## 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. |

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### Contents

2 | 15 |

Simplification | 63 |

4 | 67 |

7 | 74 |

8 | 82 |

Functions and procedures | 91 |

Sequences series and limits | 131 |

Asymptotic expansions | 179 |

Special functions | 377 |

Linear systems and Floquet theory | 437 |

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 |

Continued fractions and Pade approximants | 195 |

Linear differential equations and Greens functions | 227 |

Fourier series and systems of orthogonal functions | 267 |

Perturbation theory | 301 |

SturmLiouville systems | 342 |

Periodically driven systems | 785 |

The gamma and related functions | 834 |

845 | |

855 | |

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### Common terms and phrases

algebraic amplitude approximate solution assume asymptotic expansion behave behaviour Bessel function boundary conditions chapter coefficients complex compute consider constant continued fraction contour defined in equation delta function derived determine differential equation eigenfunctions eigenvalues eigenvectors equations of motion evalf evaluate example Exercise exponentially expression finite fixed point Fourier series frequency given gives Green's function hence increases initial conditions instance integral integrand interval iterations limit linear logistic map loop Maple procedure Mathieu functions matrix method nonlinear obtained original orthogonal oscillations Pade approximant parameter periodic orbit perturbation expansion perturbation theory phase curves plot the graph polynomials positive power series problem radius of convergence result root satisfies sequence shown in figure simple solve stable stationary points Sturm-Liouville Taylor's series turning points unperturbed unstable unstable manifold variable velocity function zero