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Introduction, General Commands | |

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Ordinary Differential Equations (ODE's) | |

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First-Order ODE's | |

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General Solutions | |

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Direction Fields | |

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Mixing Problems | |

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Integrating Factors | |

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Bernoulli's Equation | |

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RL-Circuit | |

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Problems for Chapter 1 | |

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Linear ODE's of Second and Higher Order | |

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General Solution. Initial Value Problem | |

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Mass-Spring System. Complex Roots. Damped Oscillations | |

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The Three Cases of Damping | |

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The Three Cases for an Euler-Cauchy Equation | |

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Wronskian | |

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Nonhomogeneous Linear ODE's | |

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Solution by Undetermined Coefficients | |

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Solution by Variation of Parameters | |

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Forced Vibrations. Resonance. Beats | |

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RLC-Circuit | |

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Problems for Chapter 2 | |

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Systems of Differential Equations. Phase Plane, Qualitative Methods | |

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Solving a System of ODE's by DSolve. Initial Value Problem | |

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Use of Matrices in Solving Systems of ODE's | |

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Critical Points. Node | |

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Proper Node, Saddle Point, Center, Spiral Point | |

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Pendulum Equation | |

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Nonhomogeneous System | |

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Method of Undetermined Coefficients | |

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Problems for Chapter 3 | |

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Series Solutions of Differential Equations | |

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Power Series Solutions. Plots from Them. Numerical Values | |

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Legendre Polynomials | |

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Legendre's Differential Equation | |

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Orthogonality. Fourier-Legendre Series | |

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Frobenius Method | |

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Bessel's Equation. Bessel Functions | |

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Problems for Chapter 4 | |

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Laplace Transform Method for Solving ODE's | |

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Transforms and Inverse Transforms | |

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Differential Equations | |

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Forced Vibrations. Resonance | |

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Unit Step Function (Heaviside Function), Dirac's Delta | |

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Solution of Systems by Laplace Transform | |

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Formulas on General Properties of the Laplace Transform | |

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Problems for Chapter 5 | |

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Linear Algebra, Vector Calculus | |

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Matrices, Vectors, Determinants. Linear Systems of Equations | |

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Matrix Addition, Scalar Multiplication, Matrix Multiplication. Vectors | |

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Special Matrices | |

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Changing and Composing Matrices, Accessing Entries. Submatrices | |

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Solution of a Linear System | |

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Linear Systems: A Further Case | |

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Gauss Elimination; Back Substitution | |

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Problems for Chapter 6 | |

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Matrix Eigenvalue Problems | |

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Eigenvalues, Eigenvectors, Accessing Spectrum | |

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Real Matrices with Complex Eigenvalues | |

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Orthogonal Matrices and Transformations | |

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Complex Matrices | |

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Similarity of Matrices. Diagonalization | |

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Problems for Chapter 7 | |

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Vectors in R[superscript 2] and R[superscript 3]. Dot and Cross Products. Grad, Div, Curl | |

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Vectors, Addition, Scalar Multiplication | |

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Inner Product. Cross Product | |

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Differentiation of Vectors. Curves and their Properties | |

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Gradient. Directional Derivative. Potential | |

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Divergence, Laplacian, Curl | |

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Problems for Chapter 8 | |

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Vector Integral Calculus. Integral Theorems | |

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Line Integrals | |

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Independence of Path | |

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Double Integrals. Moments of Inertia | |

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Green's Theorem in the Plane | |

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Surface Integrals. Flux | |

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Divergence Theorem of Gauss | |

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Stokes's Theorem | |

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Problems for Chapter 9 | |

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Fourier Analysis and Partial Differential Equations | |

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Fourier Series, Integrals, and Transforms | |

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Functions of Period 2[pi]. Even Functions. Gibbs Phenomenon | |

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Functions of Arbitrary Period. Odd Functions | |

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Half-Range Expansions | |

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Rectifier | |

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Trigonometric Approximation. Minimum Square Error | |

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Fourier Integral, Fourier Transform | |

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Problems for Chapter 10 | |

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Partial Differential Equations (PDE's) | |

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Wave Equation. Separation of Variables. Animation | |

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One-Dimensional Heat Equation | |

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Heat Equation, Laplace Equation | |

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Rectangular Membrane. Double Fourier Series | |

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Laplacian. Circular Membrane. Bessel Equation | |

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Problems for Chapter 11 | |

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Complex Analysis | |

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Complex Numbers and Functions. Conformal Mapping | |

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Complex Numbers. Polar Form. Plotting | |

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Equations. Roots. Sets in the Complex Plane | |

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Cauchy-Riemann Equations. Harmonic Functions | |

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Conformal Mapping | |

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Exponential, Trigonometric, and Hyperbolic Functions | |

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Complex Logarithm | |

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Problems for Chapter 12 | |

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Complex Integration | |

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Indefinite Integration of Analytic Functions | |

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Integration: Use of Path. Path Dependence | |

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Contour Integration by Cauchy's Integral Theorem and Formula | |

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Problems for Chapter 13 | |

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Power Series, Taylor Series | |

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Sequences and their Plots | |

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Convergence Tests for Complex Series | |

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Power Series. Radius of Convergence | |

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Taylor Series | |

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Uniform Convergence | |

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Problems for Chapter 14 | |

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Laurent Series. Residue Integration | |

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Laurent Series | |

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Singularities and Zeros | |

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Residue Integration | |

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Real Integrals of Rational Functions of cos and sin | |

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Improper Real Integrals of Rational Functions | |

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Problems for Chapter 15 | |

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Complex Analysis in Potential Theory | |

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Complex Potential. Related Plots | |

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Use of Conformal Mapping | |

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Fluid Flow | |

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Series Representation of Potential | |

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Mean Value Theorem for Analytic Functions | |

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Problems for Chapter 16 | |

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Numerical Methods | |

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Numerical Methods in General | |

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Loss of Significant Digits. Quadratic Equation | |

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Fixed-Point Iteration | |

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Solving Equations by Newton's Method | |

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Solving Equations by the Secant Method | |

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Solving Equations by the Bisection Method. Module | |

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Polynomial Interpolation | |

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Spline Interpolation | |

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Numerical Integration | |

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Problems for Chapter 17 | |

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Numerical Linear Algebra | |

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Gauss Elimination. Pivoting | |

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Doolittle LU-Factorization | |

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Cholesky Factorization | |

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Gauss-Jordan Elimination. Matrix Inversion | |

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Gauss-Seidel Iteration for Linear Systems | |

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Vector and Matrix Norms. Condition Numbers | |

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Fitting Data by Least Squares | |

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Approximation of Eigenvalues: Collatz Method | |

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Approximation of Eigenvalues: Power Method | |

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Approximation of Eigenvalues: QR-Factorization | |

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Problems for Chapter 18 | |

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Numerical Methods for Differential Equations | |

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Euler Method | |

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Improved Euler Method | |

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Classical Runge-Kutta Method (RK). Module | |

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Adams-Moulton Multistep Method | |

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Classical Runge-Kutta Method for Systems (RKS) | |

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Classical Runge-Kutta-Nystroem Method (RKN) | |

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Laplace Equation. Boundary Value Problem | |

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Heat Equation. Crank-Nicolson Method | |

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Problems for Chapter 19 | |

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Optimization, Graphs | |

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Unconstrained Optimization. Linear Programming | |

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Method of Steepest Descent | |

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Simplex Method of Constrained Optimization | |

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Problems for Chapter 20 | |

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No examples, no problems | |

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Probability and Statistics | |

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Data Analysis. Probability Theory | |

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Data Analysis: Mean, Variance, Standard Deviation | |

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Data Analysis: Histograms | |

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Discrete Probability Distributions | |

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Normal Distribution | |

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Problems for Chapter 22 | |

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Mathematical Statistics | |

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Random Numbers | |

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Confidence Interval for the Mean of the Normal Distribution With Known Variance | |

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Confidence Interval for the Mean of the Normal Distribution With Unknown Variance. t-Distribution | |

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Confidence Interval for the Variance of the Normal Distribution. x[superscript 2]-Distribution | |

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Test for the Mean of the Normal Distribution | |

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Test for the Mean: Power Function | |

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Test for the Variance of the Normal Distribution | |

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Comparison of Means | |

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Comparison of Variances. F-Distribution | |

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Chi-Square Test for Goodness of Fit | |

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Regression | |

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Problems for Chapter 23 | |

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References | |

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Answers to Odd-Numbered Problems | |

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Index | |