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schaum's Outline of calculus of finite differences 4e [pdf]

Schaum's Outline of Theory and Problems of Calculus of Finite Differences and Difference Equations by Muarry R. Spiegel



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Schaum's Outline of Theory and Problems of Calculus of Finite Differences and Difference Equations written by Muarry R. SpiegelPh.D. Former Profesror and Chairman Mathematics Department, Rmsd. Polytecknek Institute, Hartford Graduate Cuntre.

Schaum's Outline of Theory and Problems of Calculus of Finite Differences and Difference Equations written by Muarry R. Spiegel cover the following topics.

  • 1. THE DIFFERENCE CALCULUS 1
    Operators.
    Some Definitions Involving Operators.
    Algebra of Operators.
    The Difference Operator.
    The Translation or Shifting Operator.
    The Derivative Operator.
    The Differential Operator.
    Relationship Between Difference,
    Derivative and Differential Operators.
    General Rules of Differentiation.
    Derivatives of Special Functions.
    General Rules of the Difference Calculus.
    Factorial Polynomials.
    Stirling Numbers.
    Generalized Factorial Functions.
    Differences of Special Functions.
    Taylor Series.
    Taylor Series in Operator Form.
    The Gregory-Newton Formula.
    Leibnitz's Rule.
    Other Difference Operators.

  • 2. APPLICATIONS OF THE DIFFERENCE CALCULUS 32
    Subscript Notation. Difference Tables.
    Differences of Polynomials.
    Gregory Newton Formula in Subscript Notation.
    General Term of a Sequence or Series.
    Interpolation and Extrapolation.
    Central Difference Tables.
    Generalized
    Interpolation Formulas.
    Zig-Zag Paths and Lozenge Diagrams.
    Lagrange's vInterpolation Formula.
    Tables with Missing Entries.
    Divided Differences.
    Newton's Divided Difference Interpolation Formula.
    Inverse Interpolation.
    Approximate Differentiation.

  • 3. THE SUM CALCULUS 79
    The Integral Operator.
    General Rules of Integration.
    Integrals of Special
    Functions. Definite Integrals.
    Fundamental Theorem of Integral Calculus.
    Some Important Properties of Definite Integrals.
    Some Important Theorems of Integral Calculus.
    The Sum Operator.
    General Rules of Summation.
    Summations of Special Functions.
    Definite Sums and the Fundamental Theorem of Sum Calculus.
    Differentiation and Integration of Sums.
    Theorems on Summation Using the Subscript Notation.
    Abel's Transformation.
    Operator Methods for Summation.
    Summation of Series.
    The Gamma Function.
    Bernoulli Numbers and Polynomials.
    Important Properties of Bernoulli Numbers and Polynomials.
    Euler Numbers and Polynomials.
    Important Properties of Euler Numbers and Polynomials.

  • 4. APPLICATIONS OF THE SUM CALCULUS 121
    Some Special Methods for Exact Summation of Series. Series of Constants.
    Power Series.
    Approximate Integration.
    Error Terms in Approximate Integration Formulas.
    Gregory's Formula for Approximate Integration.
    The Euler-Maclaurin Formula.
    Error Term in Euler-Maclaurin Formula.
    Stirling's Formula for n!

  • 5. DIFFERENCE EQUATIONS 150
    Differential Equations.
    Definition of a Difference Equation.
    Order of a Difference Equation.
    Solution, General Solution and Particular Solution of a Difference Equation.
    Differential Equations as Limits of Difference Equations.
    Use of the Subscript Notation.
    Linear Difference Equations.
    Homogeneous Linear Difference Equations.
    Homogeneous Linear Difference Equations with Constant Coefficients.
    Linearly Independent Solutions.
    Solution of the Nonhomogeneous or Complete Equation.
    Methods of Finding Particular Solutions.
    Method of Undetermined Coefficients.
    Special Operator Methods.
    Method of Variation of Parameters.
    Method of Reduction of Order.
    Method of Generating Functions.
    Linear Difference Equations with Variable Coefficients.
    Sturm-Liouville Difference Equations.
    Nonlinear Difference Equations.
    Simultaneous Difference Equations.
    Mixed Difference Equations.
    Partial Difference Equations.

  • 6. APPLICATIONS OF DIFFERENCE EQUATIONS 199
    Formulation of Problems Involving Difference Equations.
    Applications to Vibrating Systems.
    Applications to Electrical Networks.
    Applications to Beams.
    Applications to Collisions.
    Applications to Probability.
    The Fibonacci Numbers.
    Miscellaneous Applications.

  • Appendix


  • A Stirling Numbers of the First Kind sk 232
    B Stirling Numbers of the Second Kind Sk 233
    C Bernoulli Numbers 234
    D Bernoulli Polynomials 235
    E Euler Numbers 236
    F Euler Polynomials 237
    G Fibonacci Numbers 238
    ANSWERS TO SUPPLEMENTARY PROBLEMS 239

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