Problems and Theorems in Analysis II

Let a ₀, a ₁, a ₂,..., a _n ,... be complex numbers not all zero. Let the power series have radius of convergence R, R>0. If R = ∞, f(z) is called an entire function. Let 0 ≦ r < R. Then the sequence tends to 0, and hence it contains a largest term, the maximum term, whose value is denoted by μ(r). Thus for n = 0, 1, 2, 3,..., r ≧ 0 [I, Ch. 3, § 3].

We investigate in this chapter real functions of the real variable x. In particular we assume that the coefficients a ₀, a ₁, a ₂,... of the polynomials a ₀ + a ₁ x +a ₂ x ² + ... + a _n x ⁿ and of the power series a ₀ + a ₁ x + a ₂ x ² + ... which we shall be considering are real. We assume further, unless the contrary is stated, that all functions are analytic in the corresponding intervals. The theorems, however, are changed only slightly or not at all if we introduce more general assumptions, e.g. the existence of derivatives up to some order. The zeros in the following are always to be counted according to their multiplicity.

Setting cos ϑ = x, the expressions are polynomials in x of degree n (the Tchebychev polynomials); the leading coefficient of T _n (x) is equal to 2 ^n-1 and that of U _n (x) is equal to 2 ⁿ , n = 1, 2, 3,....

Letx be a real number. Denote by [x] the integral part of x, i.e. the integer that satisfies the inequalities We have for example .

If we throw a heavy convex polyhedron with arbitrary interior mass distribution onto a horizontal floor then it will come to rest in a stable position on one of its faces. That is there exists for an arbitrary point P lying in the interior of the convex polyhedron one face F (at least) with the following property: The perpendicular dropped from P onto the plane in which F lies has its foot in the interior of the face F. Give a purely geometrical proof free from mechanical considerations for the existence of the face F.