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ma003 - sin x x William B Gearhart Harris S Shultz The...

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The Function William B. Gearhart Harris S. Shultz The College Mathematics Journal , March 1990, Volume 21, Number 2, pp. 90-99. William B. Gearhart received his B.S. degree in engineering physics from Cornell University, and his Ph.D degree in applied mathematics, also from Cornell University. He is currently Professor of Mathematics at the California State University, Fullerton. His research interests include approximation theory, numerical analysis, optimization theory, and mathematical modeling. Harris S. Shultz , Professor of Mathematics at California State University, Fullerton, is the author of more than forty published papers and a textbook, Mathematical Topics for Computer Instruction. He has been Co-Principal Investigator of the Southern California Mathematics Honors Institute, the Mariana Islands Mathematics Institute, and the Teacher Training Project, all funded by the National Science Foundation. In 1988 he was named Outstanding Professor at California State University, Fullerton, and in 1989 he was the recipient of the California State University Trustees’ Outstanding Professor Award. C alculus students are well aware of an important use of the function sin The common and perhaps the easiest approach to finding the derivatives of the trigonometric functions hinges on the result (1) However, few students realize that this function plays a key role in many areas of mathematics and its applications. In this paper, we briefly describe several of these roles after first presenting some elementary examples in which sin arises geometrically. These examples are accessible to calculus students and may help motivate interest in and draw attention to this function. In addition, they provide some unusual proofs of the limit (1). Properties of sin In certain fields, such as signal analysis, the function sin is often called the “sinc function.” Thus, let us set By defining sinc the function is extended to an analytic function on the real line. We shall refer to this extension also by the name sinc. The graph of sinc is shown in Figure 1, where we note that it is an even function with roots at for that for all and that lim sinc x 5 0. x ± x 0, | sinc x | < 1 ± 2, . . ., n 5 ± 1, n p , 0 5 1, sinc x 5 sin x x . x y x x / x x y x lim x 0 sin x x 5 1. x y x . C 3 sin x x
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Figure 1 The sinc function has a rapidly convergent power series representation, and even an infinite product representation [ 7 ], Also, although this integral is not absolutely convergent. However, the square of sinc is integrable on the real line with All things considered, sinc is a rather nice, well-behaved function. Properties of sinc are presented in [ 4 , sections 39–41] and [ 2 , pages 62–63]. Examples from Geometry In this section we present a few simple examples from geometry involving sin .
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