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5. Let In x"e-*dx, for any non-negative integer n. Using integration by parts, show that In = nIn-1, for n > 1, as a result show that In = n(n – 1) ...1 = n!. The integral In is a special case of the Gamma function I which can be define general for a >0 as r(a) = -1 e xª dx. You have just shown that I(a) interpolates the factorial function.

5. Let In x"e-*dx, for any non-negative integer n. Using integration by parts, show that In = nIn-1, for n > 1, as a result show that In = n(n – 1) ...1 = n!. The integral In is a special case of the Gamma function I which can be define general for a >0 as r(a) = -1 e xª dx. You have just shown that I(a) interpolates the factorial function.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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5. Let In x"e-®dx, for any non-negative integer n. Using integration by parts, show that In = nIn-1, for n > 1, as a result show that In = n(n – 1) ·. 1 = n!. The integral In is a special case of the Gamma functionI which can be defined in general for a > 0 as r(@) = xa-le=ªdx. You have just shown that I(@) interpolates the factorial function.
Transcribed Image Text:5. Let In x"e-®dx, for any non-negative integer n. Using integration by parts, show that In = nIn-1, for n > 1, as a result show that In = n(n – 1) ·. 1 = n!. The integral In is a special case of the Gamma functionI which can be defined in general for a > 0 as r(@) = xa-le=ªdx. You have just shown that I(@) interpolates the factorial function.
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