```Starting Mathematica

This is a basic introduction to Mathematica. Since Mathematica is a very powerful program```
` with many features only a small fraction of its capabilities will be discussed here. See`
` http://bartok.ucsc.edu/peter/115/math_intro/node1.html`
```
or go to Mathematica\help. Unlike conventional computer languages Mathematica can do```
` symbolic manipulation, has a huge number of built-in numerical functions, can do `
`numerical calculations to arbitrary precision, and has powerful plotting routines which `
`interface directly with the results of calculations. Mathematica can also be used for `
`programming in styles which are quite different from those of C and fortran, but we will `
`not discuss programming in Mathematica here. `
```
There are two interfaces to Mathematica: (i) a command line interface, and (ii) the ```
`notebook interface. The same commands are given in both, but the notebook interface has `
`some additional features.`
```

If one starts Mathematica in Windows  by clicking on an icon one goes straight into the ```
`notebook interface in which a separate window appears into which Mathematica commands are`
` typed.  To execute a command in Windows, you need to hold down the Shift key as well as`
` pressing Enter (just pressing Enter allows you to continue entering your command on the`
` next line but does not execute it). `
` `
```A Simple Mathematica Session

Mathematica understands the usual operators + - * / and for exponentiation. It also```
` `
`understands basic constants pi and E.  It also knows about a large number of mathematical `
` `
`functions such as Exp[x], Sin[x], Cos[x], ArcSin[x], Log[x]`
```

The following session illustrates some of these features:```
` `
` `
`x = 7`
` y = 19`
`x y`
` (x + 3) y`
` 2^x`
` Exp[ Pi]`
` `
```
Two-Dimensional Plots

The basic plotting command is Plot[f, {x, xmin, xmax}]```
```
which plots the function f(x) from xmin to xmax```
```
For example, ```
` `
` `

The command Show can also be used to combine plots. Suppose we create plots of two functions using two separate

calls to Plot, e.g.

```

Integration```
```

```
` `
` `
`Defining Functions`
` `
```

There are many functions in Mathematica but it is often very useful to be able to define ```
` `
`functions oneself. This is illustrated by the following example, which defines the `
` `
`function . f(x) = x + x^2`
` `
` `
` `
```

Note two things about the first line:. First of all the underscore after the x, which is ```
` `
`called a ``blank'', stands for ``any expression'', so when the function is called, the `
` `
`argument can be named anything, not necessarily x. Secondly, the use of :=, rather than =,`
` `
` which means that the assignment is delayed until the function is called. In the present `
` `
`example it wouldn't have made any difference if we had used =, the normal assignment`
` `
` operator, but it would if the function depended on a parameter.  `
` `
` `
```

Series Expansions
.
Mathematica can work out series of complicated functions to very high order. The command```
` `
` is Series[f, {x, x0, n}] which expands f in powers of up to n-th order, e.g. to get the`
` `
` first 20 terms in the expansion of about x=0: `
` `
` `
```

Three-Dimensional Plots

```