Defining Functions in Python

Welcome to the first lesson of our course, "Optimization with SciPy." In this lesson, we're going to lay the foundation by learning how to define functions in Python. Functions are crucial in programming as they help you organize and reuse code efficiently. As we progress in the course, you'll see how understanding functions is vital for leveraging the full power of SciPy.

Let's quickly recall python functions, as we are going to use them a lot in this course.

In Python, you define a function using the def keyword, followed by the function name and parentheses (). Inside the parentheses, you can define parameters, which are inputs to the function. Let's start with a simple function that adds two numbers:

def add_numbers(a, b):
    return a + b

print(f"3 + 4 = {add_numbers(3, 4)}")

• def add_numbers(a, b): defines a function named add_numbers that takes two parameters, a and b.
• return a + b calculates the sum of a and b and returns the result.
• When you call this function with add_numbers(3, 4), it will return 7.

Let's now explore more complex functions.

Here's how to define $f(x) = x^2 + 4x + 4$:

def quadratic_function(x):
    return x**2 + 4*x + 4

print(f"quadratic_function(2) = {quadratic_function(2)}")

• This function calculates the value of a quadratic expression x^2 + 4*x + 4.
• The ** operator is used to denote x raised to the power of 2.
• When you call this function with quadratic_function(2), it will return 16.

Let's define $f(x) = \sin{x} + \cos{x}$

import numpy as np

def trigonometric_function(x):
    return np.sin(x) + np.cos(x)

print(f"trigonometric_function(π/4) = {trigonometric_function(np.pi/4)}")

• np.sin(x) and np.cos(x) use NumPy library functions to calculate the sine and cosine of x.
• The function returns the sum of the sine and cosine of x.
• When you call this function with trigonometric_function(np.pi/4), it will return approximately 1.4142.