Problems

Evaluate the integrals: \[ \int_0^\infty \frac{x^{p-1}}{1+x+x^2}\,dx \quad (0

Explain how to distinguish the two "sides" of a bilateral surface. Define \(\iint f(x,y,z)dydz\) taken over a specified side of a given bilateral surface and show how to calculate it in terms of \(u\) and \(v\) when the equations to the surface are \[ x=\theta(u,v), \quad y=\phi(u,v), \quad z=\psi(u,v). \] Find the value of \[ \iint x^3y^3z^5dydz \] taken over the outer side of the octant of the surface of the ellipsoid \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 \] for which \(x,y,z\) are all positive.

The Green's function \(G(x,y,z)\) associated with a given closed surface \(S\) and origin \((a,b,c)\) in its interior is defined by the properties:

1. [(1)] \(G=0\) on \(S\),
2. [(2)] \(G-\frac{1}{r}\) is harmonic in the interior of \(S\), where \(r=\sqrt{\{(x-a)^2+(y-b)^2+(z-c)^2\}}\).

Prove that, if \(U(x,y,z)\) is harmonic in the interior of \(S\), then \[ U(a,b,c) = -\frac{1}{4\pi}\iint_S U\frac{\partial G}{\partial\nu}d\sigma, \] where \(\frac{\partial}{\partial\nu}\) denotes differentiation in the direction of the outward drawn normal. Find the Green's function in the case of a sphere and deduce an expression for the function which is harmonic in the interior of the sphere and takes a prescribed value \(f(x,y,z)\) on its surface.

Develop the Lagrange-Charpit method of solving a partial differential equation and illustrate by considering \[ p^2+q^2-2px-2qy+1=0. \]

Prove that two elliptic functions with the same periods, zeros and poles have a constant ratio. Assuming that the functions sn \(u\), cn \(u\), dn \(u\) have the usual periods, zeros and poles, apply this method to compare \[ \text{cn } u \text{ cn }(u+v) - \text{cn } v, \quad \text{sn } u \text{ sn }(u+v), \] proving that \[ \text{cn } u \text{ cn }(u+v) + \text{sn } u \text{ dn } v \text{ sn }(u+v) - \text{cn } v = 0. \] Deduce addition theorems for the functions sn, cn.