arbitrary values of the argument. The function is frequently written in the following manner. 0 1, 0 bt e d t b Γ += > ( 2 ) There is a special case for which one can derive an analytical expression for 1 b Γ + , namely when b is a non-negative integer. Let us consider the following special cases. Case 1: b = 0 The value of the gamma function is as follows 0 0 1 lim lim 1 1 t x t x x x ed t t e →∞ Γ = = = = Case 2: b = 1 The value of the gamma function requires that we use the integration by parts method of calculus. 0 0 0 2 lim , lim 1 t x tt t x x x te dt te dt u t dt du dv e dt v e e dt Γ = == = ⎡⎤ = + ⎢⎥ ⎣⎦ =
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Gamma Function Notes - Gamma Function Notes The gamma...

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