Chapter 11

Find the length of arc of the ellipse $x^2+\left(y^2/4\right)=1$ between $(0,2)$ and $(\frac{1}{2},\sqrt{3})$. (Note that, here $b>a;$ see Example $1.$ )

If $u=\ln (\sec \varphi +\tan \varphi )$, then $\varphi$ is a function of $u$ called the gudermannian of $u,\varphi =gdu$. Prove that: $$u=\ln \tan (\frac{\pi }{4}+\frac{\varphi }{2}),\tan \mathrm{gd}u=\sinh u,\sin \mathrm{gd}u=\tanh u,\frac{d}{du}\mathrm{gd}u=\mathrm{sech}u.$$

Identify each of the following integrals or expressions as one of the functions of this chapter and then evaluate it using appropriate formulas and/or tables. $${\int }_0^{\pi /2}(\cos x{)}^{3/2}dx$$

By transforming $${\int }_0^{\pi /2}\frac{d\varphi }{\sqrt{\cos \varphi }}$$ to one of the standard forms for an elliptic integral of the first kind, show that $$B(\frac{1}{4},\frac{1}{2})=2\sqrt{2}F(\frac{1}{\sqrt{2}},\frac{\pi }{2})=2\sqrt{2}K\left(\frac{1}{\sqrt{2}}\right)$$ $$K\left(\frac{1}{\sqrt{2}}\right)=\frac{1}{4\sqrt{\pi }}{\left[\mathrm{\Gamma }\left(\frac{1}{4}\right)\right]}^2$$ Evaluate these expressions (from $\mathrm{\Gamma }$ and $K$ tables) to check the result.

In the pendulum problem, $\theta =x\sin \sqrt{g/l}t$ is onc solution when the amplitude $x$ is small enough for the motion to be considered simple harmonic. Show that the corresponding exact solution when $\alpha$ is not small is $$\sin \frac{\theta }{2}=\sin \frac{x}{2}\mathrm{sn}\sqrt{\frac{g}{l}}t$$ where $k=\sin (\alpha /2)$ is the modulus of the elliptic function. Show that this reduces to the simple harmonic motion solution for small amplitude $\alpha$.