3.4 Hyperbolic functions

Hyperbolic functions are mathematical tools that mirror trigonometric functions but use exponentials instead of angles. They're essential in complex analysis, offering unique properties and applications in fields like physics and engineering.

These functions, including sinh, cosh, and tanh, have distinct characteristics from their trigonometric counterparts. They're not periodic, have different domain and range properties, and play crucial roles in modeling phenomena like catenary curves and relativistic velocity addition.

Definition of hyperbolic functions

• Hyperbolic functions are mathematical functions defined in terms of the exponential function $e^x$
• They are analogous to trigonometric functions but have distinct properties and applications in complex analysis
• The main hyperbolic functions are hyperbolic sine (sinh), hyperbolic cosine (cosh), and hyperbolic tangent (tanh)

Hyperbolic sine

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• Hyperbolic sine is defined as $\sinh(x) = \frac{e^x - e^{-x}}{2}$
• It is an odd function, meaning $\sinh(-x) = -\sinh(x)$
• The derivative of hyperbolic sine is hyperbolic cosine: $\frac{d}{dx}\sinh(x) = \cosh(x)$
• The integral of hyperbolic sine is $\int \sinh(x) dx = \cosh(x) + C$

Hyperbolic cosine

• Hyperbolic cosine is defined as $\cosh(x) = \frac{e^x + e^{-x}}{2}$
• It is an even function, meaning $\cosh(-x) = \cosh(x)$
• The derivative of hyperbolic cosine is hyperbolic sine: $\frac{d}{dx}\cosh(x) = \sinh(x)$
• The integral of hyperbolic cosine is $\int \cosh(x) dx = \sinh(x) + C$

Hyperbolic tangent

• Hyperbolic tangent is defined as $\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$
• It is an odd function, meaning $\tanh(-x) = -\tanh(x)$
• The derivative of hyperbolic tangent is $\frac{d}{dx}\tanh(x) = \text{sech}^2(x)$, where $\text{sech}(x)$ is the hyperbolic secant
• The integral of hyperbolic tangent is $\int \tanh(x) dx = \ln(\cosh(x)) + C$

Other hyperbolic functions

• Hyperbolic cotangent: $\coth(x) = \frac{\cosh(x)}{\sinh(x)}$
• Hyperbolic secant: $\text{sech}(x) = \frac{1}{\cosh(x)}$
• Hyperbolic cosecant: $\text{csch}(x) = \frac{1}{\sinh(x)}$

Properties of hyperbolic functions

• Hyperbolic functions have unique properties that distinguish them from trigonometric functions
• Understanding these properties is essential for solving problems and analyzing functions in complex analysis

Domain and range

• The domain of all hyperbolic functions is the entire real line, $(-\infty, \infty)$
• The range of hyperbolic sine and hyperbolic tangent is $(-\infty, \infty)$
• The range of hyperbolic cosine is $[1, \infty)$
• The range of hyperbolic cotangent is $(-\infty, -1] \cup [1, \infty)$
• The range of hyperbolic secant is $(0, 1]$
• The range of hyperbolic cosecant is $(-\infty, -1] \cup [1, \infty)$

Symmetry and periodicity

• Hyperbolic sine and hyperbolic tangent are odd functions, while hyperbolic cosine is an even function
• Unlike trigonometric functions, hyperbolic functions are not periodic
• Hyperbolic functions do not have a fixed period or repeat their values at regular intervals

Derivatives of hyperbolic functions

• The derivative of hyperbolic sine is hyperbolic cosine: $\frac{d}{dx}\sinh(x) = \cosh(x)$
• The derivative of hyperbolic cosine is hyperbolic sine: $\frac{d}{dx}\cosh(x) = \sinh(x)$
• The derivative of hyperbolic tangent is hyperbolic secant squared: $\frac{d}{dx}\tanh(x) = \text{sech}^2(x)$
• The derivative of hyperbolic cotangent is negative hyperbolic cosecant squared: $\frac{d}{dx}\coth(x) = -\text{csch}^2(x)$
• The derivative of hyperbolic secant is negative hyperbolic secant times hyperbolic tangent: $\frac{d}{dx}\text{sech}(x) = -\text{sech}(x)\tanh(x)$
• The derivative of hyperbolic cosecant is negative hyperbolic cosecant times hyperbolic cotangent: $\frac{d}{dx}\text{csch}(x) = -\text{csch}(x)\coth(x)$

Integrals of hyperbolic functions

• The integral of hyperbolic sine is hyperbolic cosine: $\int \sinh(x) dx = \cosh(x) + C$
• The integral of hyperbolic cosine is hyperbolic sine: $\int \cosh(x) dx = \sinh(x) + C$
• The integral of hyperbolic tangent is the natural logarithm of hyperbolic cosine: $\int \tanh(x) dx = \ln(\cosh(x)) + C$
• The integral of hyperbolic cotangent is the natural logarithm of hyperbolic sine: $\int \coth(x) dx = \ln(\sinh(x)) + C$
• The integral of hyperbolic secant is the inverse hyperbolic tangent: $\int \text{sech}(x) dx = \tanh^{-1}(x) + C$
• The integral of hyperbolic cosecant is the inverse hyperbolic sine: $\int \text{csch}(x) dx = \sinh^{-1}(1/x) + C$

Relationships between hyperbolic functions

• Hyperbolic functions are interconnected through various identities and relationships
• These relationships help simplify expressions and solve problems involving hyperbolic functions

Fundamental hyperbolic identities

• Pythagorean identity: $\cosh^2(x) - \sinh^2(x) = 1$
• Reciprocal identities: $\tanh^2(x) + \text{sech}^2(x) = 1$ and $\coth^2(x) - \text{csch}^2(x) = 1$
• Sum and difference formulas for hyperbolic sine and cosine: $\sinh(x \pm y) = \sinh(x)\cosh(y) \pm \cosh(x)\sinh(y)$ and $\cosh(x \pm y) = \cosh(x)\cosh(y) \pm \sinh(x)\sinh(y)$
• Double angle formulas: $\sinh(2x) = 2\sinh(x)\cosh(x)$ and $\cosh(2x) = \cosh^2(x) + \sinh^2(x)$

Hyperbolic functions vs trigonometric functions

• Hyperbolic functions are analogous to trigonometric functions but have distinct properties
• The relationships between hyperbolic and trigonometric functions can be expressed using Euler's formula: $\sinh(x) = -i\sin(ix)$ and $\cosh(x) = \cos(ix)$
• Hyperbolic functions can be derived from trigonometric functions by replacing $\theta$ with $i\theta$, where $i$ is the imaginary unit

Inverse hyperbolic functions

• Inverse hyperbolic functions are the inverses of hyperbolic functions, denoted as $\sinh^{-1}(x)$, $\cosh^{-1}(x)$, and $\tanh^{-1}(x)$
• Inverse hyperbolic sine: $\sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1})$
• Inverse hyperbolic cosine: $\cosh^{-1}(x) = \ln(x + \sqrt{x^2 - 1})$, where $x \geq 1$
• Inverse hyperbolic tangent: $\tanh^{-1}(x) = \frac{1}{2}\ln(\frac{1 + x}{1 - x})$, where $|x| < 1$

Graphs of hyperbolic functions

• Graphing hyperbolic functions helps visualize their behavior and properties
• The graphs of hyperbolic functions have distinct shapes and asymptotes

Graph of hyperbolic sine

• The graph of hyperbolic sine is an odd function that passes through the origin
• It is unbounded and monotonically increasing on the entire real line
• The graph of hyperbolic sine resembles a scaled version of the exponential function $e^x$ for positive $x$ and $-e^{-x}$ for negative $x$

Graph of hyperbolic cosine

• The graph of hyperbolic cosine is an even function with a minimum value of 1 at $x = 0$
• It is unbounded and monotonically increasing for $x > 0$ and monotonically decreasing for $x < 0$
• The graph of hyperbolic cosine resembles a vertically shifted version of the exponential function $e^x$ for both positive and negative $x$

Graph of hyperbolic tangent

• The graph of hyperbolic tangent is an odd function that passes through the origin
• It is bounded between -1 and 1 and monotonically increasing on the entire real line
• The graph of hyperbolic tangent has horizontal asymptotes at $y = -1$ and $y = 1$

Asymptotes and limits

• Hyperbolic sine and hyperbolic cosine have no horizontal asymptotes, as they are unbounded
• Hyperbolic tangent has horizontal asymptotes at $y = -1$ and $y = 1$, as $\lim_{x \to -\infty} \tanh(x) = -1$ and $\lim_{x \to \infty} \tanh(x) = 1$
• Hyperbolic cotangent has vertical asymptotes at $x = 0$ and horizontal asymptotes at $y = -1$ and $y = 1$
• Hyperbolic secant has no horizontal asymptotes but approaches 0 as $x \to \pm\infty$
• Hyperbolic cosecant has vertical asymptotes at $x = 0$ and approaches 0 as $x \to \pm\infty$

Applications of hyperbolic functions

• Hyperbolic functions have various applications in mathematics, physics, and engineering
• They are used to model and solve problems involving exponential growth, catenaries, and relativistic velocity

Catenary curves

• A catenary is the curve formed by a hanging chain or cable suspended between two points
• The shape of a catenary is described by the hyperbolic cosine function: $y = a\cosh(\frac{x}{a})$, where $a$ is a constant related to the tension and weight of the chain
• Catenaries are used in the design of suspension bridges, power lines, and architectural structures

Hyperbolic geometry

• Hyperbolic geometry is a non-Euclidean geometry that describes spaces with constant negative curvature
• Hyperbolic functions are used to define distances, angles, and trigonometric ratios in hyperbolic geometry
• The Poincaré disk and upper half-plane models of hyperbolic geometry utilize hyperbolic functions to represent hyperbolic lines and transformations

Relativistic velocity addition

• In special relativity, the addition of velocities is described using hyperbolic functions
• The relativistic velocity addition formula is $u \oplus v = \frac{u + v}{1 + \frac{uv}{c^2}}$, where $u$ and $v$ are velocities and $c$ is the speed of light
• This formula can be expressed using hyperbolic tangent: $\tanh(\alpha \oplus \beta) = \frac{\tanh(\alpha) + \tanh(\beta)}{1 + \tanh(\alpha)\tanh(\beta)}$, where $\alpha$ and $\beta$ are hyperbolic angles related to the velocities

Lorentz transformations

• Lorentz transformations are mathematical transformations that describe the relationship between space and time coordinates in different inertial reference frames in special relativity
• Hyperbolic functions are used to express Lorentz transformations, such as the Lorentz factor $\gamma = \cosh(\alpha)$ and the rapidity $\alpha = \tanh^{-1}(\frac{v}{c})$
• The Lorentz transformation equations for position and time involve hyperbolic functions: $x' = x\cosh(\alpha) - ct\sinh(\alpha)$ and $t' = -\frac{x}{c}\sinh(\alpha) + t\cosh(\alpha)$

Complex hyperbolic functions

• Hyperbolic functions can be extended to the complex plane, allowing for the analysis of complex-valued functions
• Complex hyperbolic functions have properties and identities similar to their real-valued counterparts

Definition in complex plane

• Complex hyperbolic sine: $\sinh(z) = \frac{e^z - e^{-z}}{2}$, where $z$ is a complex number
• Complex hyperbolic cosine: $\cosh(z) = \frac{e^z + e^{-z}}{2}$
• Complex hyperbolic tangent: $\tanh(z) = \frac{\sinh(z)}{\cosh(z)}$
• Other complex hyperbolic functions are defined similarly to their real-valued counterparts

Euler's formula for hyperbolic functions

• Euler's formula relates complex exponentials to trigonometric functions: $e^{ix} = \cos(x) + i\sin(x)$
• A similar relationship exists for hyperbolic functions: $e^x = \cosh(x) + \sinh(x)$
• Using Euler's formula, complex hyperbolic functions can be expressed in terms of complex exponentials: $\sinh(z) = \frac{e^z - e^{-z}}{2}$ and $\cosh(z) = \frac{e^z + e^{-z}}{2}$

Hyperbolic functions of complex arguments

• When evaluating hyperbolic functions with complex arguments, the results are complex numbers
• For example, $\sinh(a + bi) = \sinh(a)\cos(b) + i\cosh(a)\sin(b)$ and $\cosh(a + bi) = \cosh(a)\cos(b) + i\sinh(a)\sin(b)$
• The real and imaginary parts of complex hyperbolic functions can be expressed using trigonometric functions

Identities for complex hyperbolic functions

• Complex hyperbolic functions satisfy many of the same identities as their real-valued counterparts
• Pythagorean identity: $\cosh^2(z) - \sinh^2(z) = 1$
• Reciprocal identities: $\tanh^2(z) + \text{sech}^2(z) = 1$ and $\coth^2(z) - \text{csch}^2(z) = 1$
• Sum and difference formulas, double angle formulas, and other trigonometric identities can be extended to complex hyperbolic functions