Explanation A definition of the gamma function by Carl Friedrich Gauss that is valid for all real numbers, except x = 0 , x = − 1 , x = − 2....... is as follows: Γ ( x ) = lim n → ∞ n ! n x x ( x + 1 ) ( x + 2 ) .... ( x + n ) Simplify the above formula as follows: Γ ( x ) = lim n → ∞ n ! n x x ( x + 1 ) ( x + 2 ) .... ( x + n ) = lim n → ∞ ( x − 1 ) ! n ! n x ( x − 1 ) ! x ( x + 1 ) ( x + 2 ) .... ( x + n ) ( Multiply by unit 1 as ( x − 1 ) ! ( x − 1 ) ! ) = lim n → ∞ ( x − 1 ) ! n ! n x ( x + n ) ! ( z ! = z ( z − 1 ) ! ) Now, obtain the recursion formula Γ ( x + 1 ) = x Γ ( x ) by the use of above calucation as follows: Γ ( x + 1 ) = lim n → ∞ ( ( x + 1 ) − 1 ) ! n ! n ( x + 1 ) ( ( x + 1 ) + n ) ! ( x → x + 1 ) = lim n → ∞ x ! n ! n x + 1 ( x + n +

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter A, Problem 28E

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To derive: The recursion formula of gamma function Γ(x+1)=(x)Γ(x) by alternative derivation.

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Chapter A Solutions

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

Ch. A - In Problems 1 and 2 evaluate the given quantity....Ch. A - Prob. 2E Ch. A - Prob. 3E Ch. A - Prob. 4E Ch. A - Prob. 5E Ch. A - Prob. 6E Ch. A - Prob. 7E Ch. A - Prob. 8E Ch. A - Prob. 9E Ch. A - Prob. 10E

Ch. A - Prob. 11E Ch. A - Prob. 12E Ch. A - Prob. 13E Ch. A - Prob. 14E Ch. A - Prob. 15E Ch. A - Prob. 16E Ch. A - Prob. 17E Ch. A - Prob. 18E Ch. A - Prob. 19E Ch. A - Prob. 20E Ch. A - Prob. 21E Ch. A - Prob. 22E Ch. A - Prob. 23E Ch. A - Prob. 24E Ch. A - In Problems 25 and 26 use the indicated...Ch. A - Prob. 26E Ch. A - Prob. 27E Ch. A - Prob. 28E Ch. A - Prob. 29E Ch. A - Prob. 30E Ch. A - Prob. 31E Ch. A - Prob. 32E Ch. A - Prob. 33E Ch. A - Prob. 34E Ch. A - Show that the beta function is symmetric in x and...Ch. A - Prob. 36E Ch. A - Prob. 37E Ch. A - Prob. 38E Ch. A - Prob. 39E Ch. A - Prob. 40E Ch. A - Prob. 41E Ch. A - Prob. 42E Ch. A - Prob. 43E Ch. A - Prob. 44E Ch. A - Prob. 45E Ch. A - Prob. 46E Ch. A - Prob. 47E Ch. A - Prob. 48E Ch. A - Prob. 49E Ch. A - Prob. 50E Ch. A - Prob. 51E Ch. A - Prob. 52E Ch. A - Prob. 53E Ch. A - Prob. 54E Ch. A - Prob. 55E Ch. A - Prob. 56E

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To derive: The recursion formula of gamma function Γ ( x + 1 ) = ( x ) Γ ( x ) by alternative derivation.

Solution Summary: The author explains the gamma function's recursion formula by alternative derivation.

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To derive: The recursion formula of gamma function Γ(x+1)=(x)Γ(x) by alternative derivation.

Chapter A Solutions