Maple is a computer program capable of performing a wide variety of mathematical operations. It originated in the early 1980s as a computer algebra system, but today this description doesn't really do it justice. Maple has facilities for algebra, calculus, linear algebra, graphics (twoand three-dimensional plots, and animations), numerical calculations to arbitrary precision, and many other things besides. It is widely used in universities across the world, and is particularly useful for tasks that are tedious and error-prone when performed by humans, such as manipulating complicated series expansions and solving large sets of simultaneous equations. Used correctly, Maple can save time and quickly solve problems that would otherwise be intractable. Used incorrectly, it can lead to frustration, and the destruction of expensive IT equipment.

At the time of writing, the current version is Maple 2016. Versions before Maple 2015 were numbered starting from 1; the last of these was Maple 18. New features introduced in each version from Maple 4.0 onwards can be viewed using the help system (see Section 2.2). For the most part, recent changes have been relatively minor, at least as far as the material in this book is concerned. Consequently, all of the examples work with both Maple 2015 and Maple 2016. In fact, most will work in older versions as well, though naturally the number of exceptions increases the further back one goes. Two substantial new features are the dataplot command, discussed in Section 6.6, and the new rules concerning terminating characters, described in Appendix B (see also Section 2.3). Both of these were introduced in Maple 2015.

Why This Book?

This book is intended for students, teachers and researchers who will ultimately wish to use Maple for advanced applications. Here, ‘advanced’ means something more complex than evaluating a single integral, but not necessarily designing and running a simulation of the latest jet engine.

Functional programming – the use and evaluation of functions as a programming paradigm – has a long and productive history in the world of programming languages. Lisp came about in the 1950s in the search for a convenient language to represent mathematical concepts in programs, borrowing from the lambda calculus of the logician Alonzo Church. More recent languages have in turn embraced many aspects of Lisp – in addition to Lisp's offspring such as Scheme and haskell, you will find elements of functional constructs in Java, Python, Ruby, and Perl. Mathematicaitself has clear bloodlines to Lisp, including the ability to operate on data structures such as lists as single objects and in its representation of mathematical properties through rules.

Mathematica functions, unlike those in many other languages, are considered “first-class” objects, meaning that they can be used as arguments to other functions, they can be returned as values, and they can be part of many other kinds of data objects such as arrays. In addition, you can create and use functions at runtime, that is, when you evaluate an expression. This functionalstyle of programming distinguishes Mathematicafrom traditional procedural languages like C and Fortran. Facility with functional programming is therefore essential for taking full advantage of the Mathematica language to solve your computational tasks.

We start with some of the most useful functional programming constructs – higher-order functions such as Map, Apply, Thread, Select, and Outer. We then introduce iteration, a mechanism by which the output of one computation is fed as input into the next.

Programming with Mathematica

In its brief history, the world of programming has undergone a remarkable evolution. Those of us old enough to remember boxes of punch cards and batch jobs couldn't be happier about some of these changes. One could argue that the limitations, physical and conceptual,of the early programming environments helped to focus that world in a very singular manner. Eventually, efforts to overcome those limitations led to a very visible and broad transformation of the world of computer programming. We now have a plethora of languages, paradigms, and environments to choose from. At times this embarrassment of riches can be a bit overwhelming, but I think most would agree that we are fortunate to have such variety in programming languages with which to do our work.

I learned about Mathematica as I suspect many people have – after using several languages over the years, a colleague introduced me to a new and very different tool, Mathematica. I soon realized that it was going to help me in my work in ways that previous languages could not. Perhaps the most notable feature was how quickly I could translate the statement of a problem to a working program. This was no doubt due to having a functional style of programming at my fingertips but also being able to think in terms of rules and patterns seemed to fit well with my background in mathematics.

Well, Mathematica is no longer a young up-start in the programming world. It has been around now for over 25 years, making it, if not an elder statesman, certainly a mature and familiar player. And one that is used by people in fields as varied as linguistics, bioinformatics, engineering, and information theory. Like myself, many people are first introduced to it in an academic setting. Many more are introduced through a colleague at work. Still others have seen it mentioned in various media and are curious as to what it is all about. After using it to do basic or more advanced computation, most users soon find the need to extend the default set of tools that come with Mathematica.Programming is the ticket.

So what makes Mathematica such a useful programming tool? First, it is a well-designed language, one whose internal logic will be quite apparent as you get to know it.