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Anastaziya [24]
2 years ago
7

a) Γ(n) =∫[infinity]0tn−1e−tdta)Show that Γ(n+ 1) =nΓ(n) using integration by parts.b) Show that Γ(n+ 1) =n! , wherenis a positi

ve integer, by repeatedly applying the result in parta)c) Compute Γ(5).d) Prove what the value of Γ(12) should be by evaluating the gamma function forn= 1/2 and usingthe fact that∫[infinity]0e−u2du=√π2.
Mathematics
1 answer:
Nataliya [291]2 years ago
7 0

\Gamma(n)=\displaystyle\int_0^\infty t^{n-1}e^{-t}\,\mathrm dt

a. Integrate by parts by taking

u=t^{n-1}\implies\mathrm du=(n-1)t^{n-2}\,\mathrm dt

\mathrm dv=e^{-t}\,\mathrm dt\implies v=-e^{-t}

Then

\displaystyle\Gamma(n)=-t^{n-1}e^{-t}\bigg|_0^\infty+(n-1)\int_0^\infty t^{n-2}e^{-t}\,\mathrm dt

We have

\displaystyle\lim_{t\to\infty}\frac{t^{n-1}}{e^t}=0

and so

\Gamma(n)=(n-1)\displaystyle\int_0^\infty t^{n-2}e^{-t}\,\mathrm dt=(n-1)\Gamma(n-1)

or, replacing n\to n+1, \Gamma(n+1)=n\Gamma(n).

b. From the above recursive relation, we find

\Gamma(n+1)=n\Gamma(n)=n(n-1)\Gamma(n-1)=n(n-1)(n-2)\Gamma(n-2)=\cdots=n(n-1)(n-2)\cdots2\cdot1\Gamma(1)

Now,

\Gamma(1)=\displaystyle\int_0^\infty e^{-t}\,\mathrm dt=1

and so we're left with \Gamma(n+1)=n!.

c. Using the previous result, we find \Gamma(5)=4!=24.

d. If the question is asking to find \Gamma(12), then you can just use the same approach as in (c).

But if you're supposed to find \Gamma\left(\frac12\right), we have

\displaystyle\Gamma\left(\frac12\right)=\int_0^\infty t^{-1/2}e^{-t}\,\mathrm dt

Substitute

u=t^{1/2}\implies u^2=t\implies 2u\,\mathrm du=\mathrm dt

Then

\displaystyle\Gamma\left(\frac12\right)=\int_0^\infty\frac1ue^{-u^2}(2u\,\mathrm du)=2\int_0^\infty e^{-u^2}\,\mathrm du=\frac{2\sqrt\pi}2=\sqrt\pi

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