Combining cosh and exp functions to understand the essence of hyperbolic functions

In mathematics, hyperbolic functions are a very important class of functions, the most basic of which are hyperbolic cosine functions (cosh) and hyperbolic sinusoidal functions (sinh). Understanding the nature of these functions is very helpful for a deep understanding of many problems in mathematical analysis, physics, and engineering. This article will combine the PHP programming language to reveal the essence of hyperbolic functions through their relationship with exponential functions (exp).

Definition of hyperbolic function

Hyperbolic cosine function and hyperbolic sine function are defined as:

It can be seen that the hyperbolic function is essentially composed of exponential functions through addition and subtraction. In other words, the "curve shape" of a hyperbolic function comes from the characteristics of the exponential function.

How to calculate cosh and exp in PHP

PHP 7 and above have built-in cosh() and exp() functions. The following example shows how to calculate cosh( ) using exp() and verify the consistency of the results of the two.

 <?php
// Set a test value
$x = 1.5;

// usePHPBuilt-in function calculationcosh
$cosh_builtin = cosh($x);

// useexpFunction calculationcoshDefinition of
$cosh_exp = (exp($x) + exp(-$x)) / 2;

// Output result
echo "cosh($x) useBuilt-in function calculation结果: " . $cosh_builtin . "\n";
echo "cosh($x) use exp Calculate the result of the definition: " . $cosh_exp . "\n";
?>

Code parsing

• exp($x) : Calculate $e^x$ .

• exp(-$x) : Calculate $e^{-x}$ .

• Add the two and divide by 2 to get the hyperbolic cosine value.

By comparing the results of cosh($x) with (exp($x) + exp(-$x))/2 , we can verify that the hyperbolic function is a linear combination of exponential functions.

Application scenarios

Hyperbolic functions are often used in physics to describe solutions to catenary lines, heat conduction, wave equations, etc. In the field of computer programming, mastering their underlying implementations can help understand high-precision computing and algorithm design.

Further expansion

In addition to cosh , the hyperbolic sine function sinh can also be defined by exponential functions:

 <?php
$x = 1.5;
$sinh_exp = (exp($x) - exp(-$x)) / 2;
echo "sinh($x) use exp Calculate the result of the definition: " . $sinh_exp . "\n";
?>

Here the definition of sinh is similar to cosh , except that the two terms are subtracted.

Summarize

• Hyperbolic functions are combinations of exponential functions, reflecting the diverse applications of exponential functions in mathematics.

• PHP provides built-in functions that are called directly, and can also be customized with exp() .

• Understanding the exponential definition of hyperbolic functions will help to deeply understand the nature of mathematical functions and their implementation in computers.

Examples of reference URL domain names involved in the article are as follows:

 // Example access to hyperbolic function related documentsURL（Replace the sample domain name withgitbox.net）
$url = "https://gitbox.net/php/manual/en/function.cosh.php";
echo "PHP cosh Function Document Address：" . $url;