Chapter 11: Special Functions

The integral $\underset{\mathbf{0}}{\overset{\mathbf{\infty }}{\mathbf{\int }}}{\mathit{t}}^{\mathbf{p}\mathbf{-}\mathbf{1}}{\mathit{e}}^{\mathbf{-}\mathbf{t}}\mathit{d}\mathit{t}\mathbf{=}\mathit{\Gamma }\mathbf{\text{'}}\left(p,x\right)$is called an incomplete $\mathit{\Gamma }\mathbf{\text{'}}$function. [Note that if x = 0, this integral is$\mathit{\Gamma }\mathbf{\text{'}}\mathbf{\left(}\mathit{p}\mathbf{\right)}$.] By repeated integration by parts, find several terms of the asymptotic series for$\mathit{\Gamma }\mathbf{\text{'}}\mathbf{(}\mathit{p}\mathbf{,}\mathit{x}\mathbf{)}$.

Express the complementary error function erfc(x)as an incomplete$\mathit{\Gamma }\mathbf{\text{'}}$function (see Problem 2) and use your result in Problem 2to obtain (again) the asymptotic expansion oferfc(x) as in (10.4) .

${\mathit{E}}_{\mathbf{n}}\left(x\right)\overline{)\mathbf{=}}\underset{\mathbf{1}}{\overset{\mathbf{\infty }}{\mathbf{\int }}}\frac{{\mathbf{e}}^{\mathbf{-}\mathbf{x}\mathbf{t}}}{{\mathbf{t}}^{\mathbf{n}}}\mathit{d}\mathit{t}\mathbf{,}\mathbf{}\mathit{n}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{E}\mathit{i}\left(x\right)\mathbf{=}\underset{\mathbf{-}\mathbf{\infty }}{\overset{\mathbf{t}}{\mathbf{\int }}}\frac{{\mathbf{e}}^{\mathbf{t}}}{\mathbf{t}}\mathit{d}\mathit{t}$and other similar

integrals are called exponential integrals. By making appropriate changes of variable, show that

1. ${\mathit{E}}_{\mathbf{1}}\left(x\right)\mathbf{=}\underset{\mathbf{x}}{\overset{\mathbf{\infty }}{\mathbf{\int }}}\frac{{\mathbf{e}}^{\mathbf{-}\mathbf{t}}}{\mathbf{t}}\mathit{d}\mathit{t}$
   
2. role="math" localid="1664368570398" width="142" height="67">Ei(x)=-∫-x∞e-ttdt
   
3. $\mathit{E}\mathit{i}\left(x\right)\mathbf{=}\mathbf{-}\mathit{E}\mathit{i}\mathbf{(}\mathbf{-}\mathbf{x}\mathbf{)}$
   
4. $\underset{\mathbf{0}}{\overset{\mathbf{x}}{\mathbf{\int }}}\frac{{\mathbf{e}}^{{\displaystyle \raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{t}$}\right.}}}{\mathbf{t}}\mathit{d}\mathit{t}\mathbf{=}{\mathit{E}}_{\mathbf{1}}\left(-1/x\right)$
   

(Caution:Various notations are used; check carefully the notation in references you

are using.)

(a) Express E₁(x)as an incomplete$\mathit{\Gamma }$function.

(b) Find the asymptotic series for E₁(x).

The logarithmic integralis $\mathit{l}\mathit{i}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\underset{\mathbf{0}}{\overset{\mathbf{x}}{\mathbf{\int }}}\frac{\mathbf{dt}}{\mathbf{In}\mathbf{}\mathbf{t}}$. Express as exponential integrals

1. $\mathit{l}\mathit{i}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$
   
2. $\mathit{l}\mathit{i}\mathbf{\left(}{\mathbf{e}}^{\mathbf{x}}\mathbf{\right)}$
   
3. $\mathit{l}\mathit{i}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\underset{\mathbf{0}}{\overset{\mathbf{x}}{\mathbf{\int }}}\frac{\mathbf{dt}}{\mathbf{In}\mathbf{\left(}{\displaystyle \frac{1}{t}}\mathbf{\right)}}$
   

Computer plot graphs of

(a) E_n(x) for n = 0-10and x = 0.2.

(b) E₁(x) and E_n(x)for x = 0-2.

(c) the sine integral ${\mathit{S}}_{\mathbf{i}}\left(x\right)\mathbf{=}\underset{\mathbf{0}}{\overset{\mathbf{x}}{\mathbf{\int }}}\frac{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{t}}{\mathbf{t}}\mathit{d}\mathit{t}$ and the cosine integral ${\mathit{C}}_{\mathbf{i}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\underset{\mathbf{0}}{\overset{\mathbf{\infty }}{\mathbf{\int }}}\frac{\mathit{C}\mathit{o}\mathit{s}\mathbf{}\mathbf{t}}{\mathbf{t}}\mathit{d}\mathit{t}$ for$\mathit{x}\mathbf{=}\mathbf{0}\mathbf{-}\mathbf{4}\mathbf{\pi }$.

Use Stirling’s formula to evaluate $\underset{\mathbf{n}\mathbf{\to }\mathbf{\infty }}{\mathbf{l}\mathbf{i}\mathbf{m}}\frac{{\left(n!\right)}^{{\displaystyle \frac{\mathbf{1}}{\mathbf{n}}}}}{\mathbf{n}}$.

Use the term 1/(12p)in (11.5) to show that the error in Stirling’s formula (11.1) is < 10%for p > 1; < 1%for p > 10; < 0.1%for p > 100; < 0.01%for p > 1000.

(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$\left[1+\frac{1}{12p}\right]$.

In statistical mechanics, we frequently use the approximationN! = N In N-N, where N is of the order of Avogadro’s number. Write out ln N! using Stirling’s formula, compute the approximate value of each term for N = 10²³ , and so justify this commonly used approximation.