Chapter 11: Special Functions

Find the arc length of one arch of $\mathit{y}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{x}$.

A particle starting from rest at $\mathit{x}\mathbf{=}\mathbf{1}$ moves along the xaxis toward the origin.

Its potential energy is $\mathit{V}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathit{m}\mathit{l}\mathit{n}\mathit{x}$. Write the Lagrange equation and integrate it

to find the time required for the particle to reach the origin.

Express as a function

Write the integral in equation (12.7) as an elliptic integral and show that (12.8)gives its value. Hints: Write $\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{2}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathbf{\left(}\frac{\mathbf{\theta }}{\mathbf{2}}\mathbf{\right)}$ and a similar equation for$\mathit{c}\mathit{o}\mathit{s}\mathit{\alpha }$. Then make the change of variable$\mathit{x}\mathbf{=}\frac{\mathbf{sin}\mathbf{\left(}\frac{\mathbf{\theta }}{\mathbf{2}}\mathbf{\right)}}{\mathbf{sin}\mathbf{\left(}\frac{\mathbf{\alpha }}{\mathbf{2}}\mathbf{\right)}}$.

Computer plot graphs of sn u, cn u, and dn u, for several values of k, say, for example, $\mathit{k}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}}\mathbf{,}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{,}\frac{\mathbf{3}}{\mathbf{4}}\mathbf{,}\mathbf{0}\mathbf{.}\mathbf{9}\mathbf{,}\mathbf{0}\mathbf{.}\mathbf{99}\mathbf{.}$.Also plot 3D graphs of sn, cn, and dn as functions of u and k.

If$\mathit{u}\mathbf{=}\mathit{l}\mathit{n}\mathbf{(}\mathit{s}\mathit{e}\mathit{c}\mathit{\phi }\mathbf{+}\mathit{t}\mathit{a}\mathit{n}\mathit{\phi }\mathbf{)}$, then φ is a function of u called the Gudermannian of u, $\mathit{\phi }\mathbf{=}\mathit{g}\mathit{d}\mathit{u}$. Prove that: .

$\mathit{u}\mathbf{=}\mathit{l}\mathit{n}\mathit{t}\mathit{a}\mathit{n}\mathbf{(}\frac{\mathbf{\pi }}{\mathbf{4}}\mathbf{+}\frac{\mathbf{\varphi }}{\mathbf{2}}\mathbf{)}\mathbf{,}\mathit{t}\mathit{a}\mathit{n}\mathit{g}\mathit{d}\mathit{u}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\mathit{u}\mathbf{,}\mathit{s}\mathit{i}\mathit{n}\mathit{g}\mathit{d}\mathit{u}\mathbf{=}\mathit{t}\mathit{a}\mathit{n}\mathit{h}\mathit{u}\mathbf{,}\frac{\mathbf{d}}{\mathbf{du}}\mathit{g}\mathit{d}\mathit{u}\mathbf{=}\mathbf{sech}\mathit{u}$

Carry through the algebra to get the equation

$\mathit{e}\mathit{r}\mathit{f}\mathit{c}\left(x\right)\mathbf{=}\mathbf{1}\mathbf{-}\mathit{e}\mathit{r}\mathit{f}\left(x\right)\mathbf{\square }\frac{{\mathbf{e}}^{\mathbf{-}{\mathbf{x}}^{\mathbf{2}}}}{\mathbf{x}\sqrt{\mathbf{\pi }}}\left(1-\frac{1}{2x^2}+\frac{1.3}{{\left(2x^2\right)}^2}-\frac{1.3.5}{{\left(2x^2\right)}^3}+.....\right)$

Express the following integrals as $\beta$ functions, and then, by (7.1) , in terms of $\Gamma$ functions. When possible, use $\Gamma$function formulas to write an exact answer in terms of $\pi ,2$ , etc. Compare your answers with computer results and reconcile any discrepancies.

role="math" localid="1664349717981" $\underset{\mathbf{0}}{\overset{\mathbf{1}}{\mathbf{\int }}}\frac{{\mathbf{x}}^{\mathbf{4}}}{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}}}\mathit{d}\mathit{x}$

Prove that $\mathit{B}\mathbf{(}\mathit{p}\mathbf{,}\mathit{q}\mathbf{)}\mathbf{=}\mathit{B}\mathbf{(}\mathit{q}\mathbf{,}\mathit{p}\mathbf{)}$. Hint:Put$\mathit{x}\mathbf{=}\mathbf{1}\mathbf{-}\mathit{y}$in Equation (6.1).

Using (5.3) with (3.4) and (4.1), find $\mathit{\Gamma }\mathbf{(}\mathbf{3}\mathbf{/}\mathbf{2}\mathbf{)}$,$\mathit{\Gamma }\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{/}\mathbf{2}\mathbf{)}$, and$\mathit{\Gamma }\mathbf{(}\mathbf{-}\mathbf{3}\mathbf{/}\mathbf{2}\mathbf{)}$in terms of$\sqrt{\mathbf{\pi }}$.