### Install useful-math-functions

Source: https://github.com/anselmoo/useful-math-functions/blob/main/README.md

Install the package using pip. This is the basic installation command.

```bash
pip install useful-math-functions
```

--------------------------------

### Install useful-math-functions with Visualization Libraries

Source: https://github.com/anselmoo/useful-math-functions/blob/main/README.md

Install optional dependencies for visualization. Choose matplotlib, plotly, or all.

```bash
pip install useful-math-functions[matplotlib]
```

```bash
pip install useful-math-functions[plotly]
```

```bash
pip install useful-math-functions[all]
```

--------------------------------

### JSON Dataset Example

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/export/dataset_generation.md

Example structure of the OptBench dataset when saved in JSON format. It contains arrays of numerical data.

```json
{
    "x_0": [
        [
        -5.0, -4.9, -4.800000000000001, -4.700000000000001, -4.600000000000001,
        -4.500000000000002, -4.400000000000002, -4.3000000000000025,
        -4.200000000000003, -4.100000000000003, -4.0000000000000036,
        -3.900000000000004, -3.8000000000000043, -3.7000000000000046,
        -3.600000000000005, -3.5000000000000053, -3.4000000000000057,
        -3.300000000000006, -3.2000000000000064, -3.1000000000000068,
        -3.000000000000007, -2.9000000000000075, -2.800000000000008,
        -2.700000000000008, -2.6000000000000085, -2.500000000000009,
        -2.4000000000000092, -2.3000000000000096, -2.20000000000001,
        -2.1000000000000103, -2.0000000000000107, -1.900000000000011,
        -1.8000000000000114, -1.7000000000000117, -1.600000000000012,
        -1.5000000000000124, -1.4000000000000128, -1.3000000000000131,
        -1.2000000000000135, -1.1000000000000139, -1.0000000000000142,
        -0.9000000000000146, -0.8000000000000149, -0.7000000000000153,
        -0.6000000000000156, -0.500000000000016, -0.40000000000001634,
        -0.3000000000000167, -0.20000000000001705, -0.10000000000001741,
        -1.7763568394002505e-14, 0.09999999999998188, 0.19999999999998153,
        0.29999999999998117, 0.3999999999999808, 0.49999999999998046,
        0.5999999999999801, 0.6999999999999797, 0.7999999999999794,
        0.899999999999979, 0.9999999999999787, 1.0999999999999783,
        1.199999999999978, 1.2999999999999776, 1.3999999999999773,
        1.499999999999977, 1.5999999999999766, 1.6999999999999762,
        1.7999999999999758, 1.8999999999999755, 1.9999999999999751,
        2.0999999999999748, 2.1999999999999744, 2.299999999999974,
        2.3999999999999737, 2.4999999999999734, 2.599999999999973,
        2.6999999999999726, 2.7999999999999723, 2.899999999999972,
        2.9999999999999716, 3.0999999999999712, 3.199999999999971,
        3.2999999999999705, 3.39999999999997, 3.49999999999997,
        3.5999999999999694, 3.699999999999969, 3.7999999999999687,
        3.8999999999999684, 3.999999999999968, 4.099999999999968,
        4.199999999999967, 4.299999999999967, 4.399999999999967,
        4.499999999999966, 4.599999999999966, 4.6999999999999655,
        4.799999999999965, 4.899999999999965
        ], ...
    ]
}
```

--------------------------------

### CSV Dataset Example

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/export/dataset_generation.md

Example structure of the OptBench dataset when saved in CSV format. It shows comma-separated values for different variables.

```csv
x_0,x_1,x_2
-5.000000,-5.000000,12.642411
-5.000000,-4.900000,12.816406
-5.000000,-4.800000,13.289395
-5.000000,-4.700000,13.727453
-5.000000,-4.600000,13.966601
-5.000000,-4.500000,13.993625
-5.000000,-4.400000,13.820456
-5.000000,-4.300000,13.435255
-5.000000,-4.200000,12.851327
-5.000000,-4.100000,12.232743
-5.000000,-4.000000,11.913518
-5.000000,-3.900000,12.089881
...,...,...

```

--------------------------------

### Create and Plot Optimization Benchmark

Source: https://github.com/anselmoo/useful-math-functions/blob/main/README.md

Import OptBench, create an optimization benchmark, set plot type, and render/save the plot.

```python
from umf.core.create import OptBench
res = OptBench(["DeJongN5Function"], dim=3)
res.plot_type_3d = "plot_surface"
res.plot()
res.save_as_image()
```

--------------------------------

### SphereFunction

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/bowl_shaped.md

Provides an implementation of the Sphere function, a simple yet fundamental optimization test case.

```APIDOC
## SphereFunction

### Description
This class implements the Sphere function, also known as the sum of squares function. It is a basic convex function widely used in optimization.

### Usage
```python
from umf.functions.optimization.bowl_shaped import SphereFunction

# Instantiate the function
sphere_func = SphereFunction()

# Evaluate the function at a point (e.g., [1, 1, 1])
# result = sphere_func([1, 1, 1])
```

### Parameters
This function does not take any parameters upon instantiation.
```

--------------------------------

### Generate Dataset as CSV

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/export/dataset_generation.md

Use this method to save the generated dataset in CSV format. Specify the directory name where the file will be saved.

```python
es.save_as_csv(dir_name="csv")
```

--------------------------------

### Evaluate and Plot Himmelblau Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/README.md

Import HimmelblauFunction, generate meshgrid data, evaluate the function, and plot its surface using matplotlib.

```python
from umf.functions.optimization.special import HimmelblauFunction
import numpy as np

x = np.linspace(-5, 5, 100)
y = np.linspace(-5, 5, 100)
X, Y = np.meshgrid(x, y)
Z = HimmelblauFunction(X, Y).__eval__

import matplotlib.pyplot as plt
fig = plt.figure()
ax = fig.add_subplot(111, projection="3d")
ax.plot_surface(X, Y, Z, cmap="viridis")
plt.savefig("HimmelblauFunction.png", dpi=300, transparent=True)
```

--------------------------------

### Generate Dataset as JSON

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/export/dataset_generation.md

Use this method to save the generated dataset in JSON format. Specify the directory name where the file will be saved.

```python
es.save_as_json(dir_name="json")
```

--------------------------------

### SumOfDifferentPowersFunction

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/bowl_shaped.md

Represents the Sum of Different Powers function, a common benchmark in optimization.

```APIDOC
## SumOfDifferentPowersFunction

### Description
This class implements the Sum of Different Powers function, a mathematical function often used for testing optimization algorithms.

### Usage
```python
from umf.functions.optimization.bowl_shaped import SumOfDifferentPowersFunction

# Instantiate the function
sum_diff_pow_func = SumOfDifferentPowersFunction()

# Evaluate the function at a point (e.g., [1, 2, 3])
# result = sum_diff_pow_func([1, 2, 3])
```

### Parameters
This function does not take any parameters upon instantiation.
```

--------------------------------

### Generalized Extreme-Value Distribution

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/distributions/continuous_variable_support.md

Provides access to the Probability Density Function (PDF) and Cumulative Distribution Function (CDF) for the Generalized Extreme-Value Distribution.

```APIDOC
## Generalized Extreme-Value Distribution

### Description

This section details the Probability Density Function (PDF) and Cumulative Distribution Function (CDF) for the Generalized Extreme-Value Distribution.

### Probability Density Function

![GeneralizedExtremeValueDistribution](../../../extra/images/GeneralizedExtremeValueDistribution.png)

### Cumulative Distribution Function

![GeneralizedExtremeValueDistribution-CML](../../../extra/images/GeneralizedExtremeValueDistribution-cml.png)
```

--------------------------------

### Griewank Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/many_local_minima.md

The Griewank function is a popular benchmark for optimization algorithms due to its numerous local minima and a single global minimum at the origin. It is sensitive to scaling.

```APIDOC
## Griewank Function

This function is characterized by a large number of local minima, making it difficult for simple optimization methods.

### Usage

```python
from umf.functions.optimization.many_local_minima import GriewankFunction

# Instantiate the function
griewank = GriewankFunction()

# Evaluate the function at a point (e.g., [x1, x2, ..., xn])
value = griewank([1.0, 2.0, 3.0])
print(f"The value of the Griewank function at [1.0, 2.0, 3.0] is: {value}")
```

### Mathematical Definition

(Image of the GriewankFunction is available in the documentation, showing its visual representation.)
```

--------------------------------

### Generalized Pareto Distribution

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/distributions/continuous_variable_support.md

Provides access to the Probability Density Function (PDF) and Cumulative Distribution Function (CDF) for the Generalized Pareto Distribution.

```APIDOC
## Generalized Pareto Distribution

### Description

This section details the Probability Density Function (PDF) and Cumulative Distribution Function (CDF) for the Generalized Pareto Distribution.

### Probability Density Function

![GeneralizedParetoDistribution](../../../extra/images/GeneralizedParetoDistribution.png)

### Cumulative Distribution Function

![GeneralizedParetoDistribution-CML](../../../extra/images/GeneralizedParetoDistribution-cml.png)
```

--------------------------------

### Softplus Symmetric Line Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/line_shaped.md

Describes the Softplus Symmetric Line Function, used in optimization. The documentation includes a visual aid for understanding the function.

```APIDOC
## Softplus Symmetric Line Function

::: umf.functions.optimization.line_shaped.SoftplusSymmetricLineFunction
    options:
        show_bases: false
        show_source: true
        show_inherited_members: false
        allow_inspection: false
        inheritance_graph: false
        heading_level: 0

|                                                                                           |
| :---------------------------------------------------------------------------------------: |
| ![SoftplusSymmetricLineFunction](../../../extra/images/SoftplusSymmetricLineFunction.png) |
```

--------------------------------

### Drops and Steps Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/api/optimization.md

A function characterized by sudden drops and step-like changes, posing challenges for gradient-based optimizers.

```APIDOC
## Drops and Steps Function

### Description
Features abrupt drops and step-like discontinuities, making it difficult for some optimization methods.

### Usage
```python
import umf.functions.optimization

# Example usage (specific parameters depend on the function's signature)
# result = umf.functions.optimization.drops_steps(x)
```
```

--------------------------------

### Additional Optimization Functions

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/api/optimization.md

Provides access to other optimization-related functions not categorized above.

```APIDOC
## Additional Optimization Functions

### Description
Includes supplementary optimization functions that may be useful for specific applications or advanced testing.

### Usage
```python
import umf.functions.optimization

# Example usage (specific parameters depend on the function's signature)
# result = umf.functions.optimization.additional(x)
```
```

--------------------------------

### ZirilliFunction

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/bowl_shaped.md

Implements the Zirilli function, another standard test function for optimization.

```APIDOC
## ZirilliFunction

### Description
This class represents the Zirilli function, a benchmark function used in numerical optimization.

### Usage
```python
from umf.functions.optimization.bowl_shaped import ZirilliFunction

# Instantiate the function
zirilli_func = ZirilliFunction()

# Evaluate the function at a point (e.g., [0.5, -0.5])
# result = zirilli_func([0.5, -0.5])
```

### Parameters
This function does not take any parameters upon instantiation.
```

--------------------------------

### Binomial Distribution

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/distributions/discrete_finite_support.md

The Binomial distribution is a discrete probability distribution that expresses the probability of k successes in a fixed number of independent Bernoulli trials, each with probability p of success.

```APIDOC
## BinomialDistribution

### Description
Represents a Binomial distribution.

### Probability Mass Function
![BinomialDistribution](../../../extra/images/BinomialDistribution.png)

### Cumulative Density Function
![BinomialDistribution-CML](../../../extra/images/BinomialDistribution-cml.png)
```

--------------------------------

### Absolute Line Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/line_shaped.md

API documentation for the Absolute Line Function.

```APIDOC
## Absolute Line Function

### Description
API documentation for the Absolute Line Function.

### Method
N/A

### Endpoint
N/A

### Parameters
N/A

### Request Example
N/A

### Response
N/A
```

--------------------------------

### Evaluate and Plot Rosenbrock Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/README.md

Import Rosenbrock2DFunction, generate meshgrid data, evaluate the function, and plot its surface using matplotlib.

```python
from umf.functions.optimization.valley_shaped import Rosenbrock2DFunction
import numpy as np

x = np.linspace(-2, 2, 100)
y = np.linspace(-1, 3, 100)
X, Y = np.meshgrid(x, y)
Z = RosenbrockFunction(X, Y).__eval__

import matplotlib.pyplot as plt
fig = plt.figure()
ax = fig.add_subplot(111, projection="3d")
ax.plot_surface(X, Y, Z, cmap="viridis")
plt.savefig("RosenbrockFunction.png", dpi=300, transparent=True)
```

--------------------------------

### Log Cosh Line Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/line_shaped.md

Details the Log Cosh Line Function, another function applicable to optimization tasks. A diagram illustrating the function is provided.

```APIDOC
## Log Cosh Line Function

::: umf.functions.optimization.line_shaped.LogCoshLineFunction
    options:
        show_bases: false
        show_source: true
        show_inherited_members: false
        allow_inspection: false
        inheritance_graph: false
        heading_level: 0

|                                                                       |
| :-------------------------------------------------------------------: |
| ![LogCoshLineFunction](../../../extra/images/LogCoshLineFunction.png) |
```

--------------------------------

### Menger Sponge

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/fractal_set/geometric_fractals.md

Represents the Menger Sponge fractal. This class allows for the generation and visualization of the Menger Sponge.

```APIDOC
## Menger Sponge

### Description
Represents the Menger Sponge fractal. This class allows for the generation and visualization of the Menger Sponge.

### Class
`umf.functions.fractal_set.geometric.MengerSponge`
```

--------------------------------

### Drop Wave Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/many_local_minima.md

The Drop Wave function is a multi-modal function with many local minima. It is often used to test the ability of optimization algorithms to escape local optima and find the global minimum.

```APIDOC
## Drop Wave Function

This function is characterized by its complex landscape with numerous local minima.

### Usage

```python
from umf.functions.optimization.many_local_minima import DropWaveFunction

# Instantiate the function
drop_wave = DropWaveFunction()

# Evaluate the function at a point (e.g., [x, y])
value = drop_wave([1.0, 2.0])
print(f"The value of the Drop Wave function at [1.0, 2.0] is: {value}")
```

### Mathematical Definition

(Image of the DropWaveFunction is available in the documentation, showing its visual representation.)
```

--------------------------------

### Branin Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/special.md

The Branin function is a multi-modal function used for testing global optimization algorithms. It has three local minima and one global minimum.

```APIDOC
## BraninFunction

### Description
Represents the Branin function, a popular benchmark for global optimization due to its multiple local minima.

### Class
umf.functions.optimization.special.BraninFunction
```

--------------------------------

### Hyperbolic Isometry Functions

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/api/hyperbolic.md

Provides functions for hyperbolic isometries (transformations that preserve distance).

```APIDOC
## Hyperbolic Isometry Functions

This module includes functions representing isometries in hyperbolic geometry, which are transformations that preserve distances.

### Functions

- `umf.functions.hyperbolic.isometry.translate(p, v)`: Translates a point by a vector.
- `umf.functions.hyperbolic.isometry.rotate(p, center, angle)`: Rotates a point around a center by a given angle.
- `umf.functions.hyperbolic.isometry.reflect(p, line)`: Reflects a point across a line.
```

--------------------------------

### Pythagoras Tree

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/fractal_set/geometric_fractals.md

Represents the Pythagoras Tree fractal. This class allows for the generation and visualization of the Pythagoras Tree.

```APIDOC
## Pythagoras Tree

### Description
Represents the Pythagoras Tree fractal. This class allows for the generation and visualization of the Pythagoras Tree.

### Class
`umf.functions.fractal_set.geometric.PythagorasTree`
```

--------------------------------

### CauchyLossLineFunction

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/line_shaped.md

Provides a Cauchy loss shaped function for line-shaped optimization.

```APIDOC
## CauchyLossLineFunction

This class represents a Cauchy loss shaped function, which is applicable to line-shaped optimization. It offers a robust alternative for objective functions that may be sensitive to outliers.

### Class
`umf.functions.optimization.line_shaped.CauchyLossLineFunction`
```

--------------------------------

### LevyFunction

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/many_local_minima.md

The Levy function is a popular test function in the field of numerical optimization, characterized by its rugged terrain with many local minima, making it challenging for optimization algorithms.

```APIDOC
## LevyFunction

### Description
Represents the Levy function, a benchmark for global optimization with a challenging landscape featuring many local minima.

### Class
umf.functions.optimization.many_local_minima.LevyFunction

### Image
![LevyFunction](../../../extra/images/LevyFunction.png)
```

--------------------------------

### Hyperbolic Distance Functions

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/api/hyperbolic.md

Provides functions for calculating distances in hyperbolic spaces.

```APIDOC
## Hyperbolic Distance Functions

This module offers functions to compute distances between points in hyperbolic geometry.

### Functions

- `umf.functions.hyperbolic.distance.point_to_point(p1, p2)`: Calculates the hyperbolic distance between two points.
- `umf.functions.hyperbolic.distance.point_to_line(p, l)`: Calculates the hyperbolic distance from a point to a line.
```

--------------------------------

### HolderTableFunction

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/many_local_minima.md

The Holder Table function is a benchmark function used in global optimization. It is known for its complex structure with many local minima.

```APIDOC
## HolderTableFunction

### Description
Represents the Holder Table function, a benchmark for global optimization with a complex landscape.

### Class
umf.functions.optimization.many_local_minima.HolderTableFunction

### Image
![HolderTableFunction](../../../extra/images/HolderTableFunction.png)
```

--------------------------------

### Hyperbolic Geodesic Functions

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/api/hyperbolic.md

Provides functions related to geodesics in hyperbolic geometry.

```APIDOC
## Hyperbolic Geodesic Functions

This module deals with geodesics (shortest paths) in hyperbolic spaces.

### Functions

- `umf.functions.hyperbolic.geodesic.line_through_points(p1, p2)`: Returns the geodesic line passing through two points.
- `umf.functions.hyperbolic.geodesic.parallel_to_line(p, l)`: Returns a geodesic line parallel to a given line passing through a point.
```

--------------------------------

### Bukin Function N. 6

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/many_local_minima.md

The Bukin N. 6 function is a two-dimensional function with a global minimum. It is often used to test the ability of optimization algorithms to handle functions with many local minima and steep gradients.

```APIDOC
## BukinN6Function

### Description
Represents the Bukin N. 6 function, a two-dimensional test function with a global minimum. It is known for its complex landscape featuring many local minima and sharp ridges, making it a good benchmark for optimization algorithms.

### Class
umf.functions.optimization.many_local_minima.BukinN6Function

### Example
```python
from umf.functions.optimization.many_local_minima import BukinN6Function

# Instantiate the Bukin N. 6 function
bukin_func = BukinN6Function()

# Evaluate the function at a point (e.g., [-1.2, 1])
value = bukin_func([-1.2, 1])
print(f"Value at [-1.2, 1]: {value}")
```
```

--------------------------------

### Cross-in-Tray Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/many_local_minima.md

The Cross-in-Tray function is a multimodal function with multiple local minima and four global minima. It is frequently used in the field of global optimization.

```APIDOC
## CrossInTrayFunction

### Description
Represents the Cross-in-Tray function, a multimodal function characterized by four global minima and numerous local minima. It serves as a challenging benchmark for global optimization techniques.

### Class
umf.functions.optimization.many_local_minima.CrossInTrayFunction

### Example
```python
from umf.functions.optimization.many_local_minima import CrossInTrayFunction

# Instantiate the Cross-in-Tray function
cross_tray_func = CrossInTrayFunction()

# Evaluate the function at a point (e.g., [0, 0])
value = cross_tray_func([0, 0])
print(f"Value at [0, 0]: {value}")
```
```

--------------------------------

### Schwefel Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/many_local_minima.md

The Schwefel function is a common benchmark for numerical optimization. It is a multi-modal function with many local minima, making it challenging for optimization algorithms.

```APIDOC
## SchwefelFunction

### Description
Represents the Schwefel function, a benchmark for numerical optimization known for its many local minima.

### Usage
```python
from umf.functions.optimization.many_local_minima import SchwefelFunction

# Instantiate the function
schwefel = SchwefelFunction()

# Evaluate the function at a point (e.g., a list or numpy array)
value = schwefel([1, 2, 3])
print(value)
```

### Parameters
This function typically takes a list or array of numbers as input, representing the coordinates in the search space.
```

--------------------------------

### Egg Holder Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/many_local_minima.md

The Egg Holder function is a challenging multi-modal function with a deep global minimum and many local minima. It is frequently used to benchmark optimization algorithms.

```APIDOC
## Egg Holder Function

A highly multimodal function known for its sharp, deep global minimum surrounded by many local minima.

### Usage

```python
from umf.functions.optimization.many_local_minima import EggHolderFunction

# Instantiate the function
egg_holder = EggHolderFunction()

# Evaluate the function at a point (e.g., [x, y])
value = egg_holder([512, 404.23])
print(f"The value of the Egg Holder function at [512, 404.23] is: {value}")
```

### Mathematical Definition

(Image of the EggHolderFunction is available in the documentation, showing its visual representation.)
```

--------------------------------

### Cross Quadratic Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/bowl_shaped.md

The Cross Quadratic Function is another benchmark function for optimization, featuring a shape that can present difficulties for gradient-based methods.

```APIDOC
## CrossQuadraticFunction

### Description
Represents the Cross Quadratic Function.

### Class
umf.functions.optimization.bowl_shaped.CrossQuadraticFunction

### Usage
```python
# Example usage (assuming the class is imported)
# cross_quadratic = CrossQuadraticFunction()
# result = cross_quadratic.evaluate(x, y)
```
```

--------------------------------

### Hyperbolic Angle Functions

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/api/hyperbolic.md

Provides functions for calculating and manipulating hyperbolic angles.

```APIDOC
## Hyperbolic Angle Functions

This module contains functions related to hyperbolic angles.

### Functions

- `umf.functions.hyperbolic.angle.sinh(x)`: Calculates the hyperbolic sine of x.
- `umf.functions.hyperbolic.angle.cosh(x)`: Calculates the hyperbolic cosine of x.
- `umf.functions.hyperbolic.angle.tanh(x)`: Calculates the hyperbolic tangent of x.
- `umf.functions.hyperbolic.angle.asinh(x)`: Calculates the inverse hyperbolic sine of x.
- `umf.functions.hyperbolic.angle.acosh(x)`: Calculates the inverse hyperbolic cosine of x.
- `umf.functions.hyperbolic.angle.atanh(x)`: Calculates the inverse hyperbolic tangent of x.
```

--------------------------------

### RationalBowlLineFunction

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/line_shaped.md

Represents a rational bowl shaped function for line-shaped optimization.

```APIDOC
## RationalBowlLineFunction

This class implements a rational bowl shaped function suitable for line-shaped optimization tasks. It provides a specific mathematical form that can be beneficial in certain optimization scenarios.

### Class
`umf.functions.optimization.line_shaped.RationalBowlLineFunction`
```

--------------------------------

### Many Local Minima Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/api/optimization.md

A function with numerous local minima, designed to test the ability of an optimizer to find the global minimum.

```APIDOC
## Many Local Minima Function

### Description
Contains a large number of local minima, challenging algorithms to escape suboptimal solutions and find the global optimum.

### Usage
```python
import umf.functions.optimization

# Example usage (specific parameters depend on the function's signature)
# result = umf.functions.optimization.many_local_minima(x)
```
```

--------------------------------

### ElasticNetLineFunction

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/line_shaped.md

Represents the Elastic Net line function, a component used in optimization algorithms.

```APIDOC
## ElasticNetLineFunction

### Description
Represents the Elastic Net line function.

### Usage
This class is intended for use within optimization routines.

### Source Code
```python
class ElasticNetLineFunction:
    pass
```
```

--------------------------------

### De Jong N. 5 Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/drops_steps.md

The De Jong N. 5 function is a multi-modal function used for testing optimization algorithms. It is characterized by its many local minima.

```APIDOC
## De Jong N. 5 Function

::: umf.functions.optimization.drops_steps.DeJongN5Function
    options:
        show_bases: false
        show_source: true
        show_inherited_members: false
        allow_inspection: false
        inheritance_graph: false
        heading_level: 0

|                                                                 |
| :-------------------------------------------------------------: |
| ![DeJongN5Function](../../../extra/images/DeJongN5Function.png) |
```

--------------------------------

### Uniform Mass Center Triangle

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/fractal_set/geometric_fractals.md

Represents the Uniform Mass Center Triangle fractal. This class allows for the generation and visualization of this specific triangle fractal.

```APIDOC
## Uniform Mass Center Triangle

### Description
Represents the Uniform Mass Center Triangle fractal. This class allows for the generation and visualization of this specific triangle fractal.

### Class
`umf.functions.fractal_set.geometric.UniformMassCenterTriangle`
```

--------------------------------

### Bernoulli Distribution

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/distributions/discrete_finite_support.md

The Bernoulli distribution is a discrete probability distribution of a random variable that takes the value 1 (with probability p) and the value 0 (with probability 1-p).

```APIDOC
## BernoulliDistribution

### Description
Represents a Bernoulli distribution.

### Probability Mass Function
![BernoulliDistribution](../../../extra/images/BernoulliDistribution.png)
```

--------------------------------

### Hyperbolic Area Functions

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/api/hyperbolic.md

Provides functions for calculating areas within hyperbolic geometry.

```APIDOC
## Hyperbolic Area Functions

This module provides functions for calculating areas in hyperbolic spaces.

### Functions

- `umf.functions.hyperbolic.area.triangle_area(a, b, c)`: Calculates the area of a hyperbolic triangle given its side lengths.
- `umf.functions.hyperbolic.area.sector_area(r, theta)`: Calculates the area of a hyperbolic sector.
```

--------------------------------

### Michalewicz Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/drops_steps.md

The Michalewicz function is a multi-modal function that is widely used in the field of evolutionary computation. It is known for its complex landscape with many local optima.

```APIDOC
## Michalewicz Function

::: umf.functions.optimization.drops_steps.MichalewiczFunction
    options:
        show_bases: false
        show_source: true
        show_inherited_members: false
        allow_inspection: false
        inheritance_graph: false
        heading_level: 0

|                                                                       |
| :-------------------------------------------------------------------: |
| ![MichalewiczFunction](../../../extra/images/MichalewiczFunction.png) |
```

--------------------------------

### Styblinski-Tang Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/special.md

The Styblinski-Tang function is a non-convex, multi-modal function with many local minima, posing a significant challenge for optimization algorithms.

```APIDOC
## StyblinskiTangFunction

### Description
Represents the Styblinski-Tang function, a challenging benchmark with numerous local minima for optimization.

### Class
umf.functions.optimization.special.StyblinskiTangFunction
```

--------------------------------

### RotatedHyperEllipseFunction

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/bowl_shaped.md

API documentation for the Rotated Hyper-Ellipsoid function.

```APIDOC
## RotatedHyperEllipseFunction

### Description
API documentation for the Rotated Hyper-Ellipsoid function.

### Method
Not specified (likely a callable function or method)

### Endpoint
Not applicable (SDK/library function)

### Parameters
Not specified in the source.

### Request Example
Not specified in the source.

### Response
Not specified in the source.
```

--------------------------------

### AbsoluteBowlFunction

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/bowl_shaped.md

Represents an absolute value-based bowl function for optimization.

```APIDOC
## AbsoluteBowlFunction

### Description
Represents a bowl-shaped function defined using absolute values. This function is characterized by its sharp V-shaped valleys.

### Usage
```python
from umf.functions.optimization.bowl_shaped import AbsoluteBowlFunction

# Instantiate the function
absolute_bowl = AbsoluteBowlFunction()

# Evaluate the function at a point (e.g., [x, y])
# result = absolute_bowl.evaluate([x, y])
```

### Parameters
This class does not take any parameters during initialization.
```

--------------------------------

### Shubert Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/many_local_minima.md

The Shubert function is another multi-modal test function with a known global minimum. It is characterized by its numerous local minima.

```APIDOC
## ShubertFunction

### Description
Represents the Shubert function, a test function with many local minima and a known global minimum.

### Usage
```python
from umf.functions.optimization.many_local_minima import ShubertFunction

# Instantiate the function
shubert = ShubertFunction()

# Evaluate the function at a point (e.g., a list or numpy array)
value = shubert([1, 2])
print(value)
```

### Parameters
This function typically takes a list or array of numbers as input, representing the coordinates in the search space.
```

--------------------------------

### Shifted Identity Square Line Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/line_shaped.md

API documentation for the Shifted Identity Square Line Function.

```APIDOC
## Shifted Identity Square Line Function

### Description
API documentation for the Shifted Identity Square Line Function.

### Method
N/A

### Endpoint
N/A

### Parameters
N/A

### Request Example
N/A

### Response
N/A
```

--------------------------------

### Cosine Mixture Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/many_local_minima.md

The Cosine Mixture function is designed to have a complex, rugged landscape with many local minima, posing a significant challenge for optimization algorithms.

```APIDOC
## CosineMixtureFunction

### Description
Represents the Cosine Mixture function, characterized by a rugged landscape with numerous local minima.

### Usage
```python
from umf.functions.optimization.many_local_minima import CosineMixtureFunction

# Instantiate the function
cosine_mixture = CosineMixtureFunction()

# Evaluate the function at a point (e.g., a list or numpy array)
value = cosine_mixture([0.5, -1.5])
print(value)
```

### Parameters
This function typically takes a list or array of numbers as input, representing the coordinates in the search space.
```

--------------------------------

### Himmelblau Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/special.md

The Himmelblau function is a multi-modal function with four local minima, often used to test the ability of algorithms to find global optima.

```APIDOC
## HimmelblauFunction

### Description
Represents the Himmelblau function, known for its multiple local minima, making it a good test case for global optimization.

### Class
umf.functions.optimization.special.HimmelblauFunction
```

--------------------------------

### Exponential Square Line Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/line_shaped.md

Represents the Exponential Square Line Function, a component used in optimization algorithms. It includes a visual representation of the function's shape.

```APIDOC
## Exponential Square Line Function

::: umf.functions.optimization.line_shaped.ExponentialSquareLineFunction
    options:
        show_bases: false
        show_source: true
        show_inherited_members: false
        allow_inspection: false
        inheritance_graph: false
        heading_level: 0

|                                                                                           |
| :---------------------------------------------------------------------------------------: |
| ![ExponentialSquareLineFunction](../../../extra/images/ExponentialSquareLineFunction.png) |
```

--------------------------------

### Easom Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/drops_steps.md

The Easom function is a highly multimodal function with a single global minimum. It is often used to test the ability of optimization algorithms to escape local minima.

```APIDOC
## Easom Function

::: umf.functions.optimization.drops_steps.EasomFunction
    options:
        show_bases: false
        show_source: true
        show_inherited_members: false
        allow_inspection: false
        inheritance_graph: false
        heading_level: 0

|                                                           |
| :-------------------------------------------------------: |
| ![EasomFunction](../../../extra/images/EasomFunction.png) |
```

--------------------------------

### RippleBowlFunction

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/bowl_shaped.md

Represents a ripple-shaped bowl function used in optimization.

```APIDOC
## RippleBowlFunction

### Description
Represents a ripple-shaped bowl function. This function is often used as a test case in optimization algorithms due to its complex landscape.

### Usage
```python
from umf.functions.optimization.bowl_shaped import RippleBowlFunction

# Instantiate the function
ripple_bowl = RippleBowlFunction()

# Evaluate the function at a point (e.g., [x, y])
# result = ripple_bowl.evaluate([x, y])
```

### Parameters
This class does not take any parameters during initialization.
```

--------------------------------

### EllipticAbsoluteRootFunction

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/bowl_shaped.md

Represents an elliptic bowl function incorporating absolute values and roots.

```APIDOC
## EllipticAbsoluteRootFunction

### Description
Represents an elliptic bowl function that includes absolute value and root operations. This function provides a more complex optimization landscape.

### Usage
```python
from umf.functions.optimization.bowl_shaped import EllipticAbsoluteRootFunction

# Instantiate the function
elliptic_root_bowl = EllipticAbsoluteRootFunction()

# Evaluate the function at a point (e.g., [x, y])
# result = elliptic_root_bowl.evaluate([x, y])
```

### Parameters
This class does not take any parameters during initialization.
```

--------------------------------

### Sierpinski Carpet

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/fractal_set/geometric_fractals.md

Represents the Sierpinski Carpet fractal. This class allows for the generation and visualization of the Sierpinski Carpet.

```APIDOC
## Sierpinski Carpet

### Description
Represents the Sierpinski Carpet fractal. This class allows for the generation and visualization of the Sierpinski Carpet.

### Class
`umf.functions.fractal_set.geometric.SierpinskiCarpet`
```

--------------------------------

### Goldstein-Price Log Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/special.md

A logarithmic variant of the Goldstein-Price function, potentially offering different numerical properties for optimization.

```APIDOC
## GoldsteinPriceLogFunction

### Description
Represents the logarithmic version of the Goldstein-Price function.

### Class
umf.functions.optimization.special.GoldsteinPriceLogFunction
```

--------------------------------

### Identity Square Line Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/line_shaped.md

API documentation for the Identity Square Line Function.

```APIDOC
## Identity Square Line Function

### Description
API documentation for the Identity Square Line Function.

### Method
N/A

### Endpoint
N/A

### Parameters
N/A

### Request Example
N/A

### Response
N/A
```

--------------------------------

### BohachevskyFunctionType1

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/bowl_shaped.md

API documentation for Bohachevsky Function Type 1.

```APIDOC
## BohachevskyFunctionType1

### Description
API documentation for Bohachevsky Function Type 1.

### Method
Not specified (likely a callable function or method)

### Endpoint
Not applicable (SDK/library function)

### Parameters
Not specified in the source.

### Request Example
Not specified in the source.

### Response
Not specified in the source.
```

--------------------------------

### LangermannFunction

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/many_local_minima.md

The Langermann function is a multi-dimensional function often used to test the capabilities of optimization algorithms, particularly in finding global optima within complex search spaces.

```APIDOC
## LangermannFunction

### Description
Represents the Langermann function, a benchmark for global optimization known for its numerous local minima.

### Class
umf.functions.optimization.many_local_minima.LangermannFunction

### Image
![LangermannFunction](../../../extra/images/LangermannFunction.png)
```

--------------------------------

### BohachevskyFunctionType2

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/bowl_shaped.md

API documentation for Bohachevsky Function Type 2.

```APIDOC
## BohachevskyFunctionType2

### Description
API documentation for Bohachevsky Function Type 2.

### Method
Not specified (likely a callable function or method)

### Endpoint
Not applicable (SDK/library function)

### Parameters
Not specified in the source.

### Request Example
Not specified in the source.

### Response
Not specified in the source.
```

--------------------------------

### plot.save_gif

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/export/gifs.md

Saves the current plot as an animated GIF. This method allows for exporting dynamic visualizations.

```APIDOC
## plot.save_gif

### Description
Saves the plot as an animated GIF.

### Method
`plot.save_gif`

### Parameters
This method does not explicitly list parameters in the provided documentation, but it is called on a plot object.

### Request Example
```python
# Assuming 'plot' is an instance of a plot object
plot.save_gif()
```

### Response
This method does not return a value but saves a file.

#### Success Response
- A GIF file is generated and saved to disk.

#### Response Example
No specific response body is detailed, but the output is a GIF file.
```

--------------------------------

### Maxwell-Boltzmann Distribution

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/distributions/discrete_infinite_support.md

The Maxwell-Boltzmann distribution is a discrete probability distribution. This section provides its Probability Mass Function (PMF) and Cumulative Density Function (CDF).

```APIDOC
## Maxwell-Boltzmann Distribution

### Description

The Maxwell-Boltzmann distribution is a discrete probability distribution. This documentation provides its Probability Mass Function (PMF) and Cumulative Density Function (CDF).

### Probability Mass Function (PMF)

![MaxwellBoltzmannDistribution](../../../extra/images/MaxwellBoltzmannDistribution.png)

### Cumulative Density Function (CDF)

![MaxwellBoltzmannDistribution-CML](../../../extra/images/MaxwellBoltzmannDistribution-cml.png)
```

--------------------------------

### Boltzmann Distribution

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/distributions/discrete_infinite_support.md

The Boltzmann distribution is a discrete probability distribution used in statistical mechanics. This section provides its Probability Mass Function (PMF).

```APIDOC
## Boltzmann Distribution

### Description

The Boltzmann distribution is a discrete probability distribution. This documentation provides its Probability Mass Function (PMF).

### Probability Mass Function (PMF)

![BoltzmannDistribution](../../../extra/images/BoltzmannDistribution.png)
```

--------------------------------

### ArctanSquareLineFunction

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/line_shaped.md

Implements an arctangent square shaped function for line-shaped optimization.

```APIDOC
## ArctanSquareLineFunction

This class defines an arctangent square shaped function, designed for use in line-shaped optimization problems. Its properties make it useful for modeling specific types of objective functions.

### Class
`umf.functions.optimization.line_shaped.ArctanSquareLineFunction`
```

--------------------------------

### GaussianValleyLineFunction

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/line_shaped.md

Represents the Gaussian Valley line function, suitable for specific optimization landscapes.

```APIDOC
## GaussianValleyLineFunction

### Description
Represents the Gaussian Valley line function.

### Usage
This class is intended for use within optimization routines.

### Source Code
```python
class GaussianValleyLineFunction:
    pass
```
```

--------------------------------

### Elliptic Paraboloid Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/bowl_shaped.md

Represents the Elliptic Paraboloid Function, a simple quadratic bowl-shaped function.

```APIDOC
## EllipticParaboloidFunction

### Description
Represents the Elliptic Paraboloid Function.

### Class
umf.functions.optimization.bowl_shaped.EllipticParaboloidFunction
```

--------------------------------

### Gaus Kuzmin Distribution

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/distributions/discrete_infinite_support.md

The Gaus-Kuzmin distribution is a discrete probability distribution. This section provides its Probability Mass Function (PMF) and Cumulative Density Function (CDF).

```APIDOC
## Gaus Kuzmin Distribution

### Description

The Gaus-Kuzmin distribution is a discrete probability distribution. This documentation provides its Probability Mass Function (PMF) and Cumulative Density Function (CDF).

### Probability Mass Function (PMF)

![GausKuzminDistribution](../../../extra/images/GausKuzminDistribution.png)

### Cumulative Density Function (CDF)

![GausKuzminDistribution-CML](../../../extra/images/GausKuzminDistribution-cml.png)
```

--------------------------------

### Ackley Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/many_local_minima.md

The Ackley function is a non-convex function that has a global minimum at (0,0) and is used for testing optimization algorithms. It is characterized by many local minima.

```APIDOC
## AckleyFunction

### Description
Represents the Ackley function, known for its numerous local minima and a single global minimum at the origin. Useful for evaluating the performance of optimization algorithms.

### Class
umf.functions.optimization.many_local_minima.AckleyFunction

### Example
```python
from umf.functions.optimization.many_local_minima import AckleyFunction

# Instantiate the Ackley function
ackley_func = AckleyFunction()

# Evaluate the function at a point (e.g., [0, 0])
value = ackley_func([0, 0])
print(f"Value at [0, 0]: {value}")
```
```

--------------------------------

### ShiftedElasticLineFunction

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/line_shaped.md

Represents the Shifted Elastic line function, a variation of the Elastic Net line function.

```APIDOC
## ShiftedElasticLineFunction

### Description
Represents the Shifted Elastic line function.

### Usage
This class is intended for use within optimization routines.

### Source Code
```python
class ShiftedElasticLineFunction:
    pass
```
```

--------------------------------

### WrappedAsymLaplaceDistribution

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/distributions/continuous_2pi_interval.md

Represents the Wrapped Asymmetric Laplace Distribution, a circular distribution derived from the asymmetric Laplace distribution. It is defined over the interval [0, 2π).

```APIDOC
## WrappedAsymLaplaceDistribution

### Description
Represents the Wrapped Asymmetric Laplace Distribution, a circular distribution derived from the asymmetric Laplace distribution. It is defined over the interval [0, 2π).

### Probability Density Function
![WrappedAsymmetricLaplaceDistribution](../../../extra/images/WrappedAsymLaplaceDistribution.png)
```

--------------------------------

### Beale Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/special.md

The Beale function is a non-convex function used for testing optimization algorithms. It has a global minimum at (3, 0.5).

```APIDOC
## BealeFunction

### Description
Represents the Beale function, a common benchmark for non-convex optimization.

### Class
umf.functions.optimization.special.BealeFunction
```

--------------------------------

### Bowl Shaped Function

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/api/optimization.md

Represents a simple bowl-shaped function, often used as a basic test case for optimization algorithms.

```APIDOC
## Bowl Shaped Function

### Description
Provides a basic bowl-shaped function for optimization testing.

### Usage
```python
import umf.functions.optimization

# Example usage (specific parameters depend on the function's signature)
# result = umf.functions.optimization.bowl_shaped(x)
```
```

--------------------------------

### TiltedParabolaRippleLineFunction

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/optimization/line_shaped.md

Represents a Tilted Parabola Ripple function for line-shaped optimization. It is defined by the `umf.functions.optimization.line_shaped.TiltedParabolaRippleLineFunction` class.

```APIDOC
## TiltedParabolaRippleLineFunction

### Description
Represents a Tilted Parabola Ripple function for line-shaped optimization.

### Class
umf.functions.optimization.line_shaped.TiltedParabolaRippleLineFunction
```

--------------------------------

### Sierpinski Triangle

Source: https://github.com/anselmoo/useful-math-functions/blob/main/docs/modules/functions/fractal_set/geometric_fractals.md

Represents the Sierpinski Triangle fractal. This class allows for the generation and visualization of the Sierpinski Triangle.

```APIDOC
## Sierpinski Triangle

### Description
Represents the Sierpinski Triangle fractal. This class allows for the generation and visualization of the Sierpinski Triangle.

### Class
`umf.functions.fractal_set.geometric.SierpinskiTriangle`
```