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Improper Integral

  • Thread starter eestep
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  • #1
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Homework Statement


Gamma function is defined for all x>0 by rule
[tex]\Gamma[/tex](x)=[tex]\int[/tex]0[tex]\infty[/tex]tx-1e-tdt
Find a simple expression for [tex]\Gamma[/tex](n) for positive integers n. Answer is [tex]\Gamma[/tex](n)=(n-1)!


Homework Equations





The Attempt at a Solution


[tex]\int[/tex]0[tex]\infty[/tex]tn-1e-tdt=-tn-1e-t-[tex]\int[/tex](n-1)tn-2(-e-t)dt=-tn-1e-t+(n-1)[tex]\int[/tex]tn-2e-tdt
u=tn-1 du=(n-1)tn-2dt
dv=e-tdt v=[tex]\int[/tex]e-tdt=-e-t
 

Answers and Replies

  • #2
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Use mathematical induction together with integration by parts.
 
  • #3
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I appreciate the advice!
 
  • #4
SammyS
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Homework Statement


Gamma function is defined for all x>0 by rule
[tex]\Gamma(x)=\int_0^\infty\, t^{x-1}\,e^{-t}\,dt[/tex]
Find a simple expression for [tex]\Gamma[/tex](n) for positive integers n. Answer is [tex]\Gamma[/tex](n)=(n-1)!

Homework Equations



The Attempt at a Solution



[tex]\Gamma(n)=\int_0^\infty\, t^{n-1}\,e^{-t}\,dt=-t^{n-1}\,e^{-t}-\int(n-1)t^{n-2}(-e^{-t})dt=-t^{n-1}e^{-t}\ +\ (n-1)\int t^{n-2}e^{-t}dt
[/tex]
You are missing your limits of integration after doing integration by parts.

Click on the expression at the right to see the LaTeX code that produced it: [tex]\left[a^{-x}\right]_{\sqrt{2}}^{\infty}[/tex]  .
 

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