# Open Math Notes

## Resources and inspiration for math instruction and learning

Welcome to AMS Open Math Notes, a repository of freely downloadable mathematical works hosted by the American Mathematical Society as a service to researchers, faculty and students. Open Math Notes includes:

• Draft works including course notes, textbooks, and research expositions. These have not been published elsewhere and are subject to revision.
• Items previously published in the Journal of Inquiry-Based Learning in Mathematics, a refereed journal
• Refereed publications at the AMS

Visitors are encouraged to download and use any of these materials as teaching and research aids, and to send constructive comments and suggestions to the authors.

## Open Math Notes Advisory Board:

• Karen Vogtmann, Chair | University of Warwick
• Tom Halverson | Macalester College
• Andrew Hwang | College of the Holy Cross
• Robert Lazarsfeld | Stony Brook University
• Mary Pugh | University of Toronto

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Showing 1 - 5 of 219 result(s)
 Lectures on Modular Forms and Hecke Operators Much enhanced notes by the second author of a course given by the first author in 1996 Kenneth Ribet · University of California, Berkeley · William Stein · CoCalc · Date posted: November 16, 2022 Send feedback to the author(s) Numerical Mathematics II: Numerical Solution of Ordinary Differential Equations We study explicit and implicit methods for initial value problems.For general Runge-Kutta methods, Butcher's theory of rooted treesis presented in detail, proving Butcher's theorem on conditions for high orders of consistency and convergence. Further topics includecollocation methods, stability theory, stiff equations, asymptotic error expansions, extrapolation methods, and stepsize control. Peter Philip · LMU Munich · Date posted: October 31, 2022 Send feedback to the author(s) Vector Fields and Differential Forms Vector fields and differential forms have very different properties. However a given volume element allows vector fields to be converted into twisted differential (n-1)-forms via contraction. Also a metric tensor allows vector fields to be converted into 1-forms. The ultimate theory works with an arbitrary metric tensor in n dimensions. The exposition includes insightful pictures along with extensive computations. William Faris · University of Arizona · Date posted: October 10, 2022 Send feedback to the author(s) Introduction to Functional Analysis Functional analysis is one of the important fields of mathematics. The focus of the first part of this book is on the presentation of the fundamental concepts in functional analysis: Banach space, linear operator, Hilbert space and dual space. The emphasis in the second part is on the main theorems in functional analysis and the third part is devoted to compact operators which play a crucial role in the theory of integral equations and partial differential equations and in mathematical physics. Len Meas · Royal University of Phnom Penh · Date posted: September 11, 2022 Send feedback to the author(s) Complex Variables and Functions These notes cover standard undergraduate material from the theory of complex-valued functions, including the Cauchy–Riemann equations, Cauchy’s Theorem, the residue theorem, power series and Laurent series, the classification of singularities, and the Riemann sphere. Students are assumed to have studied multivariable calculus and to have some familiarity with set theory and proofs, but no prior experience in analysis is required. Some historical and cultural background is included in footnotes. Joshua Bowman · Pepperdine University · Date posted: August 23, 2022 · Date revised: September 5, 2022 Send feedback to the author(s)

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Showing 1 - 5 of 219 result(s)