Generalized Sine Functions, Complexified

SummaryGeneralized sine functions, primarily viewed as real-valued functions, are studied herein as complex analytic functions. Their natural domains of analyticity are identified and their mapping properties are detailed. Some implications of this theory for calculus and geometry are explored.MSC:: 30-02 Notes* For z near 1, factor (1−z4)3/4 into g(z)(z−1)3/4 where g is continuous and nonzero at 1. Thus, it suffices to track the phase change of (z−1)3/4 as z traverses an arc in V¯4 from 1−ϵ to 1+ϵ.* The Schwarz reflection principle also guarantees that the extended function is analytic on [O,B4).* At the other half of the vertices, i.e., the translates of the ω3kA3’s, the extended sin 3 is bounded and hence analytic by Riemann’s principle of removable singularity.Additional informationNotes on contributorsPisheng DingPISHENG DING (MR Author ID: 784635) studied physics and mathematics as an undergraduate at the legendary City College of New York, where Joseph Bak kindled in him a keen interest in complex analysis. He earned his doctorate in 2003 from the Courant Institute under the direction of Sylvain Cappell. So that his family could be under one roof, he joined Illinois State University in 2010 as an adjunct and has since remained in this position. While deploring the state of mathematics education in the United States, he currently enjoys teaching his 6th-grade daughter Elaine authentic Euclidean geometry.Sunil K. CheboluSUNIL CHEBOLU (MR Author ID: 781874) received his Ph.D. in mathematics from the University of Washington in 2005. After completing a three-year postdoctoral fellowship at the University of Western Ontario in Canada, he joined the faculty at Illinois State University in 2008. Although his primary research interests lie in algebra and number theory, he embraces all areas of mathematics. During his spare time, he enjoys playing his guitar or observing deep-sky objects through his telescope.