Getting Started with Sage

Functions

Functions are a way to encapsulate and modularize data processing. Data can be passed to a function using arguments. The function performs some kind of operation, and (optionally) returns a result.

Timeforaction–callingfunctions

We've already seen many simple examples of calling functions. Now, we'll use the plot function to illustrate more advanced ways to call functions. Evaluate the following code:

var('x')

sinc(x) = sin(x) / x

plot(sinc, (x, -10, 10))

The result should look like this:

Now, let's customize our plot:

plot(sinc, x, xmin=-15, xmax=15, thickness=2, color='red', legend_label='sinc')

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The customized plot should look like this:

What just happened?

In the first part of the example, we defined a callable symbolic expression that represents the sinc function. This function has important applications in signal processing and information theory. We plotted the function using a simple call to the plot function. When calling a function with multiple arguments, it is important to put the arguments in the right order. The first argument to plot is the callable symbolic expression. The second argument is known as a tuple, which we'll learn about in the next chapter. The tuple contains the independent variable, the minimum value of the plotting domain, and the maximum value of the domain.

In the second part of the example, we used keyword arguments to customize the plot. The arguments we used in previous examples are called positional arguments. Positional arguments are required, and they must occur in the correct order. A keyword argument is optional—if a keyword argument is not specified in the function call, it takes on a default value. If keywords are used, the arguments can be placed in any order. However, keyword

arguments must come after all the positional arguments. In general, a function is called using the syntax:

result = function_name(argument_1, argument_2, … , argument_n, keyword=value)

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Getting Started with Sage

The number of positional arguments in the function call must match the number in the definition. The function does not have to return a value, and you don't have to assign its return value to a variable. In simple cases, it's possible to pass optional arguments without keywords by passing the optional arguments in the right order. However, this is discouraged, because it makes the code less readable and more prone to bugs.

Sage has very sophisticated plotting capabilities, which we will cover in Chapter 6. If you are interested in learning more now, evaluate the command plot? in a worksheet cell or in the interactive shell to get help on the plot function.

Haveagohero–makesomemoreplots

Use the plot command to make plots of some of the built-in mathematical functions listed in the next section. Practice using keyword arguments to customize the plots. Then, use the built-in functions to define a callable symbolic expression, and plot it.

Built-infunctions

A vast number of functions are pre-defined in Sage. Even more are available through Python modules, which we will learn about in the next chapter. For now, here is a brief summary of the most commonly used mathematical functions, and how to access them in Sage:

Numericalapproximations

Any numerical type in Sage can be converted to a real number with the numerical_approx function (this function can also be abbreviated as n or N). For example:

print(pi) print(numerical_approx(pi)) print(type(numerical_approx(pi))) print(numerical_approx(pi, prec=16)) print(numerical_approx(pi, digits=5))

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The result should look like this:

The numerical_approx function accepts three arguments. The first argument, which is mandatory, is the item to be converted to a real number. The keyword argument prec can be used to specify the number of bits of precision for the real number. Alternately, the keyword argument digits can be used to specify the number of digits of precision.

Theresetandrestorefunctions

It's possible to accidentally re-define a built-in function or constant. For example, the letters i and n are commonly used as names for counting variables in loops. Fortunately, the restore function can be used to restore predefined global variables (such as i and n) to their default values. Here's a short example:

print(e + i * 5)

i = 10 e = 5

print(e + i * 5)

restore('e i') print(e + i * 5)

The result should look like this:

If you call restore without any arguments, it will restore all the predefined variables to their default values. Another useful function is called reset. This function deletes all the variables you have defined, restores all global variables to their default values, and resets the interfaces to other computer algebra systems.

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If you start getting strange results from your calculations, you may have accidentally re-defined a built-in function or constant. Try calling the reset() function and running the calculation again. Remember that reset will delete any variables or functions that you may have defined, so your calculation will have to start over from the beginning.

Definingyourownfunctions

Sage allows you to define your own functions using Python syntax. This will be very useful for keeping your code organized, especially as we move into writing longer programs.

Timeforaction–definingandusingyourownfunctions

Let's return to the series RC circuit that we have been using as an example. We will now define a function that computes the voltage across the capacitor. You can enter the following code in an input cell in a worksheet, or on the command line. When you type the colon at the end of the first line and press Enter, the cursor will automatically indent the lines that follow. Make sure that you consistently indent each line inside the function.

print('Voltage at t=' + str(n(t, digits=4)) + 's is ' + str(n(v, digits=4)) + 'V')

This block of code produces the following output:

Voltage at t=1.000s is 36.79V

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