c$.$ If $p$ is a positive integer $n,$ show that $(n+1)=n!$ Since $(p)$ is also defined when $p$ is not an integer, this function provides an extension of the factorial function to nonintegral values of the independent variable. Note that it is also consistent to define $01.$ d$.$ Show that, for $p>0,p(p+1)(p+2)cdots(p+n-1)=\frac{(p+n)}{(p)}$ Thus $(p)$ can be determined for all positive values of $p$ if $(p)$ is known in a single interval of unit length $-$ say, $(\frac{1}{2})=\sqrt[\phantom{2}]{}(\frac{3}{2})(\frac{11}{2})0.Itis$ possible $to$ show that $(\frac{1}{2})=\sqrt[\phantom{2}]{}.$ Find $(\frac{3}{2})$ and $(\frac{11}{2}).$