## Applied MathematicsThis 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. |

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### Contents

Determinants | 1 |

11 Cramers Rules | 3 |

12 Gaussian Elimination | 4 |

13 Special Determinants | 7 |

Matrices | 13 |

21 Several Theorems | 14 |

22 Linear Equations | 15 |

23 Inverse of a Matrix | 17 |

653 Mapping | 172 |

Markov Averaging | 177 |

72 Speckle | 183 |

73 Inhomogeneous Broadening | 188 |

Fourier Transforms | 195 |

812 Half Space | 197 |

813 Finite Systems in 1D | 199 |

82 Laplace Transforms | 202 |

24 Eigenvalues and Eigenvectors | 18 |

25 Unitary Transformations | 23 |

26 NonHermitian Matrices | 25 |

27 A Special Matrix | 32 |

28 GramSchmidt | 34 |

29 Chains | 38 |

Group Theory | 47 |

32 Group Representations | 51 |

33 Characters | 54 |

34 Direct Product Groups | 58 |

35 Basis Functions | 60 |

36 Angular Momentum | 61 |

37 Products of Representations | 64 |

382 Representations | 65 |

383 Matrix Elements | 67 |

39 Double Groups | 68 |

Complex Variables | 73 |

42 Analytic Functions | 75 |

43 Multivalued Functions | 80 |

44 Contour Integrals | 84 |

45 Meromorphic Functions | 99 |

46 Higher Poles | 100 |

47 Integrals Involving Branch Cuts | 101 |

48 Approximate Evaluation of Integrals | 108 |

481 Steepest Descent | 109 |

482 Saddle Point Integrals | 110 |

Series | 119 |

52 Convergence | 121 |

53 Laurent Series | 126 |

54 Meromorphic Functions | 128 |

55 Asymptotic Series | 129 |

56 Summing Series | 132 |

57 Fade Approximants | 135 |

Conformal Mapping | 141 |

62 Mapping | 144 |

63 Examples | 149 |

64 SchwartzChristoffel Transformations | 157 |

65 van der Pauw | 167 |

651 Currents | 168 |

652 Resistance | 170 |

83 Wavelets | 205 |

831 Continuous Wavelet Transform | 207 |

832 Discrete Transforms | 210 |

Equations of Physics | 217 |

91 Boundary and Initial Conditions | 218 |

921 Moment Equations | 219 |

922 Diffusion Equations | 223 |

93 Solving Differential Equations | 225 |

932 Inhomogeneous Linear Equations | 227 |

933 Nonlinear Equations | 229 |

94 Elliptic Integrals | 235 |

One Dimension | 237 |

102 Diffusion Equation | 240 |

103 Wave Equation | 252 |

Two Dimensions | 265 |

1112 Diffusion Equation | 268 |

1113 Wave Equation | 272 |

112 Polar Coordinates | 273 |

1121 Laplaces Equation | 274 |

1122 Helmholtz Equation | 279 |

1123 Hankel Transforms | 286 |

Three Dimensions | 297 |

122 Cylindrical Coordinates | 299 |

123 Spherical Coordinates | 305 |

1231 Laplaces Equation | 306 |

1232 Diffusion and Wave Equations | 309 |

124 Problems Inside a Sphere | 315 |

125 Vector Wave Equation | 319 |

1252 Boundary Conditions | 320 |

Odds and Ends | 333 |

1311 Continued Fractions | 337 |

1312 Solving Equations With Series | 339 |

132 Orthogonal Polynomials | 343 |

1321 Parabolic Cylinder Functions | 344 |

1323 Laguerre Polynomial | 346 |

133 SturmLiouville Theory | 347 |

134 Greens Functions | 352 |

135 Singular Integral Equations | 357 |

365 | |

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