Chapter 1: Mathematical PreliminariesSection 1-1:
Infinite Series
Section 1-2:
Series of FunctionsSection 1-3:
Binomial TheoremSection 1-4:
Mathematical InductionSection 1-5:
Operations on Series Expansions of FunctionsSection 1-6:
Some Important SeriesSection 1-7:
Vectors
Section 1-8:
Complex Numbers and Functions
Section 1-9:
Derivatives and ExtremaSection 1-10:
Evaluation of Integrals
Section 1-11:
Dirac Delta Function
Exercise 2
Exercise 4
Exercise 8
Exercise 9
Exercise 12
Exercise 13
Exercise 14
Exercise 15
Exercise 16
Chapter 2: Determinants and MatricesSection 2-1:
Determinants
Section 2-2:
Matrices
Chapter 3: Vector AnalysisSection 3-2:
Vectors in 3-D Space
Section 3-3:
Coordinate Transformations
Section 3-4:
Rotations in R3Section 3-5:
Differential Vector OperatorsSection 3-6:
Differential Vector Operators: Further PropertiesSection 3-7:
Vector IntegrationSection 3-8:
Integral TheoremsSection 3-9:
Potential Theory
Section 3-10:
Curvilinear CoordinatesExercise 3
Exercise 6
Exercise 8
Exercise 11
Exercise 14
Chapter 4: Tensors and Differential FormsCOMING SOON
Section 5-1:
Vectors in Function SpacesSection 5-2:
Gram-Schmidt Orthogonalization
Section 5-3:
OperatorsSection 5-4:
Self-Adjoint OperatorsSection 5-5:
Unitary OperatorsSection 5-6:
Transformations of OperatorsSection 5-7:
InvariantsExercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Chapter 6: Eigenvalue ProblemsSection 6-2:
Matrix Eigenvalue Problems
Section 6-4:
Hermitian Matrix Diagonalization
Section 6-5:
Normal Matrices
Chapter 7: Ordinary Differential EquationsSection 7-2:
First-Order Equations
Section 7-3:
ODEs with Constant Coefficients
Section 7-4:
Second-Order Linear ODEsSection 7-5:
Series Solutions—Frobenius' MethodSection 7-6:
Other SolutionsSection 7-7:
Inhomogeneous Linear ODEsSection 7-8:
Nonlinear Differential EquationsExercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 11
Exercise 13
Exercise 14
Exercise 15
Exercise 16
Exercise 17
Exercise 18
Chapter 8: Sturm-Liouville TheorySection 8-2:
Hermitian Operators
Section 8-3:
ODE Eigenvalue Problems
Section 8-4:
Variation Method
Exercise 3
Exercise 5
Exercise 8
Exercise 10
Chapter 9: Partial Differential EquationsSection 9-2:
First-Order Equations
Section 9-3:
Second-Order EquationsSection 9-4:
Separation of Variables
Section 9-5:
Laplace and Poisson EquationsSection 9-6:
Wave EquationSection 9-7:
Heat-Flow, or Diffusion PDEExercise 3
Exercise 4
Exercise 5
Exercise 6
Chapter 10: Green's FunctionsCOMING SOON
Chapter 11: Complex Variable TheorySection 11-2:
Cauchy-Riemann Conditions
Section 11-3:
Cauchy's Integral TheoremSection 11-4:
Cauchy's Integral FormulaSection 11-5:
Laurent ExpansionSection 11-6:
SingularitiesSection 11-7:
Calculus of ResiduesSection 11-8:
Evaluation of Definite Integrals
Section 11-9:
Evaluation of SumsSection 11-10:
Miscellaneous TopicsExercise 2
Exercise 4
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 11
Exercise 12
Chapter 12: Further Topics in AnalysisCOMING SOON
Chapter 13: Gamma FunctionSection 13-1:
Definitions, Properties
Section 13-2:
Digamma and Polygamma FunctionsSection 13-3:
The Beta FunctionSection 13-4:
Stirling's SeriesSection 13-5:
Riemann Zeta FunctionSection 13-6:
Other Related FunctionsExercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 11
Exercise 12
Exercise 13
Exercise 14
Exercise 15
Exercise 16
Exercise 17
Exercise 18
Exercise 19
Exercise 20
Exercise 21
Exercise 22
Exercise 23
Chapter 14: Bessel FunctionsSection 14-1:
Bessel Functions of the First Kind, Jv (x)
Section 14-2:
OrthogonalitySection 14-3:
Neumann Functions, Bessel Functions of The Second KindSection 14-4:
Hankel FunctionsSection 14-5:
Modified Bessel Functions, Iv (x) and Kv (x)Section 14-6:
Asymptotic ExpansionsSection 14-7:
Spherical Bessel FunctionsExercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 11
Exercise 12
Exercise 13
Exercise 14
Exercise 15
Exercise 16
Exercise 17
Exercise 18
Exercise 19
Exercise 20
Exercise 21
Exercise 22
Exercise 23
Exercise 24
Exercise 25
Exercise 26
Exercise 27
Exercise 28
Exercise 29
Chapter 15: Legendre FunctionsCOMING SOON
Chapter 16: Angular MomentumCOMING SOON
Section 17-1:
Introduction to Group Theory
Section 17-2:
Representation of GroupsSection 17-3:
Symmetry and PhysicsSection 17-4:
Discrete GroupsSection 17-5:
Direct ProductsSection 17-6:
Symmetric GroupSection 17-7:
Continuous GroupsSection 17-8:
Lorentz GroupSection 17-9:
Lorentz Covariance of Maxwell's EquationsExercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Chapter 18: More Special FunctionsSection 18-1:
Hermite FunctionsSection 18-2:
Applications of Hermite Functions
Section 18-3:
Laguerre FunctionsSection 18-4:
Chebyshev PolynomialsSection 18-5:
Hypergeometric FunctionsSection 18-6:
Confluent Hypergeometric FunctionsSection 18-7:
DilogarithmSection 18-8:
Elliptic IntegralsExercise 1
Exercise 2
Exercise 3
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Chapter 19: Fourier SeriesSection 19-1:
General Properties
Section 19-2:
Applications of Fourier Series
Section 19-3:
Gibbs PhenomenonExercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 11
Exercise 12
Exercise 13
Exercise 14
Exercise 15
Exercise 16
Chapter 20: Integral TransformsSection 20-2:
Fourier Transform
Section 20-3:
Properties of Fourier TransformsSection 20-4:
Fourier Convolution TheoremSection 20-5:
Signal-Processing ApplicationsSection 20-6:
Discrete Fourier TransformSection 20-7:
Laplace TransformsSection 20-8:
Properties of Laplace TransformsSection 20-9:
Laplace Convolution TheoremSection 20-10:
Inverse Laplace TransformExercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 9
Exercise 10
Exercise 11
Exercise 12
Exercise 13
Exercise 14
Exercise 15
Exercise 16
Chapter 21: Integral EquationsCOMING SOON
Chapter 22: Calculus of VariationsSection 22-1:
Euler Equation
Section 22-2:
More General Variations
Section 22-3:
Constrained Minima/MaximaSection 22-4:
Variation with ConstraintsExercise 1
Exercise 2
Exercise 3
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 11
Exercise 13
Exercise 14
Exercise 15
Chapter 23: Probability and StatisticsSection 23-1:
Probability: Definitions, Simple Properties
Section 23-2:
Random VariablesSection 23-3:
Binomial DistributionSection 23-4:
Poisson DistributionSection 23-5:
Gauss' Normal DistributionSection 23-6:
Transformations of Random VariablesSection 23-7:
StatisticsExercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
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