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ALTERNATIVE APPROACHES TO CLASSICAL ELEMENTARY AND ELLIPTIC FUNCTIONS

Haiduc, Florian

Abstract Details

2007, MS, Kent State University, College of Arts and Sciences / Department of Mathematical Sciences.
A careful examination of most calculus textbooks reveals a gap in the foundational treatment of the trigonometric functions; for pedagogical reasons, they rely on geometric axioms and intuition. By contrast, in a later course in real variables, only the axioms for the real numbers are allowed in the development of the trigonometric functions, although this means that higher structures (such as power series or the Fundamental Theorem of Calculus) must first be developed. In the first part of this thesis we examine five of the most common alternative definitions of the classic elementary functions (radical/power, exponential/logarithmic, and trigonometric/inverse-trigonometric functions). In particular we show how they may be developed using recurrence relations and only the basic axioms for the real numbers, and we also discuss the more commonly seen alternative developments using power series, inverse of integrals, functional equations, and differential equations. The primary idea is to show that, by starting from any of these definitions, each of the main properties of a function may be deduced. To illustrate the extension of these ideas more exotic functions (so-called special functions), in the second part of the thesis we examine various historial approaches to elliptic functions due to Abel, Gauss, Jacobi, and Weierstrass.
Morley Davidson (Advisor)

Recommended Citations

Citations

  • Haiduc, F. (2007). ALTERNATIVE APPROACHES TO CLASSICAL ELEMENTARY AND ELLIPTIC FUNCTIONS [Master's thesis, Kent State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=kent1195571229

    APA Style (7th edition)

  • Haiduc, Florian. ALTERNATIVE APPROACHES TO CLASSICAL ELEMENTARY AND ELLIPTIC FUNCTIONS. 2007. Kent State University, Master's thesis. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=kent1195571229.

    MLA Style (8th edition)

  • Haiduc, Florian. "ALTERNATIVE APPROACHES TO CLASSICAL ELEMENTARY AND ELLIPTIC FUNCTIONS." Master's thesis, Kent State University, 2007. http://rave.ohiolink.edu/etdc/view?acc_num=kent1195571229

    Chicago Manual of Style (17th edition)