Advanced Engineering Mathematics

Incluye contenido y apéndice

1. Determinants and Matrices. -- Determinants. -- Matrices. -- Systems of Linear Equations. -- Characteristic-value Problems. -- 2. Ordinary Differential Equations of the First Order -- Introduction. -- Fundamental Definitions. -- Separable First-order Equations. -- Applications of First-order Differential Equations. -- 3. Linear Diferential Equations with Constant Cofficients -- The General Linear Second-order Equation. -- The Homogeneous Linear Equation with Constant Coefficients. -- The Nonhomogeneous Equation. -- Particular integrals by the Method of Variation of Parameters. -- Equations of Higher Order. -- Applications. -- 4. Simultaneous Linear Differential Equations -- Introduction. -- The Reduction of a System to a Single Equation. -- Complementary Functions and Particular Integrals for Systems of Equations. -- 5. Finite Differences -- The Differences of a Function. -- Interpolation Formulas. -- Numerical Differentiation and Integration. -- The Numerical Solution of Differential Equations. -- Difference Equations. -- The Method of Least Squares. -- 6. Mechanical and Electrical Circuits -- Introduction. -- Systems with One Degree of Freedom. -- The Translational-mechanical System. -- The Series-electrical Circuit. -- Systems with Several Degrees of Fredom. -- Electromechanical Analogies. -- 7. Fourier Series and Integrals. -- Introduction. -- The Euler Coefficients. -- Change of Interval. -- Half-range Expansions. -- Alternative Forms of Fourier Series. -- Applications. -- Harmonic Analysis. -- The Fourier Integral as the Limit of a Fourier Series. -- From the Fourier Integral to the Laplace Transform. -- 8. The Laplace Transformation -- Theoretical Preliminaries. -- The General Method. -- The Transforms of Special Functions. -- Further General Theorems. -- The Heaviside Expansion Teorems. -- Transforms of Periodic Functions. -- Convolution and the Duhamel Formulas. -- 9. Partial Differential Equations -- Introduction. -- The Derivation of Equations. -- The D'Alembert Solution of the Wave Equation. -- Separation of Variables. -- Orthogonal Functions and the General Expansion Problem. -- Further Applications. -- Laplace Transform Methods. -- 10. Bessel Functions and Legendre Polynomials -- Theoretical Preliminaries. -- The Series Solution of Bessel's Equation. -- Modified Bessel Functions. -- Equations Reducible to Bessel's Equation. -- Identities for the Bessel Functions. -- Orthogonality of the Bessel Functions. -- Applications of Bessel Functions. -- Legendre Polynomials. -- 11. Vector Analysis. -- The Algebra of Vectors. -- Vector Functions of One Variable. -- The Operator V. -- Line, Surface, and Volume Integrals. -- Integral Theorems. -- Further Applications. -- 12. Analytic Functions of a Complex Variable -- Introduction. -- Algebraic Preliminaries. -- The Geometric Representation of Complex Numbers. -- Absolute Values. -- Functions of a Complex Variable. -- Analytic Functions. -- The Elementary Functions of z. -- Integration in the Complex Plane. -- 13. Infinite Series in the Complex Plane -- Series of Complex Terms. -- Taylor's Expansion. -- Laurent's Expansion. -- 14. The Theory of Residues -- The Residue Theorem. -- The Evaluation of Real Definite Integrals. -- The Complex Inversion Integral. -- Stability Criteria. -- 15. Conformal Mapping. -- The Geometrical Representation of Functions of z. -- Conformal Mapping. -- The Bilinear Transformation. -- The Schwarz-Christoffel Transformation.