Hyperbolic functions

Math helps us study shapes.

Some math uses a circle. Other math uses a curve called a hyperbola.

This new math uses a special angle. This angle is based on an area.

These math tools help us understand many things. They help us study how heat moves. They also help us study how water flows. These shapes are found in our world. Math is a great way to learn more.

Math helps us study shapes.

Most people know about circles. We use math to find parts of a circle. We call these trigonometric functions.

But some math uses a different curve. This curve is called a hyperbola. We use hyperbolic functions to study this shape.

These functions work with a special angle. This angle is based on an area.

These tools are very useful. They help us understand how heat moves. They also help us study how water flows. Scientists use them to study physics. They even help us understand how gravity works. For example, a hanging chain makes a shape called a catenary. Hyperbolic functions can describe this curve.

People have studied these ideas for a long time. Isaac Newton thought about them in 1687. Later, Vincenzo Riccati gave them their names in 1757. Today, we use them to solve many hard puzzles in science.

Mathematics helps us describe the shapes of the world.

Most people learn about circles first. We use trigonometric functions to measure parts of a circle. However, math also looks at a different curve called a hyperbola.

Hyperbolic functions are like cousins to those circle functions. Instead of using a circle, they use a hyperbola. They work with a special kind of angle. This angle is measured by the area of a shape called a hyperbolic sector.

This area tells us how much the angle has grown.

These functions work in many different ways. You can find them in many math equations. For example, they help solve equations about cubes. They also help with Laplace's equation. This equation is very important in physics. It helps scientists study how heat moves from place to place. It also helps them study how water flows in fluids.

They even help explain how electricity and magnetism work. One famous use is describing a catenary. A catenary is the curve a heavy chain makes when it hangs freely between two points.

People have been curious about these shapes for hundreds of years. Gerardus Mercator used them around 1566 for his maps. Isaac Newton wrote about them in his famous book in 1687. Later, Vincenzo Riccati helped name them in 1757. He used specific names for the hyperbolic functions. In 1759, Daviet de Foncenex showed how they connect to circle functions. Then, Johann Heinrich Lambert organized them in the 1760s.

He changed the names to the ones we use today. He gave credit to Riccati for the original ideas.

There are several main hyperbolic functions to know. The two most basic ones are hyperbolic sine and hyperbolic cosine.

You can also use hyperbolic tangent. Other versions include hyperbolic cotangent, hyperbolic secant, and hyperbolic cosecant.

These functions are closely tied to the exponential function. For instance, hyperbolic sine is the odd part of an exponential function. Hyperbolic cosine is the even part. This connection makes them very powerful for solving hard math problems.

Hyperbolic functions connect to things you might already know. They relate to the way things grow or change. They also connect to complex numbers. In complex analysis, these functions appear when we use imaginary angles.

They can even be seen in the way light or energy moves. Even though they seem different from circles, they follow many of the same rules. This makes them a natural part of how we study geometry and science. They turn the curves of a hyperbola into something we can measure and understand.

{ "text": "Hyperbolic functions are mathematical tools that act as analogues to ordinary trigonometric functions. While standard trigonometry describes the properties of a circle, hyperbolic functions describe the properties of a hyperbola.

Specifically, they relate to the right half of a unit hyperbola. Just as circular functions use a circular angle, hyperbolic functions use a hyperbolic angle. The magnitude of this hyperbolic angle is defined as the area of its hyperbolic sector.

This connection allows mathematicians to extend the concepts of geometry beyond simple circles.\n\nThe basic hyperbolic functions are hyperbolic sine (sinh) and hyperbolic cosine (cosh).

These two functions can be defined using the exponential function, $e^u$. Hyperbolic sine is the odd part of the exponential function, calculated as half the difference between $e^u$ and $e^{-u}$.

Hyperbolic cosine is the even part, calculated as the average of $e^u$ and $e^{-u}$.

From these two, other functions are derived, including hyperbolic tangent (tanh), hyperbolic cotangent (coth), hyperbolic secant (sech), and hyperbolic cosecant (csch).

These functions can also be expressed as solutions to specific differential equations.\n\nIn complex analysis, hyperbolic functions arise when applying sine and cosine to imaginary angles. For example, $\sinh(z)$ is equal to $-i \sin(iz)$. This relationship shows how these functions are deeply connected to circular trigonometry through the imaginary unit $i$.

The hyperbolic sine and cosine are considered entire functions. This means they are well-behaved throughout the whole complex plane. Consequently, the other derived hyperbolic functions are meromorphic. This term describes functions that are analytic except at certain isolated points called poles. By the Lindemann–Weierstrass theorem, these functions yield transcendental values for any non-zero algebraic argument.\n\nThe history of these functions spans several centuries of mathematical discovery. The first known use of hyperbolic trigonometry is attributed to Gerardus Mercator around 1566. He used it to solve problems related to his map projection. Later, in 1687, Isaac Newton suggested the similarity between circular and hyperbolic sectors in his work, *Principia Mathematica*. In 1757, Vincenzo Riccati formally introduced the term \"hyperbolic functions.\" He used the abbreviations $\text{sin}$ and $\text{cos}$ for circular functions and $\text{sinh}$ and $\text{cosh}$ for hyperbolic ones.

By the 1760s, Johann Heinrich Lambert systematized these functions and provided their exponential expressions. Lambert is responsible for the specific abbreviations used in modern mathematics.\n\nHyperbolic functions are vital for solving many types of equations in physics and engineering. They appear in the solutions of linear differential equations, such as the equation defining a catenary. A catenary is the specific curve formed by a flexible chain hanging freely under its own weight. They are also used to solve cubic equations and Laplace's equation in Cartesian coordinates. Laplace's equation is a fundamental tool used in electromagnetic theory, heat transfer, and fluid dynamics. Furthermore, they are used to express Lorentz boosts as hyperbolic rotations within the framework of special relativity. They also help express the angle of parallelism in hyperbolic geometry.\n\nOne notable property involves the relationship between the hyperbolic cosine and arc length. The area under the curve of the hyperbolic cosine over a finite interval is always equal to the arc length of that interval. Additionally, the hyperbolic tangent is the unique solution to the differential equation $y' = 1 - y^2$ with the initial condition $y(0) = 0$. There are also many identities that mirror trigonometric identities. Osborn's rule provides a way to convert trigonometric identities into hyperbolic ones. This rule involves expanding the identity and switching the signs of terms containing a product of two hyperbolic sines.\n\nUltimately, hyperbolic functions represent a significant expansion of trigonometry. They allow for the measurement of complex curves and growth patterns that circles cannot describe. They connect the world of exponential growth to the world of geometric shapes. Whether studying the flow of fluids or the behavior of light, these functions provide the necessary language. They bridge the gap between real-valued geometry and the complex plane. This makes them an essential part of the mathematical toolkit for understanding the physical universe.", "media": [ "File:Circular and hyperbolic angle.svg", "File:Hyperbolic functions-2.svg", "File:sinh cosh tanh.svg", "File:Hyperbolic and exponential; sinh.svg", "File:Hyperbolic and exponential; cosh.svg", "File:csch sech coth.svg", "File:Complex Sinh.jpg", "File:Complex Cosh.jpg" ] }