# Question (The gamma function) The gamma function $$\Gamma(x)$$ is defined by the improper integral $\Gamma(x)=\int_{0}^{\infty} t^{x-1} e^{-t} d t$ ( $$\Gamma$$ is the Greek capital letter gamma.) (a) Show that the integral converges for $$x>0$$. (b) Use integration by parts to show that $$\Gamma(x+1)=x \Gamma(x)$$ for $$x>0$$.>(c) Show that $$\Gamma(n+1)=n$$ ! for $$n=0,1,2, \ldots$$ (d) Given that $$\int_{0}^{\infty} e^{-x^{2}} d x=\frac{1}{2} \sqrt{\pi}$$, show that $$\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$$ and $$\Gamma\left(\frac{3}{2}\right)=\frac{1}{2} \sqrt{\pi}$$. In view of (c), $$\Gamma(x+1)$$ is often written $$x$$ ! and regarded as a real-valued extension of the factorial function. Some scientific calculators (in particular, HP calculators) with the factorial function $$n$$ ! built in actually calculate the gamma function rather than just the integral factorial. Check whether your calculator does this by asking it for $$0.5$$ !. If you get an error message, it's not using the gamma function.

(The gamma function) The gamma function $$\Gamma(x)$$ is defined by the improper integral
$\Gamma(x)=\int_{0}^{\infty} t^{x-1} e^{-t} d t$
( $$\Gamma$$ is the Greek capital letter gamma.)
(a) Show that the integral converges for $$x>0$$.
(b) Use integration by parts to show that $$\Gamma(x+1)=x \Gamma(x)$$ for $$x>0$$.>(c) Show that $$\Gamma(n+1)=n$$ ! for $$n=0,1,2, \ldots$$
(d) Given that $$\int_{0}^{\infty} e^{-x^{2}} d x=\frac{1}{2} \sqrt{\pi}$$, show that $$\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$$ and $$\Gamma\left(\frac{3}{2}\right)=\frac{1}{2} \sqrt{\pi}$$.
In view of (c), $$\Gamma(x+1)$$ is often written $$x$$ ! and regarded as a real-valued extension of the factorial function. Some scientific calculators (in particular, HP calculators) with the factorial function $$n$$ ! built in actually calculate the gamma function rather than just the integral factorial. Check whether your calculator does this by asking it for $$0.5$$ !. If you get an error message, it&qpos;s not using the gamma function.
Transcribed Image Text: (The gamma function) The gamma function $$\Gamma(x)$$ is defined by the improper integral $\Gamma(x)=\int_{0}^{\infty} t^{x-1} e^{-t} d t$ ( $$\Gamma$$ is the Greek capital letter gamma.) (a) Show that the integral converges for $$x>0$$. (b) Use integration by parts to show that $$\Gamma(x+1)=x \Gamma(x)$$ for $$x>0$$.>(c) Show that $$\Gamma(n+1)=n$$ ! for $$n=0,1,2, \ldots$$ (d) Given that $$\int_{0}^{\infty} e^{-x^{2}} d x=\frac{1}{2} \sqrt{\pi}$$, show that $$\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$$ and $$\Gamma\left(\frac{3}{2}\right)=\frac{1}{2} \sqrt{\pi}$$. In view of (c), $$\Gamma(x+1)$$ is often written $$x$$ ! and regarded as a real-valued extension of the factorial function. Some scientific calculators (in particular, HP calculators) with the factorial function $$n$$ ! built in actually calculate the gamma function rather than just the integral factorial. Check whether your calculator does this by asking it for $$0.5$$ !. If you get an error message, it's not using the gamma function.
Transcribed Image Text: (The gamma function) The gamma function $$\Gamma(x)$$ is defined by the improper integral $\Gamma(x)=\int_{0}^{\infty} t^{x-1} e^{-t} d t$ ( $$\Gamma$$ is the Greek capital letter gamma.) (a) Show that the integral converges for $$x>0$$. (b) Use integration by parts to show that $$\Gamma(x+1)=x \Gamma(x)$$ for $$x>0$$.>(c) Show that $$\Gamma(n+1)=n$$ ! for $$n=0,1,2, \ldots$$ (d) Given that $$\int_{0}^{\infty} e^{-x^{2}} d x=\frac{1}{2} \sqrt{\pi}$$, show that $$\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$$ and $$\Gamma\left(\frac{3}{2}\right)=\frac{1}{2} \sqrt{\pi}$$. In view of (c), $$\Gamma(x+1)$$ is often written $$x$$ ! and regarded as a real-valued extension of the factorial function. Some scientific calculators (in particular, HP calculators) with the factorial function $$n$$ ! built in actually calculate the gamma function rather than just the integral factorial. Check whether your calculator does this by asking it for $$0.5$$ !. If you get an error message, it's not using the gamma function.