Discrete Taylor Transform and Inverse Transform

Revolutionize the calculation of mixed derivatives with this groundbreaking text Transform and inverse transform techniques, such as the Fourier transform and the Laplace transform, enable scientists and engineers to conduct research and design in transformed domains where the work is simpler, after which the results can be converted back into the real domain where they can be applied or actualized. This latter stage in the process, the inverse transform, ordinarily poses significant challenges. New transform/inverse transform techniques carry extraordinary potential to produce revolutionary new science and engineering solutions. Discrete Taylor Transform and Inverse Transform presents the groundbreaking discovery of a new transform technique. Placing a novel emphasis on the “position variable” and “derivative operator” as main actors, the Discrete Taylor Transform and Inverse Transform (D-TTIT) will facilitate the calculation of mixed derivatives of multivariate functions to any desired order. The result promises to create new applications not only in its allied fields of quantum physics and quantum engineering, but potentially much more widely. Readers will also find: Discussion of possible applications in electrical engineering, acoustics, photonics, and many more Analysis of functions depending on one, two, or three independent variables Tools for theoreticians and practitioners to design their own algorithms for solving specific boundary-value problems Discrete Taylor Transform and Inverse Transform is ideal for any scientific or engineering professional looking to understand a cutting-edge research and design tool.

About the Author xv Preface xvii Introduction 1 1 Toy Model I-1: {−Δ, 0, Δ} 19 2 Toy Model I-2:{0, Δ, 2Δ} 31 3 Toy Model I-3: {−2Δ, −Δ, 0} 39 4 Toy Model I-4: {−Δ, 0, Δ} 47 5 Toy Model I-5: {−2Δ, −Δ, 0, Δ, 2Δ} 59 6 Toy Model I-7: {−3Δ, −2Δ, −Δ, 0, Δ, 2Δ, 3Δ} 79 7 Self-consistent Expressions for |D (n) > 111 8 Toy Model I-3: {Δ −1 , Δ 0 , Δ 1 } 125 9 Toy Model I-5: {Δ −2 , Δ −1 , Δ 0 , Δ 1 , Δ 2 } 165 10 Toy Model I-6: {Δ −3 , Δ −2 , Δ −1 , Δ 0 , Δ 1 , Δ 2 , Δ 3 } 207 11 Toy Model I-7: {Δ −3 , Δ −2 , Δ −1 , Δ 0 , Δ 1 , Δ 2 , Δ 3 } 231 12 Toy Model II: {{−Δ 1 , 0, Δ 1 }, {−Δ 2 , 0, Δ 2 }} 283 13 Toy Model III: {−Δ 1 , Δ 1 }×{−Δ 2 , Δ 2 }×{−Δ 3 , Δ 3 } 317 14 Solidification and Further Refinements 527 Appendix A The Canonical Matrix C 3×3 and Its Inverse 609 Appendix B The Canonical Matrix C 3×3 and Its Inverse Revisited 615 Appendix C The Canonical Matrix C 4×4 and Its Inverse 621 Appendix D The Canonical Matrix C 5×5 635 Appendix E The Canonical Matrix C 7×7 643 Index 657