## Introduction to Hyperbolic Functions:

Have you ever wondered if there are mathematical functions that are just as important as trigonometric and exponential functions? The answer is yes, and they are called hyperbolic functions.

Hyperbolic functions are comparable to other mathematical functions as they relate certain variables to others. In this article, we will introduce you to hyperbolic functions and discuss important formulas as well as work with the hyperbolic tangent function.

## Definition of Hyperbolic Function:

Hyperbolic functions are related to the unit hyperbola in the same way that trigonometric functions are related to the unit circle. The unit hyperbola is simply the graph of the equation xy=1.

The hyperbolic functions are defined in terms of the exponential function, e^x. They describe certain hyperbolic angles and the corresponding hyperbolic distances on the hyperbola as opposed to circular angles and distances on the unit circle.

## Important Formulas for Hyperbolic Functions:

Hyperbolic sine, cosine, and tangent functions are calculated using the following formulas:

- sinh(x) = (e^x – e^-x) / 2 ;
- cosh(x) = (e^x + e^-x) / 2 ;
- tanh(x) = (e^x – e^-x) / (e^x + e^-x)

### Other important formulas that are derived from the above formulas are:

- cosh^2(x) – sinh^2(x) = 1;
- sech^2(x) = 1 – tanh^2(x);
- coth(x) = 1 / tanh(x)

## Hyperbolic Tangent Function:

The most commonly used hyperbolic function is the hyperbolic tangent function or tanh(x). Like any other mathematical function, it has a defined set of inputs and an output.

It is available in the NumPy library that is specifically designed for scientific computing in Python. NumPy.tanh() function is used to calculate the hyperbolic tangent of an input array.

## Working with Hyperbolic Tangent Function:

### Syntax of NumPy.tanh() method:

The syntax of the NumPy.tanh() method is as follows:

`numpy.tanh(x, /, out=None, *, where=True, casting=same_kind, order=K, dtype=None, subok=True[, signature, extobj])`

### Parameters of NumPy.tanh() method:

The NumPy.tanh() method has the following parameters:

- x: input array for which the hyperbolic tangent needs to be calculated
- out(optional): Output array to store the result
- where(optional): It specifies what elements to compute
- dtype(optional): Desired output data type

### Method 1: Passing a Single-Valued Parameter:

We will begin with a simple example of passing a single value to tanh(x), which returns a single output. The output of this method is a float data type.

### Method 2: Passing Multi-Valued Parameter:

When an array is passed as input to tanh(x), where each element of the array is calculated, resulting in an array of the same dimension as that of the input.

### Method 3: Providing an Optional Output Variable:

The NumPy.tanh() method returns a new array with the same dimension as that of the input.

However, users can specify an output array to store the result if they wish to avoid memory allocation overhead.

### Method 4: Passing a Complex Variable as a Parameter:

The NumPy.tanh() function only returns a float data type, so passing a complex variable directly to it results in a TypeError.

To perform the hyperbolic tangent of a complex value, we can make use of the cmath library.

### Representation of Numpy.tanh() in Graph:

Finally, we’ll represent the output obtained from the tanh(x) function on a graph.

We will use the Matplotlib library to plot the graph. The graph displays the typical sigmoid curve that is related to hyperbolic functions, with the y-axis representing the function output and the x-axis giving the input value.

## Conclusion:

Hyperbolic functions are essential in mathematics, especially in scientific computing. They are used in various fields such as engineering, physics, and finance to name a few.

The hyperbolic tangent function is one of the most commonly used hyperbolic functions and is vital in neural networks and machine learning. By understanding the basic concepts and working with the hyperbolic tangent function using the NumPy library, one can begin to apply these functions in diverse fields and applications.

## Recap of the Article:

In this article, we introduced hyperbolic functions and how they are related to the unit hyperbola. We also discussed important formulas for hyperbolic sine, cosine, and tangent functions.

We then focused on the most commonly used hyperbolic function, the hyperbolic tangent, and explored working with it using the NumPy library. We covered various methods to pass input and obtain the output, along with graphing the output using Matplotlib.

## Implementation of Other Hyperbolic Functions:

Apart from the hyperbolic tangent function, there are several other hyperbolic functions that have their unique uses in mathematics and beyond.

In this section, we will introduce hyperbolic cosecant, secant, and cotangent functions, along with their formulas and implementation in Python.

### Hyperbolic Cosecant:

The hyperbolic cosecant function, csch(x), is the ratio of the inverse of the hyperbolic sine function to its argument.

### It can be calculated using the following formula:

csch(x) = 1 / sinh(x)

Its graph is similar to that of the reciprocal of the sine function when graphed over the real number line. We can implement the csch(x) function in Python by defining the function as:

```
import numpy as np
def csch(x):
return 1/np.sinh(x)
```

### Hyperbolic Secant:

The hyperbolic secant function, sech(x), is the ratio of the reciprocal of the hyperbolic cosine function to its argument. It can be calculated using the following formula:

sech(x) = 1 / cosh(x)

Its graph is similar to that of the reciprocal of the cosine function when graphed over the real number line.

We can implement the sech(x) function in Python by defining the function as:

```
import numpy as np
def sech(x):
return 1/np.cosh(x)
```

### Hyperbolic Cotangent:

The hyperbolic cotangent function, coth(x), is the ratio of the hyperbolic cosine function to the hyperbolic sine function. It can be calculated using the following formula:

coth(x) = cosh(x) / sinh(x)

Its graph is similar to that of the reciprocal of the tangent function when graphed over the real number line.

We can implement the coth(x) function in Python by defining the function as:

```
import numpy as np
def coth(x):
return np.cosh(x)/np.sinh(x)
```

All these hyperbolic functions have the same properties and definitions as trigonometric functions. They have their own unique formulas, graphs, and uses, making them essential in various fields of mathematics and beyond.

The PyTorch and TensorFlow libraries also contain several functions explicitly designed for work with these mathematical functions.

Hyperbolic functions find applications in various fields, including statistics, engineering, machine learning, and physics.

For instance, the hyperbolic tangent function is used widely in neural networks and decision-making models. The hyperbolic functions also find practical applications in fields such as finance, where they are instrumental in modeling interest rates as well as volatility.

## Conclusion:

Hyperbolic functions have a vital role in mathematical and scientific computing. In this article, we discussed the definition of hyperbolic functions, their primary formulas, and the implementation of the hyperbolic tangent function using the NumPy library.

We also explored the implementation and usage of other hyperbolic functions such as hyperbolic cosecant, secant, and cotangent functions. These functions have essential applications in fields such as engineering, statistics, finance, and machine learning.

Therefore, understanding these functions’ properties and how to implement and use them in practical applications is critical to mastering scientific computing capabilities.

In summary, hyperbolic functions play a vital role in mathematical and scientific computing, comparable to trigonometric and exponential functions.

The hyperbolic tangent function is the most commonly used of the hyperbolic functions and can be implemented using the NumPy library in Python. In addition, hyperbolic cosecant, secant, and cotangent functions also have critical applications in various fields such as finance, engineering, and machine learning, and are implemented in Python using Numpy.

By understanding the properties and applications of these functions, individuals can gain necessary tools to master scientific computing capabilities and apply them in practical applications. Overall, hyperbolic functions remain a crucial topic that should be mastered by individuals working in the fields of math, science, and engineering.