1816

by Carl Edward Skeen

2003 · University Press of Kentucky

The year 1816 found America on the cusp of political, social, cultural, and economic modernity. Celebrating its fortieth year of independence, the country's sense of self was maturing. Americans, who had emerged from the War of 1812 with their political system intact, embraced new opportunities. For the first time, citizens viewed themselves not as members of a loose coalition of states but as part of a larger union. This optimism was colored, however, by bizarre weather. Periods of extreme cold and severe drought swept the northern states and the upper south throughout 1816, often referred to as The Year Without a Summer. Faced with thirty-degree summer temperatures, many farmers migrated west in search of better weather and more fertile farmlands. In 1816, historian C. Edward Skeen illuminates this unique year of national transition. Politically, the era of good feelings allowed Congress to devise programs that fostered prosperity. Social reform movements flourished. This election year found the Federalist party in its death throes, seeking cooperation with the nationalistic forces of the Republican party. Movement west, maturation of political parties, and increasingly conte

The Fractional Trigonometry

by Carl F. Lorenzo, Tom T. Hartley

2016 · John Wiley & Sons

Addresses the rapidly growing field of fractional calculus and provides simplified solutions for linear commensurate-order fractional differential equations The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science is the result of the authors’ work in fractional calculus, and more particularly, in functions for the solutions of fractional differential equations, which is fostered in the behavior of generalized exponential functions. The authors discuss how fractional trigonometry plays a role analogous to the classical trigonometry for the fractional calculus by providing solutions to linear fractional differential equations. The book begins with an introductory chapter that offers insight into the fundamentals of fractional calculus, and topical coverage is then organized in two main parts. Part One develops the definitions and theories of fractional exponentials and fractional trigonometry. Part Two provides insight into various areas of potential application within the sciences. The fractional exponential function via the fundamental fractional differential equation, the generalized exponential function, and R-function relationships are discussed in addition to the fractional hyperboletry, the R1-fractional trigonometry, the R2-fractional trigonometry, and the R3-trigonometric functions. The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science also: Presents fractional trigonometry as a tool for scientists and engineers and discusses how to apply fractional-order methods to the current toolbox of mathematical modelers Employs a mathematically clear presentation in an e ort to make the topic broadly accessible Includes solutions to linear fractional differential equations and generously features graphical forms of functions to help readers visualize the presented concepts Provides effective and efficient methods to describe complex structures The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science is an ideal reference for academic researchers, research engineers, research scientists, mathematicians, physicists, biologists, and chemists who need to apply new fractional calculus methods to a variety of disciplines. The book is also appropriate as a textbook for graduate- and PhD-level courses in fractional calculus. Carl F. Lorenzo is Distinguished Research Associate at the NASA Glenn Research Center in Cleveland, Ohio. His past positions include chief engineer of the Instrumentation and Controls Division and chief of the Advanced Controls Technology and Systems Dynamics branches at NASA. He is internationally recognized for his work in the development and application of the fractional calculus and fractional trigonometry. Tom T. Hartley, PhD, is Emeritus Professor in the Department of Electrical and Computer Engineering at The University of Akron. Dr Hartley is a recognized expert in fractional-order systems, and together with Carl Lorenzo, has solved fundamental problems in the area including Riemann’s complementary-function initialization function problem. He received his PhD in Electrical Engineering from Vanderbilt University.