Prove that if (1) \(f_n(z)\) is, for every positive integral value of \(n\), analytic in a region \(T\), and (2) \(\Sigma f_n(z)\) is uniformly convergent throughout any closed domain \(D\) interior to \(T\), then the sum \(f(z)\) of the series is analytic in \(T\). Deduce that the sum of a power series represents an analytic function inside its circle of convergence. Prove that the function \(\Sigma n^{-s}\), where \(s=\sigma+it\), is analytic for \(\sigma>1\).
Prove that if \(s_n\) is the sum of the first \(n\) terms of the Fourier series of a continuous and periodic function \(f(x)\), then \[ \frac{s_1+s_2+\dots+s_n}{n} \] tends uniformly to \(f(x)\) when \(n\to\infty\). State any extensions of this theorem with which you are familiar. What are the principal theorems concerning the convergence of Fourier series? How are they related to the theorem just stated?
Find from first principles the differential coefficient of \(\tan x\). If \(\tan y = \{(e^x+1)/(e^x-1)\}^{1/2}\), prove that \[ \frac{d^2y}{dx^2} = 1 + 12\left(\frac{dy}{dx}\right)^2\left\{1+4\left(\frac{dy}{dx}\right)^2\right\}. \]
Trace the curve \(y=e^{1/x}\). Find the inflexions and the asymptotes.
Prove the following formulae for the radius of curvature at any point of a plane curve \[ \text{(i) } r\frac{dr}{dp}, \quad \text{(ii) } \left\{u^2+\left(\frac{du}{d\theta}\right)^2\right\}^{3/2} / \left\{u^3\left(u+\frac{d^2u}{d\theta^2}\right)\right\} \quad (u=1/r). \] Prove that the distance between the origin and the centre of curvature at any point of \(r^n=a^n\cos n\theta\) is \[ \{a^{2n}+(n^2-1)r^{2n}\}^{1/2} / \{(n+1)r^{n-1}\}. \]
Prove that the area of one loop of the curve \(x^4-2xy a^2+a^2y^2=0\) is \(\frac{1}{6}a^2\).
Show that \((y-c)^2+\frac{1}{2}(x-c)^3=0\) is a family of solutions of \[ y+\frac{1}{2}p^3-(x+p^2)=0, \quad \text{where } p=\frac{dy}{dx}. \] Find the envelope of the family, and show that \(y=x\) is a cusp locus.
One of Sir Walter Scott's novels.
The Turkish Empire.
A league of Nations.