2.8 Predefined Functions

The possibility of defining new functions is fundamental for increasing the language’s flexibility and its ability to adapt to the problems we want to solve. The new functions, however, must be defined in terms of others that were either defined by the user or, at most, predefined in the language.

As we will see, Julia has a vast set of predefined functions. In many cases, they suffice for what we want to do. Nevertheless, we should not restrain from defining new functions whenever we deem it necessary.

In this table we see a set of mathematical functions predefined in Julia. Note that, due to syntax limitations (which are also present in other programming languages), there are cases in which Julia uses a notation that differs from the traditional mathematical notation. For example, the square root function \(\sqrt{x}\) is written as sqrt(x).

The name sqrt is a contraction of the words square and root

Similar mechanisms are used for several other functions, includind the absolute value function \(|x|\) (written abs(x)), the exponentiation \(x^y\) (written x^y), and the remainder between the numbers \(m\) and \(n\) (written m%n). This table shows some of the equivalencies between the invocations of Julia’s functions and the correspondent mathematics invocations.

Some mathematical functions predefined in Julia.
Function

Arguments

Result
abs

A number

The absolute value of the argument.
sin

A number

The sine of the argument (in radians).
cos

A number

The cosine of the argument (in radians).
atan

One or two numbers

With only one argument, the arc tangent of the argument (in radians). With two arguments, the arc tangent of the division between the first and the second, where the sign of the arguments is used to determine the quadrant.
sqrt

A number

The square root of the argument.
exp

A number

The exponential value with base \(e\) of the argument.
^

Two numbers

The first argument raised to the power of the second argument.
log

One or two arguments

With one argument, the natural logarithm of the argument. With two arguments, the logarithm of the second argument with the first argument as its base.
max

Multiple Numbers

The highest argument.
min

Multiple Numbers

The lowest argument.
round

A number

Rounds the argument to the nearest integer.
floor

A number

Rounds down the argument to the nearest integer.
ceil

A number

Rounds up the argument to the nearest integer.

Julia’s predefined math functions.
Julia

Mathematics
abs(x)

\(|x|\)
sin(x)

\(\sin x\)
cos(x)

\(\cos x\)
atan(x)

\(\arctan x\)
atan(y, x)

\(\arctan \frac{y}{x}\)
sqrt(x)

\(\sqrt{x}\)
exp(x)

\(e^x\)
x^y

\(x^y\)
log(x)

\(\log_e x\)
floor(x)

\(\lfloor x\rfloor\)
ceil(x)

\(\lceil x\rceil\)

2.8.1 Exercises 5

2.8.1.1 Question 13

Translate the following mathematical expressions into Julia’s notation:

\(\sqrt{\frac{1}{\log 2^{\left|(3-9\log 25)\right|}}}\)
\(\frac{\cos^4 \frac{2}{\sqrt 5}}{\arctan 3}\)
\(\frac{1}{2} + \sqrt 3 + \sin^{\frac{5}{2}} 2\)

2.8.1.2 Question 14

Translate the following Julia expressions into mathematical notation:

log(sin(2^4+floor(atan(pi))/sqrt(5)+pi))
cos(cos(cos(0.5)))^5
sin(cos(sin(pi/3)/3)/3)

2.8.1.3 Question 15

Define the function odd that, for a given number, evaluates if it is odd, i.e., if the remainder of that number when divided by two is one.

2.8.1.4 Question 16

The area \(A\) of a pentagon inscribed in a circle of radius \(r\) is given by the following expression: \[A = \frac{5}{8}r^2\sqrt{10 + 2\sqrt{5}}\] Define a function that calculates this area and test it with values of your choice.

2.8.1.5 Question 17

Define a function that calculates the volume of an ellipsoid with semi-axes \(a\), \(b\) and \(c\). This volume can be obtained by using the formula: \(V=\frac{4}{3}\pi a b c\)

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1	Preface
2	Programming
3	Modeling
4	Recursion
5	Randomness
6	Data Structures
7	Complex Shapes
8	Transformations
9	Higher-Order Functions
10	Parametric Representation
11	Epilogue

2.1	Programming Languages
2.2	The Julia Language
2.3	Language Elements
2.4	Combinations
2.5	Defining Functions
2.6	Names
2.7	Debugging
2.8	Predefined Functions
2.9	Arithmetic in Julia
2.10	Name Evaluation
2.11	Conditional Expressions
2.12	Predicates
2.13	Predicates with a Variable Number of Arguments
2.14	Recognizers
2.15	Logical Operators
2.16	Selection
2.17	Local Variables
2.18	Global Variables
2.19	Modules

Function	Arguments	Result
abs	A number	The absolute value of the argument.
sin	A number	The sine of the argument (in radians).
cos	A number	The cosine of the argument (in radians).
atan	One or two numbers	With only one argument, the arc tangent of the argument (in radians). With two arguments, the arc tangent of the division between the first and the second, where the sign of the arguments is used to determine the quadrant.
sqrt	A number	The square root of the argument.
exp	A number	The exponential value with base \(e\) of the argument.
^	Two numbers	The first argument raised to the power of the second argument.
log	One or two arguments	With one argument, the natural logarithm of the argument. With two arguments, the logarithm of the second argument with the first argument as its base.
max	Multiple Numbers	The highest argument.
min	Multiple Numbers	The lowest argument.
round	A number	Rounds the argument to the nearest integer.
floor	A number	Rounds down the argument to the nearest integer.
ceil	A number	Rounds up the argument to the nearest integer.

Julia		Mathematics
abs(x)		\(\|x\|\)
sin(x)		\(\sin x\)
cos(x)		\(\cos x\)
atan(x)		\(\arctan x\)
atan(y, x)		\(\arctan \frac{y}{x}\)
sqrt(x)		\(\sqrt{x}\)
exp(x)		\(e^x\)
x^y		\(x^y\)
log(x)		\(\log_e x\)
floor(x)		\(\lfloor x\rfloor\)
ceil(x)		\(\lceil x\rceil\)