\(A, A', B, B'\) are four points on a line, and \(BT, B'T'\) are tangents to a conic passing through \(A\) and \(A'\). Show that \(TT'\) cuts the line in a double point of the involution \((AA', BB')\). How can you determine the other double point?
Express the left-hand side of the equation \[ x^4+8x^3-12x^2+104x-20=0 \] as the product of two quadratics with rational coefficients, and solve the equation.
The function \(\mu(n)\) is defined as being equal to 0 when \(n\) contains any squared factor, to 1 when \(n=1\), and to \((-1)^{\nu}\) when \(n=p_1 p_2 \dots p_{\nu}\), \(p_1, p_2, \dots, p_{\nu}\) being different primes. Prove that \[ \sum \mu(d)=0, \] the summation being extended to all divisors \(d\) of a given number \(N\).
Prove that if \(u_n(x)\) is a continuous function of \(x\) for \(a \le x \le b\), and \(\sum_{0}^{\infty} u_n(x)\) is uniformly convergent for \(a \le x \le b\), then \[ \int_a^b \left\{\sum_0^\infty u_n(x)\right\}dx = \sum_0^\infty \int_a^b u_n(x)dx. \] Prove that, if \(\alpha, \beta\), and \(\delta\) are positive, then \[ \sum_{n=0}^{\infty} \int_0^\delta (\alpha e^{-n\alpha x} - \beta e^{-n\beta x})dx = \log\frac{1-e^{-\alpha\delta}}{1-e^{-\beta\delta}} - \log\frac{\alpha}{\beta}, \] but \[ \int_0^\delta \sum_{n=0}^\infty (\alpha e^{-n\alpha x} - \beta e^{-n\beta x})dx = \log\frac{1-e^{-\alpha\delta}}{1-e^{-\beta\delta}}. \] Explain the discrepancy in the results.
Show that if \(t=u+iv = f(x+iy) = f(z)\), where \(f\) is an analytic function, and \(F\) is a real function of \(u\) and \(v\), with continuous second derivatives, then \[ \left(\frac{\partial^2 F}{\partial x^2}\right) + \left(\frac{\partial^2 F}{\partial y^2}\right) = M^2 \left\{\left(\frac{\partial^2 F}{\partial u^2}\right) + \left(\frac{\partial^2 F}{\partial v^2}\right)\right\}, \quad \frac{\partial^2 F}{\partial x \partial y} = M^2 \left(\frac{\partial^2 F}{\partial u \partial v}\right), \] where \(M = \left|\frac{dt}{dz}\right|\).
Obtain the solution of the equation \[ \frac{d^2 y}{dx^2} + \frac{1}{x}\frac{dy}{dx} + \left(1-\frac{n^2}{x^2}\right)y=0, \] where \(n\) is not an integer, in the form \[ A J_n(x) + B J_{-n}(x), \] where \[ J_n(x) = (\frac{1}{2}x)^n \sum_{v=0}^\infty \frac{(-1)^v (\frac{1}{2}x)^{2v}}{v!\Gamma(n+1+v)}. \] Show that \[ \frac{i e^{\frac{1}{2}(n+1)\pi i}}{\sin n\pi}\{e^{-n\pi i}J_n(ix) - J_{-n}(ix)\} \] is real for all real values of \(x\).
Show that a right circular cone can be drawn to touch three consecutive osculating planes of a curve in space, that its semivertical angle is \[ \arctan(\sigma/\rho) \] and that the direction cosines of its axis are \[ \frac{l'\sigma-l\rho}{\sqrt{(\rho^2+\sigma^2)}}, \frac{m'\sigma-m\rho}{\sqrt{(\rho^2+\sigma^2)}}, \frac{n'\sigma-n\rho}{\sqrt{(\rho^2+\sigma^2)}}; \] \(l,m,n\) being the direction cosines of the tangent and \(l', m', n'\) those of the binormal.