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This is known as the geometric series formula, and is used in a variety of problems. we use the first definition its that strange sum of all the powers of 2. Example (duplication formula). Prove that Γ(n)Γ(n + 1/2) = 21−2n. √ π Γ(2n). Solution. By the definition of beta function, we have. B(n,n) = 2. Using Gamma Function to Evaluate Integrals. Example(1). Evaluate: I = 0 ∞ x. 6 e. -2x dx. Solution: Letting y = 2x, we get. I = (1/128) 0 ∞. This book consists of a selection of problems, each with a solution worked out in detail, dealing with the properties and applications of the Gamma function For any positive real number α: Γ(12)=√π. Answer the following questions: Find the value of the following integral: I=∫∞0x6e−5xdx. To find Γ(72), we Two fundamental properties of the gamma function are that. Γ (z+1) = zΓ (z) where p0 = 1.84742 is the unique solution of the equation 2Γ ( p+1.