2.1. Mathematical functions

MATLAB offers many predefined mathematical functions for technical computing which contains a large set of mathematical functions. Typing help elfun and help specfun calls up full lists of elementary and special functions respectively.

MATLAB is a powerful programming environment that provides users with a variety of mathematical functions that can be used for computations. These functions are predefined, meaning they are built into MATLAB and ready to use without any additional programming. To see a full list of elementary functions, you can use the commands 'help elfun' and for special functions, 'help specfun'. This helps users understand the capabilities of MATLAB in handling mathematical operations.

Think of MATLAB like a toolbox for a mechanic. Just as a mechanic has a variety of tools available to fix cars, MATLAB has a toolbox filled with functions that can help complete various mathematical tasks. By looking up the tools (or functions) it has, users can solve their mathematical problems efficiently.

There is a long list of mathematical functions that are built into MATLAB. These functions are called built-ins. Many standard mathematical functions, such as sin(x), cos(x), tan(x), ex, ln(x), are evaluated by the functions sin, cos, tan, exp, and log respectively in MATLAB. Table 2.1 lists some commonly used functions, where variables x and y can be numbers, vectors, or matrices.

MATLAB includes a variety of built-in mathematical functions that users can utilize. These functions cover a range of basic mathematical operations, such as trigonometric functions (like sine, cosine, and tangent), exponential functions, and logarithmic functions. The special variable types in MATLAB mean that users can work with not only single numbers but also arrays (vectors and matrices) to perform calculations on multiple values at the same time.

Imagine you are baking cookies, and you need to measure out different ingredients. The built-in functions in MATLAB are like the measuring cups and spoons that allow you to accurately measure sugar, flour, and butter (the numbers) all at once, efficiently mixing them to create perfect cookie dough (the results of your calculations).

Table 2.1: Elementary functions cos(x) Cosine abs(x) Absolute value sin(x) Sine sign(x) Signum function tan(x) Tangent max(x) Maximum value acos(x) Arc cosine min(x) Minimum value asin(x) Arc sine ceil(x) Round towards +∞ atan(x) Arc tangent floor(x) Round towards −∞ exp(x) Exponential round(x) Round to nearest integer sqrt(x) Square root rem(x) Remainder after division log(x) Natural logarithm angle(x) Phase angle

log10(x) Common logarithm conj(x) Complex conjugate

The table presents a selection of elementary functions that are readily available in MATLAB. Each function serves a specific mathematical purpose. For instance, 'cos(x)' computes the cosine of an angle in radians, 'abs(x)' gives the absolute value of a number, and 'sqrt(x)' calculates the square root. Users need to familiarize themselves with these functions to efficiently implement them in their calculations in MATLAB.

Think of these functions as different types of calculators for various tasks. Just as you might use a scientific calculator for advanced mathematics (like calculating sine or cosine) and a basic calculator for simple arithmetic (like finding maximum or minimum values), MATLAB provides a comprehensive set of tools that can handle both complex and simple mathematical operations with ease.

In addition to the elementary functions, MATLAB includes a number of predefined constant values. A list of the most common values is given in Table 2.2.

Table 2.2: Predefined constant values pi The π number, π = 3.14159... i,j The imaginary unit i, √1− Inf The infinity, ∞ NaN Not a number

MATLAB also provides a collection of predefined constants that are frequently used in mathematical calculations. For example, pi is a constant representing the value of π, and it can be directly used in mathematical expressions. The imaginary unit i is also predefined in MATLAB, which allows users to seamlessly work with complex numbers. Additionally, constants such as Inf (infinity) and NaN (not a number) help in representing mathematical concepts that cannot be represented by regular numbers.

Think of these constants like common ingredients in a recipe. Just as you might always need flour or sugar when baking, certain constants like π and the imaginary unit are frequently needed in mathematical formulas. By having these constants predefined, MATLAB saves time—just like having your ingredients pre-measured and ready to go allows you to bake faster.

We illustrate here some typical examples which relate to the elementary functions previously defined.

As a first example, the value of the expression y = e^(asin(x)) + 10√y, for a = 5, x = 2, and y = 8 is computed by

a = 5; x = 2; y = 8; y = exp(-a)sin(x)+10sqrt(y) y = 28.2904

The subsequent examples are

log(142) ans = 4.9558

log10(142) ans = 2.1523

Note the difference between the natural logarithm log(x) and the decimal logarithm (base 10) log10(x).

In this section, we find concrete examples of how to use the functions discussed earlier. We calculate a more complex expression that involves the sine function, the exponential function, and the square root. This example shows how MATLAB can handle various calculations in one line of code. Additionally, we explore logarithmic functions, illustrating the difference between natural logarithms and common logarithms. This highlights the importance of understanding which function to use depending on the mathematical scenario.

Imagine you are a scientist conducting an experiment. The equations you use are like the recipes for mixing your chemicals. With MATLAB, you can quickly calculate the outcomes based on the variables you have, just as a scientist would mix known quantities to observe a reaction. For example, the expression y = e^(asin(x)) + 10√y could represent a calculation for determining a compound's reaction based on different factors.

Notes: • Only use built-in functions on the right-hand side of an expression. Reassigning the value to a built-in function can create problems. • There are some exceptions. For example, i and j are pre-assigned to √1. However, one or both of i or j are often used as loop indices. • To avoid any possible confusion, it is suggested to use instead ii or jj as loop indices.

This notes section provides important guidelines when using functions in MATLAB to prevent potential issues. It emphasizes the importance of using built-in functions as intended and avoiding unintended variable assignments to those functions. Moreover, it suggests that while MATLAB uses i and j for imaginary units, they can also be used as loop indices, which might confuse users. Thus, it's better to use alternative letter combinations if you're creating loops within your code.

Imagine organizing a team to finish a project. If you assign team members names that are too similar or the same as project names, it can lead to confusion. By taking steps to clearly differentiate between individual responsibilities and project tasks, you prevent mix-ups. Similarly, following MATLAB's guidelines helps maintain clarity in your calculations and avoids conflicts within your code.

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