Advanced Mathematical Methods with MapleThe last five years have seen an immense growth in the use of symbolic computing and mathematical software packages such as Maple. The first three chapters of this book provide a user-friendly introduction to computer-assisted algebra with Maple. The rest of the book then develops these techniques and demonstrates the use of this technology for deriving approximate solutions to differential equations (linear and nonlinear) and integrals. In each case, the mathematical concepts are comprehensively introduced, with an emphasis on understanding how solutions behave and why various approximations can be used. Where appropriate, the text integrates the use of Maple to extend the utility of traditional approximation techniques. Advanced Mathematical Methods with Maple is the ideal companion text for advanced undergraduate and graduate students of mathematics and the physical sciences. It incorporates over 1000 exercises with different levels of difficulty, for which solutions are provided on the Internet. |
Contents
Functions and procedures | 22 |
Simplification | 63 |
Sequences series and limits | 131 |
Asymptotic expansions | 179 |
Continued fractions and Padé approximants | 195 |
Linear differential equations and Greens functions | 227 |
Fourier series and systems of orthogonal functions | 267 |
Perturbation theory | 301 |
Integrals and their approximation | 478 |
Stationary phase approximations | 526 |
Uniform approximations for differential equations | 573 |
Dynamical systems I | 628 |
periodic orbits | 673 |
Discrete Dynamical Systems | 727 |
2220 | 834 |
| 845 | |
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Common terms and phrases
a₁ algebraic amplitude asymptotic expansion b₁ behave behaviour Bessel function boundary conditions chapter coefficients command compute consider constant continued fraction convergence d˛y defined in equation delta function derived differential equation dx f(x eigenfunctions eigenvalues equations of motion evalf evaluate example Exercise exponentially expression finite fixed point Fourier series frequency function f(x given gives Green's function hence initial conditions instance integral integrand interval iterations limit linear Maple procedure Mathieu functions matrix method obtained original orthogonal oscillations Padé approximant parameter periodic orbit perturbation theory phase curves plot the graph polynomials power series problem radius of convergence result root sequence shown in figure sin(x solve stable stationary points Sturm-Liouville Taylor's series unstable unstable manifold variable zero