91Ó°ÊÓ

Use Stirling’s formula to evaluate$\underset{\mathbf{n}\mathbf{→}\mathbf{∞}}{\mathbf{l}\mathbf{i}\mathbf{m}}\frac{\mathbf{\left(}\mathbf{2}\mathbf{n}\mathbf{\right)}\mathbf{!}\sqrt{\mathbf{n}}}{{\mathbf{2}}^{\mathbf{2}\mathbf{n}}{\mathbf{(}\mathbf{n}\mathbf{!}\mathbf{)}}^{\mathbf{2}}}$.

Use Stirling’s formula to evaluate$\underset{\mathbf{n}\mathbf{→}\mathbf{∞}}{\mathbf{l}\mathbf{i}\mathbf{m}}\frac{\mathbf{(}\mathbf{n}\mathbf{+}{\displaystyle \frac{3}{2}}\mathbf{)}\mathbf{!}}{\sqrt{\mathbf{n}}\mathbf{(}\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{!}}$.

Use equations (3.4) and (11.5) to show that $\mathit{Γ}\mathbf{\left(}\mathit{p}\mathbf{\right)}\mathbf{□}{\mathit{p}}^{\mathbf{p}}{\mathit{e}}^{\mathbf{-}\mathbf{p}}\sqrt{\frac{\mathbf{2}\mathbf{π}}{\mathbf{p}}}\left(1+\frac{1}{12p}+...\right)$.

The function $\mathit{ψ}\left(p\right)\mathbf{=}\frac{\mathbf{d}}{\mathbf{d}\mathbf{p}}\mathit{I}\mathit{n}\mathit{Γ}\left(p\right)$ is called the digamma function, and the polygama functions are defined by${\mathit{ψ}}^{\mathbf{n}}\left(p\right)\mathbf{=}\frac{{\mathbf{d}}^{\mathbf{n}}}{\mathbf{d}{\mathbf{p}}^{\mathbf{n}}}\mathit{ψ}\left(p\right)$. [Warning: Some authors define $\mathit{ψ}\left(p\right)$as $\frac{\mathbf{d}}{\mathbf{d}\mathbf{p}}\mathit{I}\mathit{n}\left(p!\right)\mathbf{=}\frac{\mathbf{d}}{\mathbf{d}\mathbf{p}}\mathit{I}\mathit{n}\mathit{Γ}\left(p+1\right)$ ).]

(a) Show that $\mathit{ψ}\left(p+1\right)\mathbf{=}\mathit{ψ}\left(p\right)\mathbf{+}\frac{\mathbf{1}}{\mathbf{p}}$. Hint: See (3.4).

(b) Use Problem 6 to obtain$\mathit{ψ}\left(p\right)\mathbf{□}\mathit{I}\mathit{n}\left(p\right)\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}\mathbf{p}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{12}{\mathbf{p}}^{\mathbf{2}}}\mathbf{.}\mathbf{.}\mathbf{.}$.

Expand the integrands of Kand E[see ( 12.3 )] in power series in${\mathit{k}}^{\mathbf{2}}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{θ}$(assuming small k ), and integrate term by term to find power series approximations for the complete elliptic integrals Kand E.

The following expression occurs in statistical mechanics:

$\mathit{P}\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\left(np+u\right)\mathbf{!}\left(nq-u\right)\mathbf{!}}{\mathit{p}}^{\mathbf{n}\mathbf{p}\mathbf{+}\mathbf{u}}{\mathit{q}}^{\mathbf{n}\mathbf{q}\mathbf{-}\mathbf{u}}\mathbf{.}$

Use Stirling’s formula to show that

$\frac{\mathbf{1}}{\mathbf{P}}\mathbf{â–¡}{\mathit{x}}^{\mathbf{n}\mathbf{p}\mathbf{x}}{\mathit{y}}^{\mathbf{n}\mathbf{q}\mathbf{y}}\sqrt{\mathbf{2}\mathbf{Ï€²Ô\pm è\pm ç³\ae ²â}}\mathbf{}\mathbf{,}\mathbf{}\mathit{w}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{}\mathit{x}\mathbf{=}\mathbf{1}\mathbf{+}\frac{\mathbf{u}}{\mathbf{n}\mathbf{p}}\mathbf{,}\mathit{y}\mathbf{=}\mathbf{1}\mathbf{-}\frac{\mathbf{u}}{\mathbf{n}\mathbf{q}}\mathbf{,}\mathit{p}\mathbf{+}\mathit{q}\mathbf{=}\mathbf{1}\mathbf{.}$

Hint: Show that${\mathbf{\left(}\mathit{n}\mathit{p}\mathbf{\right)}}^{\mathbf{n}\mathbf{p}\mathbf{+}\mathbf{u}}{\mathbf{\left(}\mathit{n}\mathit{q}\mathbf{\right)}}^{\mathbf{n}\mathbf{q}\mathbf{-}\mathbf{u}}\mathbf{=}{\mathit{n}}^n{\mathit{p}}^{\mathbf{n}\mathbf{p}\mathbf{+}\mathbf{u}}{\mathit{q}}^{\mathbf{n}\mathbf{q}\mathbf{-}\mathbf{u}}$.

In Problem 4 to 13, identify each of the integral as an elliptic (see Example 1 and 2). Learn the notation of your computer program (see Problem 3) and then evaluate the integral by computer.

10. $\underset{\mathbf{0}}{\overset{\frac{\mathbf{π}}{\mathbf{4}}}{\mathbf{∫}}}\frac{\mathbf{1}}{\sqrt{\mathbf{4}\mathbf{-}\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\mathbf{θ}}}\mathit{d}\mathit{θ}$.

In Problem 4 to 13, identify each of the integral as an elliptic (see Example 1 and 2). Learn the notation of your computer program (see Problem 3) and then evaluate the integral by computer.

11. $\underset{\frac{-\mathrm{π}}{2}}{\overset{\frac{3\mathrm{π}}{8}}{∫}}\frac{1}{\sqrt{1-{\displaystyle \frac{9}{10}}{\sin }^2\mathrm{θ}}}d\mathrm{θ}$

In Problem 4 to 13, identify each of the integral as an elliptic (see Example 1 and 2). Learn the notation of your computer program (see Problem 3) and then evaluate the integral by computer.

12. $\underset{\mathbf{0}}{\overset{\frac{\mathbf{1}}{\mathbf{2}}}{\mathbf{∫}}}\frac{\sqrt{\mathbf{100}\mathbf{-}{\mathbf{t}}^{\mathbf{2}}}}{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{t}}^{\mathbf{2}}}}\mathit{d}\mathit{t}$

In Problem 4 to 13, identify each of the integral as an elliptic (see Example 1 and 2). Learn the notation of your computer program (see Problem 3) and then evaluate the integral by computer.

13. $\underset{\frac{\mathbf{-}\mathbf{1}}{\mathbf{2}}}{\overset{\frac{\mathbf{3}}{\mathbf{4}}}{\mathbf{∫}}}\frac{\sqrt{\mathbf{9}\mathbf{-}\mathbf{4}{\mathbf{t}}^{\mathbf{2}}}}{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{t}}^{\mathbf{2}}}}\mathit{d}\mathit{t}$