Question

Find an error in a program. Consider the following program for computing $$f(x)=e^{rx}\sin (mx)+e^{sx}\sin (nx)$$ def $f(x,m,n,r,s):$ return expsin $(x,r,m)+\mathrm{expsin}(x,s,n)$ $x=2.5$ print $f(x,0.1,0.2,1,1)$ from math import exp, sin def expsin $(x,p,q):$ return $\exp \left(p^{\ast x}\right)\ast \sin \left(q^{\ast x}\right)$ Running this code results in NameError: global name 'expsin' is not defined What is the problem? Simulate the program flow by hand or use the debugger to step from line to line. Correct the program.

Short answer

The issue was the order of function definitions; `expsin` was moved above `f`. Corrected syntax errors in `expsin`.

Step-by-step solution

Understand the error

The error message `NameError: global name 'expsin' is not defined` suggests that the function `expsin` is being called before it is defined. This is because in Python, functions must be defined before they are used in the code, usually preceding any calls to it in the script.

Check the code execution order

Notice that the call to `f(x, 0.1, 0.2, 1, 1)` is happening before the `expsin(x, p, q)` function is defined in the script. In Python, functions should be declared before they are used, so we need to move the function definition above the call.

Fix the function definition order

Move the `expsin(x, p, q)` function definition above the `f(x, m, n, r, s)` function, ensuring that the `expsin` function is defined before it is called within `f`.

Correct syntax errors

Correct the syntax inside the `expsin` function: replace `p^{* x}` with `p * x`, and `q^{* x}` with `q * x`. Use Python multiplication instead of the syntax that seems like exponentiation. Change them to `p * x` and `q * x`, respectively.

Final corrected code

The corrected code should look like this: ```python from math import exp, sin def expsin(x, p, q): return exp(p * x) * sin(q * x) def f(x, m, n, r, s): return expsin(x, r, m) + expsin(x, s, n) x = 2.5 print(f(x, 0.1, 0.2, 1, 1)) ```

Key concepts

Function Definition

In Python, a function is defined using the `def` keyword followed by the function name and a set of parentheses containing any parameters. This line ends with a colon, and the function's contents follow on indented lines. For example:

• `def my_function(arg1, arg2):`
• `# Code block here`

This is important because Python script reads top to bottom. If a function is called before it is defined, Python raises a `NameError`. In the case of the provided code, the function `expsin` is called in `f` before its own definition, leading to the error.

To avoid this, ensure all functions are defined before they are used in any operation or another function. This means arranging your code logically so that function definitions come first.

Syntax Errors

Syntax errors are mistakes in the use of the language. These can include missing colons, incorrect function calls, or wrong operators. In the given programming problem, a syntax error is observed in `expsin` where incorrect syntax for multiplication is used.

Instead of using `^{* x}`, which is incorrect and leads to errors, proper multiplication should be written as `* x`. For example, if you want to multiply two variables, it should be `a * b` without using any caret or additional symbols. Careful attention to syntax helps prevent simple errors that can stop a program from running.

Debugging

Debugging is a critical skill in programming. It involves identifying, analyzing, and correcting issues in your code. Common errors like the `NameError` in our code occur because something is being called that hasn't been defined. This specific type of error is often due to incorrect function call sequences.

For effective debugging:

• Run your code gradually, stepping through it line by line to understand the flow.
• Check variable and function names for typos or mismatches.
• Ensure functions are defined before they are used.

By taking logical steps and examining your code's execution order, you can troubleshoot and fix problems systematically given you proper understanding.

Mathematical Functions

In programming, mathematical functions such as those provided in Python's `math` module are invaluable. These include functions like `exp(x)` for exponential calculations and `sin(x)` for sine calculations. In the module imported in the example, these functions are used to calculate expressions for `f(x)` and `expsin(x, p, q)`.

Remember that in mathematical expressions, you're often combining several calculations:

• Exponential: `exp(p * x)` computes $e^{p\cdot x}$.
• Sine: `sin(q * x)` calculates the sine of $q\cdot x$ radians.

Incorrect use or calling these functions without appropriate mathematical understanding or syntax can result in logical or runtime errors. Approach composing these expressions step-by-step and ensure you validate each part of the equations used in your functions.