Trignometric Sums And Their Applications

In its second installment, Innovative Integrals and Their Applications II explores multidimensional integral identities, unveiling powerful techniques for attacking otherwise intractable integrals, thus demanding ingenuity and novel approaches. This volume focuses on novel approaches for evaluating definite integrals, with the aid of tools such as Mathematica as a means of obtaining useful results. Building upon the previous methodologies, this volume introduces additional concepts such as interchanging the order of integration, permutation symmetry, and the use of pairs of Laplace transforms and Fourier transforms, offering readers a comprehensive array of integral identities. The content further elucidates the techniques of permutation symmetry and extends the multivariate substitution approach to integrals with finite limits of integration. These insights culminate in a collection of integral identities involving gamma functions, incomplete beta functions, Bessel functions, polylogarithms, and the Meijer G-function. Additionally, readers will encounter applications of error functions, inverse error functions, hypergeometric functions, the Lambert W-function, elliptic integrals, Jacobi elliptic functions, and the Riemann zeta function, among many others, with a focus on their relevance in various scientific disciplines and cutting-edge technologies. Each chapter in this volume concludes with many interesting exercises for the reader to practice.

This volume is dedicated to ferroelectrics and their applications. Ferroelectricity is a symptom of inevitable electrical polarization changes in materials without external electric field interference. Ferroelectricity is a phenomenon exhibited by crystals with a spontaneous polarization and hysteresis effects associated with dielectric changes when an electric field is given.

This book highlights the evolution of, and novel challenges currently facing, nanomaterials science, nanoengineering, and nanotechnology, and their applications and development in the biological and biomedical fields. It details different nanoscale and nanostructured materials syntheses, processing, characterization, and applications, and considers improvements that can be made in nanostructured materials with their different biomedical applications.

This book highlights the evolution of, and novel challenges currently facing, nanomaterials science, nanoengineering, and nanotechnology, and their applications and development in the biological and biomedical fields. It details different nanoscale and nanostructured materials syntheses, processing, characterization, and applications, and considers improvements that can be made in nanostructured materials with their different biomedical applications.