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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 | |