Introduction to Methods of Applied Mathematics

1 Introduction to Methods of Applied MathematicsorAdvanced Mathematical Methods for Scientists and EngineersSean MauchApril 26, 2001 Contents1 Anti-Copyright222 Advice to Teachers.. Acknowledgments.. Warnings and Disclaimers.. Suggested Use.. About the Title..25I Algebra13 Sets and Sets.. Single Valued Functions.. Inverses and Multi-Valued Functions.. Transforming Equations..94 Vectors.. and Vectors.. Kronecker Delta and Einstein Summation Convention.. Dot and Cross Product.. Sets of Vectors in n Dimensions.. Exercises.. Hints.. Solutions..29II Calculus365 Differential Limits of Functions.. Continuous Functions.. The Derivative.. Implicit Differentiation.. Maxima and Minima.. Mean Value Theorems.. : Using Taylor s Theorem to Approximate .. : Finite Difference Schemes.. L Hospital s Rule.

2 Exercises.. Hints.. Solutions..816 Integral The Indefinite Integral.. The Definite Integral.. The Fundamental Theorem of Integral Calculus.. Techniques of Integration.. Fractions.. Improper Integrals.. Exercises.. Hints.. Solutions..1257 Vector Vector Functions.. Gradient, Divergence and Curl.. Exercises.. Hints.. Solutions..145 III Functions of a Complex Variable1508 Complex Complex Numbers.. The Complex Plane.. Polar Form.. Arithmetic and Vectors.. Integer Exponents.. Rational Exponents.. Exercises.. Hints.. Solutions..17939 Functions of a Complex Curves and Regions.. Cartesian and Modulus-Argument Form.. Graphing Functions of a Complex Variable.. Trigonometric Functions.. Inverse Trigonometric Functions.. Branch Points.. Exercises.. Hints.. Solutions.

3 25810 Analytic Complex Derivatives.. Cauchy-Riemann Equations.. Harmonic Functions.. Singularities.. Categorization of Singularities.. Isolated and Non-Isolated Singularities.. Exercises.. Hints.. Solutions..33411 Analytic Analytic Continuation.. Analytic Continuation of Sums.. Analytic Functions Defined in Terms of Real Variables.. Polar Coordinates.. Analytic Functions Defined in Terms of Their Real or Imaginary Parts.. Exercises.. Hints.. Solutions..37612 Contour Integration and Cauchy s Line Integrals.. Under Construction.. Cauchy s Theorem.. Indefinite Integrals.. Contour Integrals.. Exercises.. Hints.. Solutions..39813 Cauchy s Integral Cauchy s Integral Formula.. The Argument Theorem.. Rouche s Theorem.. Exercises.. Hints.. Solutions..41814 Series and Series of Constants.. Definitions.. Special Series.. Convergence Tests.

4 Uniform Convergence.. Tests for Uniform Convergence.. Uniform Convergence and Continuous .. Uniformly Convergent Power Series.. Integration and Differentiation of Power Series.. Taylor Series.. Newton s Binomial .. Laurent Series.. Exercises.. Hints.. Solutions..46515 The Residue The Residue Theorem.. Cauchy Principal Value for Real Integrals.. The Cauchy Principal Value.. Cauchy Principal Value for Contour Integrals.. Integrals on the Real Axis.. Fourier Integrals.. Fourier Cosine and Sine Integrals.. Contour Integration and Branch Cuts.. Exploiting Symmetry.. Wedge Contours.. Box Contours.. Definite Integrals Involving Sine and Cosine.. Sums..553IV Ordinary Differential Equations63416 First Order Differential Notation.. One Parameter Families of Functions.. Exact Equations.. Separable Equations.. Homogeneous Coefficient Equations.

5 The First Order, Linear Differential Equation.. Homogeneous Equations.. Inhomogeneous Equations.. Variation of .. Initial Conditions.. Piecewise Continuous Coefficients and Inhomogeneities.. Well-Posed Problems.. Equations in the Complex Plane.. Ordinary Points.. Regular Singular Points.. Irregular Singular Points.. The Point at Infinity.. Exercises.. Hints..68317 First Order Systems of Differential Matrices and Jordan Canonical Form.. Systems of Differential Equations.. Exercises.. Hints.. Solutions..72718 Theory of Linear Ordinary Differential Nature of Solutions.. Transformation to a First Order System.. The Wronskian.. Derivative of a .. The Wronskian of a Set of .. The Wronskian of the Solutions to a Differential Equation.. Well-Posed Problems.. The Fundamental Set of Solutions.. Adjoint Equations.. Exercises.. Hints.

6 Solutions..78019 Techniques for Linear Differential Constant Coefficient Equations.. Second Order Equations.. Higher Order Equations.. Real-Valued Solutions.. Euler Equations.. Real-Valued Solutions.. Exact Equations.. Equations Without Explicit Dependence on y.. Reduction of Order.. *Reduction of Order and the Adjoint Equation.. Exercises.. Hints.. Solutions..81720 Techniques for Nonlinear Differential Bernoulli Equations.. Riccati Equations.. Exchanging the Dependent and Independent Variables.. Autonomous Equations.. *Equidimensional-in-x Equations.. *Equidimensional-in-y Equations.. *Scale-Invariant Equations.. Exercises.. Hints..86621 Transformations and Canonical The Constant Coefficient Equation.. Normal Form.. Second Order Equations.. Higher Order Differential Equations.. Transformations of the Independent Variable.. Transformation to the form u + a(x) u = 0.

7 Transformation to a Constant Coefficient Equation.. Integral Equations.. Initial Value Problems.. Boundary Value Problems.. Exercises.. Hints.. Solutions..89722 The Dirac Delta Derivative of the Heaviside Function.. The Delta Function as a Limit.. Higher Dimensions.. Non-Rectangular Coordinate Systems.. Exercises.. Hints.. Solutions..91223 Inhomogeneous Differential Particular Solutions.. Method of Undetermined Coefficients.. Variation of Parameters.. Second Order Differential Equations.. Higher Order Differential Equations.. Piecewise Continuous Coefficients and Inhomogeneities.. Inhomogeneous Boundary Conditions.. Eliminating Inhomogeneous Boundary Conditions.. Separating Inhomogeneous Equations and Inhomogeneous Boundary Conditions.. Existence of Solutions of Problems with Inhomogeneous Boundary Conditions.. Green Functions for First Order Equations.

8 Green Functions for Second Order Equations.. Green Functions for Sturm-Liouville Problems.. Initial Value Problems.. Problems with Unmixed Boundary Conditions.. Problems with Mixed Boundary Conditions.. Green Functions for Higher Order Problems.. Fredholm Alternative Theorem..98624 Difference Introduction .. Exact Equations.. Homogeneous First Order.. Inhomogeneous First Order.. Homogeneous Constant Coefficient Equations.. Reduction of Order.. Exercises.. Hints.. Solutions..104225 Series Solutions of Differential Ordinary Points.. Taylor Series Expansion for a Second Order Differential Equation.. Regular Singular Points of Second Order Equations.. Indicial Equation.. The Case: Double Root.. The Case: Roots Differ by an Integer.. Irregular Singular Points.. The Point at Infinity.. Exercises.. Hints.. Solutions..108926 Asymptotic Asymptotic Relations.

9 Leading Order Behavior of Differential Equations.. Integration by Parts.. Asymptotic Series.. Asymptotic Expansions of Differential Equations.. The Parabolic Cylinder ..113527 Hilbert Linear Spaces.. Inner Products.. Norms.. Linear .. Orthogonality.. Gramm-Schmidt Orthogonalization.. Orthonormal Function Expansion.. Sets Of Functions.. Least Squares Fit to a Function and Completeness.. Relation.. Operators..116928 Self Adjoint Linear Adjoint Operators.. Self-Adjoint Operators.. Exercises.. Hints.. Solutions..117729 Self-Adjoint Boundary Value Summary of Adjoint Operators.. Formally Self-Adjoint Operators.. Self-Adjoint Problems.. Self-Adjoint Eigenvalue Problems.. Inhomogeneous Equations.. Exercises.. Hints.. Solutions..11931230 Fourier An Eigenvalue .. Fourier .. Least Squares Fit.. Fourier Series for Functions Defined on Arbitrary Ranges.

10 Fourier Cosine Series.. Fourier Sine Series.. Complex Fourier Series and Parseval s Theorem.. Behavior of Fourier Coefficients.. Gibb s Phenomenon.. and Differentiating Fourier Series..124131 Regular Sturm-Liouville Derivation of the Sturm-Liouville Form.. Properties of Regular Sturm-Liouville Problems.. Solving Differential Equations With Eigenfunction Expansions.. Exercises.. Hints.. Solutions..131732 Integrals and Uniform Convergence of Integrals.. The Riemann-Lebesgue Lemma.. Cauchy Principal Value.. Integrals on an Infinite Domain.. Singular Functions..13451333 The Laplace The Laplace Transform.. The Inverse Laplace Transform.. F(s) with Poles.. f(s) with Branch Points.. Asymptotic Behavior of F(s).. Properties of the Laplace Transform.. Constant Coefficient Differential Equations.. Systems of Constant Coefficient Differential Equations.