Applied Mathematics

This volume is a textbook for a year-long graduate level course in All research universities have applied mathematics for scientists and engineers. such a course, which could be taught in different departments, such as mathematics, physics, or engineering. I volunteered to teach this course when I realized that my own research students did not learn much in this course at my university. Then I learned that the available textbooks were too introduc­ tory. While teaching this course without an assigned text, I wrote up my lecture notes and gave them to the students. This textbook is a result of that endeavor. When I took this course many, many, years ago, the primary references were the two volumes of P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953). The present text returns the contents to a similar level, although the syllabus is quite different than given in this venerable pair of books.

1. Determinants.- 1.1. Cramer’s Rule.- 1.2. Gaussian Elimination.- 1.3. Special Determinants.- 2. Matrices.- 2.1. Several Theorems.- 2.2. Linear Equations.- 2.3. Inverse of a Matrix.- 2.4. Eigenvalues and Eigenvectors.- 2.5. Unitary Transformations.- 2.6. Hon-Hermitian Matrices.- 2.7. A Special Matrix.- 2.8. Gram-Schmidt.- 2.9. Chains.- 3. Group Theory.- 3.1. Basic Properties of Groups.- 3.2. Group Representations.- 3.3. Characters.- 3.4. Direct Product Groups.- 3.5. Basis Functions.- 3.6. Angular Momentum.- 3.7. Products of Representations.- 3.8. Quantum Mechanics.- 3.9. Double Groups.- 4. Complex Variables.- 4.1. Introduction.- 4.2. Analytic Functions.- 4.3. Multivalued Functions.- 4.4. Contour Integrals.- 4.5. Meromorphic Functions.- 4.6. Higher Poles.- 4.7. Integrals Involving Branch Cuts.- 4.8. Approximate Evaluation of Integrals.- 5. Series.- 5.1. Taylor Series.- 5.2. Convergence.- 5.3. Laurent Series.- 5.4. Meromorphic Functions.- 5.5. Asymptotic Series.- 5.6. Summing Series.- 5.7. Padé Approximants.- 6. Conformal Mapping.- 6.1. Laplace’s Equation.- 6.2. Mapping.- 6.3. Examples.- 6.4. Schwartz-Christoffel Transformations.- 6.5. van der Pauw.- 7. Markov Averaging.- 7.1. Random Walk.- 7.2. Speckle.- 7.3. Inhomogeneous Broadening.- 8. Fourier Transforms.- 8.1. Fourier Transforms.- 8.2. Laplace Transforms.- 8.3. Wavelets.- 9. Equations of Physics.- 9.1. Boundary and Initial Conditions.- 9.2. Boltzmann Equation.- 9.3. Solving Differential Equations.- 9.4. Elliptic Integrals.- 10. One Dimension.- 10.1. Introduction.- 10.2. Diffusion Equation.- 10.3. Wave Equation.- 11. Two Dimensions.- 11.1. Rectangular Coordinates.- 11.2. Polar Coordinates.- 12. Three Dimensions.- 12.1. Cartesian Coordinates.- 12.2. Cylindrical Coordinates.- 12.3. Spherical Coordinates.- 12.4.Problems Inside a Sphere.- 12.5. Vector Wave Equation.- 13. Odds and Ends.- 13.1. Hypergeometric Functions.- 13.2. Orthogonal Polynomials.- 13.3. Sturm-Liouville.- 13.4. Green’s Functions.- 13.5. Singular Integral Equations.