Question

(a) To see the results in Problem 1graphically, computer plot the percentage error in Stirling’s formula as a function of p for values of p = 1-1000. Make separate plots, say for p = 1-10, 10-100, 100-1000, to make it easier to read values from your plots.

(b) Repeat part (a) for the percentage error in (11.5) using two terms of the asymptotic series, that is, Stirling’s formula times$\left[1+\frac{1}{12p}\right]$.

Short answer

The graph is shown below.

a)

b)

Step-by-step solution

Given Information

The gamma function is$\Gamma \left(p+1\right)=p!$.

Definition of Gamma function.

Gamma function behaves similarly to a factorial for natural numbers (a discrete set), its extension to positive real numbers (a continuous set) allows it to be used to model situation involving continuous change.

Gamma function$\mathit{\Gamma }\mathbf{\left(}\mathit{p}\mathbf{\right)}\mathbf{=}\underset{\mathbf{0}}{\overset{\mathbf{\infty }}{\mathbf{\int }}}{\mathit{x}}^{\mathbf{p}\mathbf{-}\mathbf{1}}{\mathit{e}}^{\mathbf{-}\mathbf{x}}\mathit{d}\mathit{x}$.

Plot the graph for part (a).

The gamma function is $\Gamma \left(p+1\right)=p!$.

Find the relative error using Sterling’s formula.

localid="1664876138184">R=xxe-x2πx1+112x-xxe-x2πxxxe-x2πx1+112x=112x1+112x=1+112x=1001+12x%

The graph of the function is shown below.

Plot the graph for part (b).

The gamma function is $\Gamma \left(p+1\right)=p!$.

Find the relative error using Sterling’s formula.

$\begin{array}{rcl}R& =& \frac{\left|x^x{e}^{-x}\sqrt{2\mathrm{\pi x}}\left(1+{\displaystyle \frac{1}{12x}+\frac{1}{288x^2}}\right)-x^x{e}^{-x}\sqrt{2\mathrm{\pi x}}\left(1+\frac{1}{12x}\right)\right|}{x^x{e}^{-x}\sqrt{2\mathrm{\pi x}}\left(1+\frac{1}{12x}+{\displaystyle \frac{1}{288x^2}}\right)}\\ & =& \frac{{\displaystyle \frac{1}{288x^2}}}{\left(1+\frac{1}{12x}+{\displaystyle \frac{1}{288x^2}}\right)}\\ & =& \left(\frac{1}{1+24x+288x^2}\right)\\ & =& \frac{100}{1+24x+288x^2}\%\end{array}$

The graph of the function is shown below.

Hence, the graph has been plotted.