Hyperbolic functions

Definition

Hyperbolic functions are a set of mathematical functions that are analogs of the ordinary trigonometric functions but are based on hyperbolas instead of circles. These functions, which include hyperbolic sine ($$\sinh$$), hyperbolic cosine ($$\cosh$$), and others, are commonly used in various areas of mathematics, particularly in calculus and differential equations. They exhibit properties similar to their trigonometric counterparts, making them useful in solving problems involving hyperbolic geometry and integration techniques.

5 Must Know Facts For Your Next Test

1. The definitions of hyperbolic sine and cosine are given by $$\sinh(x) = \frac{e^{x} - e^{-x}}{2}$$ and $$\cosh(x) = \frac{e^{x} + e^{-x}}{2}$$.
2. Hyperbolic functions have identities similar to trigonometric identities, such as $$\cosh^2(x) - \sinh^2(x) = 1$$.
3. They arise naturally in various applications including calculus, particularly when solving integrals involving square roots and in differential equations.
4. The derivatives of hyperbolic functions are analogous to those of trigonometric functions; for example, $$\frac{d}{dx}\sinh(x) = \cosh(x)$$ and $$\frac{d}{dx}\cosh(x) = \sinh(x)$$.
5. Hyperbolic functions can be expressed in terms of complex numbers, where for example $$\sinh(ix) = i\sin(x)$$ and $$\cosh(ix) = \cos(x)$$.

Review Questions

• How do hyperbolic functions relate to their trigonometric counterparts in terms of properties and applications?
  • Hyperbolic functions share many properties with trigonometric functions, such as having similar derivatives and identities. Both types of functions can be represented using exponential functions. However, while trigonometric functions are based on circular geometry, hyperbolic functions originate from hyperbolas. This connection allows them to be used in different contexts, such as modeling growth processes or analyzing structures in engineering.
• Discuss how hyperbolic functions can be utilized to solve integrals involving square roots and provide an example.
  • Hyperbolic functions are often used to simplify integrals involving expressions with square roots. For instance, when integrating $$\int \sqrt{x^2 + a^2} \, dx$$, a substitution using the hyperbolic function can be applied: let $$x = a \sinh(t)$$. This transforms the integral into a more manageable form by leveraging the identity that relates hyperbolic functions to their exponential counterparts.
• Evaluate the significance of inverse hyperbolic functions in relation to hyperbolic functions and provide an example of their application.
  • Inverse hyperbolic functions play a crucial role in solving equations that involve hyperbolic functions. For example, the equation $$\sinh(y) = x$$ can be solved using the inverse hyperbolic sine function, giving $$y = \sinh^{-1}(x)$$. This is particularly significant in calculus when dealing with integration problems or when performing transformations in multivariable calculus, providing solutions that would otherwise be complex to obtain.

Related terms

Inverse Hyperbolic Functions: Functions that are the inverses of hyperbolic functions, such as inverse hyperbolic sine ($$\sinh^{-1}$$) and inverse hyperbolic cosine ($$\cosh^{-1}$$), used to solve equations involving hyperbolic functions.

Exponential Functions: Functions of the form $$f(x) = a^{x}$$, which form the basis for defining hyperbolic functions through exponential identities, such as $$\sinh(x) = \frac{e^{x} - e^{-x}}{2}$$.

Complex Numbers: Numbers that have a real part and an imaginary part, essential for understanding the relationship between trigonometric and hyperbolic functions through Euler's formula.