Introduction to Methods of Applied Mathematics - BGU

1 Introduction to Methods of Applied MathematicsorAdvanced Mathematical Methods for Scientists and EngineersSean seanJanuary 24, Advice to Teachers.. Acknowledgments.. Warnings and Disclaimers.. Suggested Use.. About the Title..xxviiI Algebra11 Sets and Sets.. Single Valued Functions.. Inverses and Multi-Valued Functions.. Transforming Equations.. Exercises.. Hints.. Solutions..16i2 Vectors.. Scalars and Vectors.. The Kronecker Delta and Einstein Summation Convention.. The Dot and Cross Product.. Sets of Vectors in n Dimensions.. Exercises.. Hints.. Solutions..40II Calculus473 Differential Limits of Functions.. Continuous Functions.. The Derivative.. Implicit Differentiation.. Maxima and Minima.. Mean Value Theorems.. Application: Using Taylor s Theorem to Approximate .. Application: Finite Difference Schemes.

2 L Hospital s Rule.. Exercises.. Limits of Functions.. Continuous Functions.. The Derivative.. Implicit Differentiation.. Maxima and Minima.. Mean Value Theorems.. L Hospital s Rule.. Hints.. Solutions.. Quiz.. Quiz Solutions..1144 Integral The Indefinite Integral.. The Definite Integral.. Definition.. Properties.. The Fundamental Theorem of Integral Calculus.. Techniques of Integration.. Partial Fractions.. Improper Integrals.. Exercises.. The Indefinite Integral.. The Definite Integral.. The Fundamental Theorem of Integration.. Techniques of Integration.. Improper Integrals.. Hints.. Solutions.. Quiz.. Quiz Solutions..1515 Vector Vector Functions.. Gradient, Divergence and Curl.. Exercises.. Hints.. Solutions.. Quiz.. Quiz Solutions..178 III Functions of a Complex Variable1796 Complex Complex Numbers.

3 The Complex Plane.. Polar Form.. Arithmetic and Vectors.. Integer Exponents.. Rational Exponents.. Exercises.. Hints.. Solutions..2117 Functions of a Complex Curves and Regions.. The Point at Infinity and the Stereographic Projection.. A Gentle Introduction to Branch Points.. Cartesian and Modulus-Argument Form.. Graphing Functions of a Complex Variable.. Trigonometric Functions.. Inverse Trigonometric Functions.. Riemann Surfaces.. Branch Points.. Exercises.. Hints.. Solutions..3028 Analytic Complex Derivatives.. Cauchy-Riemann Equations.. Harmonic Functions.. Singularities.. Categorization of Singularities.. Isolated and Non-Isolated Singularities.. Application: Potential Flow.. Exercises.. Hints.. Solutions..3999 Analytic Analytic Continuation.. Analytic Continuation of Sums.. Analytic Functions Defined in Terms of Real Variables.

4 Polar Coordinates.. Analytic Functions Defined in Terms of Their Real or Imaginary Parts.. Exercises.. Hints.. Solutions..45710 Contour Integration and the Cauchy-Goursat Line Integrals.. Contour Integrals.. Maximum Modulus Integral Bound.. The Cauchy-Goursat Theorem.. Contour Deformation.. Morera s .. Indefinite Integrals.. Fundamental Theorem of Calculus via Primitives.. Line Integrals and Primitives.. Contour Integrals.. Fundamental Theorem of Calculus via Complex Calculus.. Exercises..48311 Cauchy s Integral Cauchy s Integral Formula.. The Argument Theorem.. Rouche s Theorem.. Exercises.. Hints.. Solutions..51112 Series and Series of Constants.. Definitions.. Special Series.. Convergence Tests.. Uniform Convergence.. Tests for Uniform Convergence.. Uniform Convergence and Continuous .. Uniformly Convergent Power Series.

5 Integration and Differentiation of Power Series.. Taylor Series.. Newton s Binomial .. Laurent Series.. Exercises.. Series of Constants.. Uniform Convergence.. Uniformly Convergent Power Series.. Integration and Differentiation of Power Series.. Taylor Series.. Laurent Series.. Hints.. Solutions..58213 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..686viiIV Ordinary Differential Equations77214 First Order Differential Notation.. Example Problems.. Growth and Decay.. One Parameter Families of Functions.

6 Integrable Forms.. Separable Equations.. Exact Equations.. Homogeneous Coefficient Equations.. 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.. Additional Exercises.. Solutions..844viii15 First Order Linear Systems of Differential Introduction .. Using Eigenvalues and Eigenvectors to find Homogeneous Solutions.. Matrices and Jordan Canonical Form.. Using the Matrix Exponential.. Exercises.. Hints.. Solutions..87216 Theory of Linear Ordinary Differential Exact Equations.. Nature of Solutions.. Transformation to a First Order System.. The Wronskian.

7 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.. Additional Exercises.. Hints.. Solutions..92917 Techniques for Linear Differential Constant Coefficient Equations.. Second Order Equations.. Real-Valued Solutions.. Higher Order Equations.. Euler Equations.. Real-Valued Solutions.. Exact Equations.. Equations Without Explicit Dependence on y.. Reduction of Order.. *Reduction of Order and the Adjoint Equation.. Additional Exercises.. Hints.. Solutions..96018 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.

8 Exercises.. Hints..100619 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.. Transformation to a Constant Coefficient Equation.. Integral Equations.. Initial Value Problems.. Boundary Value Problems.. Exercises.. Hints.. Solutions..103520 The Dirac Delta Derivative of the Heaviside Function.. The Delta Function as a Limit.. Higher Dimensions.. Non-Rectangular Coordinate Systems.. Exercises.. Hints.. Solutions..105221 Inhomogeneous Differential Particular Solutions.. Method of Undetermined Coefficients.. Variation of Parameters.. Second Order Differential Equations.. Higher Order Differential Equations.. Piecewise Continuous Coefficients and Inhomogeneities.

9 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.. 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.. Solutions..116522 Difference Introduction .. Exact Equations.. Homogeneous First Order.. Inhomogeneous First Order.. Homogeneous Constant Coefficient Equations.. Reduction of Order.. Exercises.. Hints.. Solutions..118123 Series Solutions of Differential Ordinary Points.. Taylor Series Expansion for a Second Order Differential Equation.

10 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.. Quiz.. Quiz Solutions..124924 Asymptotic Asymptotic Relations.. Leading Order Behavior of Differential Equations.. Integration by Parts.. Asymptotic Series.. Asymptotic Expansions of Differential Equations.. The Parabolic Cylinder ..127225 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..130526 Self Adjoint Linear Adjoint Operators.. Self-Adjoint Operators.. Exercises.. Hints.. Solutions..131327 Self-Adjoint Boundary Value Summary of Adjoint Operators.