Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Real and Complex Analysis: Solutions to Problems in Amer. Math. Monthly, Math. Magazine, College Math. J., Elemente der Math., Crux Math., EMS Newsletter, Math. Gazette

Raymond Mortini

TL;DR

Real and Complex Analysis: Solutions to Problems in AMM, Math. Magazine, College Math. J., Elemente der Math., Crux Math., EMS Newsletter, and Math. Gazette is a curated collection of analytic problem-solutions, emphasizing explicit calculations of challenging integrals, sums, and products, as well as inequalities and functional equations. The author and collaborators employ a diverse toolkit—residue calculus, Euler–Mittag–Leffler representations, Beta/Gamma identities, and monotone/convergence arguments—to obtain closed-form evaluations and sharp bounds, including results like $I(f)=\int_0^1 x^3 f(x)^2 dx - \int_0^1 x^2 f(x)^3 dx \le \frac{2}{81}$. The collection serves as an educational resource for undergraduates and graduates, and as an archival record of collaborative work and historical mathematical practices in recreational and real/complex analysis. Overall, the work demonstrates how classical analytic techniques yield precise, sometimes surprising, evaluations in concrete problems across multiple journals.

Abstract

In this arxiv-post I present my solutions (published or not) to Problems that appeared in Amer. Math. Monthly, Math. Magazine, Elemente der Mathematik and CRUX, that were mostly done in collaboration with Rudolf Rupp. Some of them (including a few own proposals which were published) were also done in cooperation with Rainer Brück, Bikash Chakraborty, Pamela Gorkin, Gerd Herzog, Jérôme Noël, Peter Pflug and Amol Sasane.

Real and Complex Analysis: Solutions to Problems in Amer. Math. Monthly, Math. Magazine, College Math. J., Elemente der Math., Crux Math., EMS Newsletter, Math. Gazette

TL;DR

Real and Complex Analysis: Solutions to Problems in AMM, Math. Magazine, College Math. J., Elemente der Math., Crux Math., EMS Newsletter, and Math. Gazette is a curated collection of analytic problem-solutions, emphasizing explicit calculations of challenging integrals, sums, and products, as well as inequalities and functional equations. The author and collaborators employ a diverse toolkit—residue calculus, Euler–Mittag–Leffler representations, Beta/Gamma identities, and monotone/convergence arguments—to obtain closed-form evaluations and sharp bounds, including results like . The collection serves as an educational resource for undergraduates and graduates, and as an archival record of collaborative work and historical mathematical practices in recreational and real/complex analysis. Overall, the work demonstrates how classical analytic techniques yield precise, sometimes surprising, evaluations in concrete problems across multiple journals.

Abstract

In this arxiv-post I present my solutions (published or not) to Problems that appeared in Amer. Math. Monthly, Math. Magazine, Elemente der Mathematik and CRUX, that were mostly done in collaboration with Rudolf Rupp. Some of them (including a few own proposals which were published) were also done in cooperation with Rainer Brück, Bikash Chakraborty, Pamela Gorkin, Gerd Herzog, Jérôme Noël, Peter Pflug and Amol Sasane.
Paper Structure (13 sections, 16 theorems, 1514 equations, 248 figures)

This paper contains 13 sections, 16 theorems, 1514 equations, 248 figures.

Table of Contents

  1. American Math. Monthly
  2. The calculations
  3. Mathematics Magazine
  4. An analytic proof
  5. Appendix
  6. Remarks
  7. College Math. J.
  8. Elemente der Mathematik
  9. Nichtnegative Parameter
  10. Die restlichen Fälle
  11. Crux Mathematicorum
  12. EMS Newsletter
  13. Math. Gazette

Key Result

Theorem 1

Suppose that $D\subseteq \mathbb R$ is an interval, that $H:[a,b[\times D \to \mathbb R$ is locally Lipschitz, and that $u,v: [a,b[\to D$ are differentiable functions satisfying Then $u(x)\leq v(x)$ for $x\in [a,b[$.

Figures (248)

  • Figure 1: The curves $y_2= \frac{1}{\sqrt 3}{\rm arsinh}\sqrt{\frac{1}{4}e^{-6x} -\cos^2 (\sqrt 3 x)}$ and $y_1=2\cos(\sqrt 3 x)+e^{-3x}$ and $s_0=\frac{\pi}{2\sqrt 3}$
  • Figure 2: The curves $y=|\pm \sqrt 3\sqrt{4-e^{-6x}}-3 e^{-3x}|$
  • Figure 3: $k\leq j\leq 2k+1$, $k=0,1,2,3,4$, or $\lfloor j/2\rfloor\leq k\leq j$ for $j=0,1,2,3,4$
  • Figure 4: $k\leq j\leq 2k$, $k=0,1,2,3,4$, or $\lceil j/2\rceil\leq k\leq j$ for $j=0,1,2,3,4$
  • Figure 5: The domain of variation of $\arg w$
  • ...and 243 more figures

Theorems & Definitions (39)

  • Theorem 1
  • Theorem 2
  • proof
  • proof
  • proof
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 29 more