State and prove Taylor's Theorem with Lagrange's form of remainder. Shew that, if \(s\) is any positive number and \(n\) any positive integer, then \[ \int_0^\infty \frac{e^{-sx}dx}{\sqrt{1+x^2}} = \sum_{v=0}^{n-1} \frac{(-1)^v c_v}{s^{2v+1}} + \theta_n(s) \frac{(-1)^n c_n}{s^{2n+1}}, \] where \[ c_0=1, \quad c_v = 1^2 . 3^2 \dots (2v-1)^2 \quad (v \ge 1), \] and \(\theta_n(s)\) satisfies the inequality \(0 < \theta_n(s) < 1\). Give reasons for the existence of any integrals which you use.
A sequence of numbers \(a_1, a_2, \dots\), all different from \(-1\), is such that \[ a_n = \frac{\gamma}{n} + b_n, \] where \(\gamma\) is a constant and \(\Sigma|b_n|\) is convergent. Shew, by considering the infinite product \[ \prod \left\{(1+a_n)(1-\frac{\gamma}{n})\right\} \] or otherwise, that \[ P_n = \prod_{v=1}^n (1+a_v) \sim An^\gamma \] as \(n\to\infty\), \(A\) being a constant different from zero. If further \(b_n=O(n^{-1-\delta})\), where \(0<\delta<1\), prove that \[ P_n = An^\gamma + O(n^{\gamma-\delta}) \quad (n\to\infty). \] Discuss the convergence, for all real values of \(\gamma, \alpha, \beta\), and all values (real or complex) of \(x\), of the series \[ \Sigma \frac{x^n}{n^\gamma}, \quad \Sigma \frac{\alpha(\alpha+1)(\alpha+2)\dots(\alpha+n-1)}{\beta(\beta+1)(\beta+2)\dots(\beta+n-1)}x^n, \] distinguishing between absolute and conditional convergence.
Shew that, if \(\Sigma u_n(x)\) is uniformly convergent over the infinite range \(x \ge a\), and if, for each \(n\), \(u_n(x)\) tends to a limit \(u_n\) as \(x\to\infty\), then \(\Sigma u_n\) is convergent, and \[ \Sigma u_n(x) \to \Sigma u_n, \quad \text{as } x\to\infty. \] Prove that \(e^y(1-y)\) is a decreasing function of \(y\) in the interval \(0 \le y \le 1\), and deduce that, as \(x\to\infty\), \[ \sum_{n=1}^{[x]} a_n e^n \left(1-\frac{n}{x}\right)^x \to \sum_{n=1}^\infty a_n, \] provided that the series on the right is convergent. Here \([x]\) denotes the integral part of \(x\).
Obtain the complete solution of the equation \[ x\frac{d^2y}{dx^2} + (\gamma+1)\frac{dy}{dx}-xy=0, \] in the form of series of ascending powers of \(x\), \(\gamma\) being any real number.
Give an account of Lagrange's method of solving the linear partial differential equation \[ Pp+Qq=R, \] explaining the geometrical interpretation of the method. Solve the equation \[ yp-xq=c \quad (c>0). \] Describe the general nature of the surfaces represented by this equation.
A function \(f(z)\) is regular (holomorphic) in the domain \(D\) obtained by excluding from the \(z\)-plane the two regions defined by
\begin{align*}
x < -\frac{1}{2}-\delta, \quad y \ge 0 \\
x \ge +\frac{1}{2}+\delta, \quad y \le 0
\end{align*}
(\(z=x+iy\)) (\(\delta\) being a positive constant), and satisfies throughout \(D\) the inequality
\[
|f(z)|
Prove the addition formula \[ \wp(u+v) = \frac{1}{4}\left(\frac{\wp'u-\wp'v}{\wp u-\wp v}\right)^2 - \wp u - \wp v \] for the Weierstrassian elliptic function \(\wp u\), and deduce a formula for \(\wp(2u)\). Shew that if \(\wp u\) has primitive periods \(2\omega_1, 2\omega_2\), and invariants \(g_2, g_3\), then \(\wp\left(\frac{2\omega_1}{3}\right)\) is a root of the equation \[ 48x^4-24g_2x^2-48g_3x-g_2^2=0. \] Find the other roots of this equation.
If \(f(x) = (x+1)(2x^2-x+1)^{1/2}(x-1)^{-1/2}\) prove that \(f(x) = f(\{1-x\}/\{1+x\})\). Show that the equation \(f(x) = f(3)\) has two real and two imaginary roots, giving the values of each.
Prove that the product of any set of integers, each of which can be expressed as the sum of the squares of two integers, is equal to the sum of the squares of two integers. Express 7540 as the sum of the squares of two integers.
Show that, if \(a_1, a_2, \dots, a_m\) are distinct prime numbers other than unity, the number of solutions in integers (including unity) of the equation \(x_1 x_2 x_3 \dots x_n = a_1 a_2 a_3 \dots a_m\) is \(n^m\). Show also that the number of solutions in which at least one of the \(x\)'s is unity is \[ n! \left\{ \frac{(n-1)^m}{(n-1)! 1!} - \frac{(n-2)^m}{(n-2)! 2!} + \dots + (-)^{n-2} \frac{1^m}{1!(n-1)!} \right\}. \]