Hyperbolic Functions

Introduction

Just as the circular functions sine and cosine parametrize the unit circle, the hyperbolic functions sinh and cosh parametrize the unit hyperbola x² - y² = 1. While circular functions arise from circular motion, hyperbolic functions appear naturally in the study of hanging chains, special relativity, and many differential equations.

The hyperbolic functions satisfy identities remarkably parallel to their circular counterparts, with key sign changes reflecting the geometry of hyperbolas versus circles. Understanding these parallels illuminates both families of functions.

Unlike circular functions, hyperbolic functions are not periodic. They grow or decay exponentially, since they are defined in terms of exponential functions.

Definitions

The basic hyperbolic functions are defined in terms of the exponential function.

Hyperbolic Sine and Cosine

sinh(x)=(ex−e(−x))/2

cosh(x)=(ex+e(−x))/2

Note that sinh is the odd part of ex (sinh(−x)=−sinh(x) while cosh is the even part (cosh(−x)=cosh(x). Together they decompose the exponential:

ex=cosh(x)+sinh(x)

Other Hyperbolic Functions

The remaining hyperbolic functions are defined analogously to their circular counterparts:

tanh(x)=sinh(x)/cosh(x)=(ex−e(−x))/(ex+e(−x))

coth(x)=cosh(x)/sinh(x)=(ex+e(−x))/(ex−e(−x)),x≠0

sech *x=1/cosh(x)=2/(ex+e(−x))

csch *x=1/sinh(x)=2/(ex−e(−x)),x≠0

Fundamental Identity

The defining identity of hyperbolic functions parallels the Pythagorean identity for circular functions, but with a crucial sign change:

cosh2(x)−sinh2(x)=1

Proof: Expanding using the definitions:

cosh2(x)−sinh2(x)=((ex+e(−x))/2)2−((ex−e(−x))/2)2

=(e(2*x)+2+e(−2*x))/4−(e(2*x)−2+e(−2*x))/4=4/4=1

This identity means (cosh(x),sinh(x)) lies on the hyperbola x2−y2=1 just as (cos(θ),sin(θ)) lies on the circle x2+y2=1

Related Identities

Dividing the fundamental identity by cosh2(x) or sinh2(x) yields:

1−tanh2(x)=sech2(x)

coth2(x)−1=csch2(x)

Addition Formulas

The addition formulas for hyperbolic functions closely mirror those for circular functions:

sinh(x+y)=sinh(x)*cosh(y)+cosh(x)*sinh(y)

cosh(x+y)=cosh(x)*cosh(y)+sinh(x)*sinh(y)

Note the plus sign in the cosh formula where the cosine addition formula has a minus. This sign difference arises from the hyperbolic versus circular geometry.

tanh(x+y)=(tanh(x)+tanh(y))/(1+tanh(x)*tanh(y))

Double Angle Formulas

Setting y=x in the addition formulas:

sinh(2)*x=2*sinh(x)*cosh(x)

cosh(2)*x=cosh2(x)+sinh2(x)=2*cosh2(x)−1=1+2*sinh2(x)

Derivatives

The derivatives of hyperbolic functions are elegantly simple:

d()/d(x)*sinh(x)=cosh(x)

d()/d(x)*cosh(x)=sinh(x)

Unlike circular functions where d()/d(x)*(cos(x))=−sin(x) here we have d()/d(x)*(cosh(x))=sinh(x) with no negative sign. This makes cosh x its own second derivative partner.

d()/d(x)*tanh(x)=sech2(x)

d()/d(x)*coth(x)=−csch2(x)

d()/d(x)*sech *x=−sech *x*tanh(x)

d()/d(x)*csch *x=−csch *x*coth(x)

Inverse Hyperbolic Functions

Since hyperbolic functions are defined via exponentials, their inverses can be expressed as logarithms.

arsinh *x=ln(x+√(,x2+1)),x∈R

arcosh *x=ln(x+√(,x2−1)),x≥1

artanh *x=1/2*ln((1+x)/(1−x)),|x|<1

Derivatives of Inverse Functions

d()/d(x)*arsinh *x=1/√(,x2+1)

d()/d(x)*arcosh *x=1/√(,x2−1),x>1

d()/d(x)*artanh *x=1/(1−x2),|x|<1

Connection to Circular Functions

Hyperbolic and circular functions are intimately connected through complex numbers. Using Euler's formula e(i*x)=cos(x)+i*sin(x)

sinh(i*x)=i*sin(x)

cosh(i*x)=cos(x)

Conversely:

sin(x)=−i*sinh(i*x)

cos(x)=cosh(i*x)

This explains why hyperbolic identities mirror circular identities with sign changes—they are related by multiplying the argument by i

Applications

The Catenary

A hanging chain or cable under gravity forms a curve called a catenary, described by:

y=a*cosh(x/a)

where a depends on the chain's weight density and tension. The Gateway Arch in St. Louis is an inverted catenary.

Special Relativity

In special relativity, rapidities (hyperbolic angles) add linearly even though velocities do not. If rapidity φ is defined by tanh(φ)=v/c then:

γ=cosh(φ),γ*v/c=sinh(φ)

Lorentz transformations become hyperbolic rotations, paralleling how ordinary rotations involve circular functions.

Differential Equations

The equation y″=y has solutions y=A*ex+B*e(−x)=C*cosh(x)+D*sinh(x) Hyperbolic functions are natural solutions when the characteristic equation has real roots.

Summary

Hyperbolic functions are defined using exponentials: sinh(x)=(ex−e(−x))/2 and cosh(x)=(ex+e(−x))/2 They parametrize the hyperbola x2−y2=1 analogous to how circular functions parametrize the circle.

The fundamental identity is cosh2(x)−sinh2(x)=1 (note the minus sign, contrasting with cos2(x)+sin2(x)=1. Addition formulas and other identities parallel circular ones with characteristic sign changes.

Derivatives are particularly elegant: d()/d(x)*(sinh(x))=cosh(x) and d()/d(x)*(cosh(x))=sinh(x) with no minus signs. Inverse hyperbolic functions are expressible as logarithms.

The connection sinh(i*x)=i*sin(x) and cosh(i*x)=cos(x) reveals hyperbolic and circular functions as different aspects of the same complex-analytic structure. Applications include the catenary curve, special relativity, and differential equations with real characteristic roots.

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions defined using the exponential function rather than the unit circle. They arise naturally in many areas of mathematics and physics, including the description of catenary curves, relativistic velocity addition, and solutions to certain differential equations.

The hyperbolic functions are named for their relationship to the hyperbola, just as trigonometric functions relate to the circle.

Definitions

The two fundamental hyperbolic functions are hyperbolic sine and hyperbolic cosine:

sinh(x)=(ex−e(−x))/2

cosh(x)=(ex+e(−x))/2

The other hyperbolic functions are defined in terms of these:

tanh(x)=sinh(x)/cosh(x)=(ex−e(−x))/(ex+e(−x))

coth(x)=cosh(x)/sinh(x)=(ex+e(−x))/(ex−e(−x))

sech *x=1/cosh(x)=2/(ex+e(−x))

csch *x=1/sinh(x)=2/(ex−e(−x))

Fundamental Identity

The hyperbolic analog of the Pythagorean identity is:

cosh2(x)−sinh2(x)=1

Note the minus sign, in contrast to the plus sign in the circular identity cos2(x)+sin2(x)=1 This reflects the difference between circles and hyperbolas.

Related identities include:

1−tanh2(x)=sech2(x)

coth2(x)−1=csch2(x)

Graphs and Properties

Hyperbolic Sine

$\sinh x$ is an odd function: $\sinh(-x) = -\sinh(x)$. It passes through the origin, is negative for $x < 0$, and positive for $x > 0$. The function increases without bound in both directions, growing like $e^x/2$ for large positive $x$. ### Hyperbolic Cosine $\cosh x$ is an even function: $\cosh(-x) = \cosh(x)$. It has a minimum value of 1$ at $x = 0$ and increases in both directions. Its graph is called a catenary, the shape of a hanging chain. ### Hyperbolic Tangent $\tanh x$ is an odd function bounded between $-1$ and 1$. It has horizontal asymptotes at $y = 1$ and $y = -1$. The function is commonly used in neural networks as an activation function. ## Derivatives The derivatives of hyperbolic functions follow patterns similar to trigonometric functions: $$

\frac{d}{dx}(\sinh x) = \cosh x

\frac{d}{dx}(\cosh x) = \sinh x

\frac{d}{dx}(\tanh x) = \text{sech}^2 x

\frac{d}{dx}(\coth x) = -\text{csch}^2 x

\frac{d}{dx}(\text{sech } x) = -\text{sech } x \tanh x

\frac{d}{dx}(\text{csch } x) = -\text{csch } x \coth x

$$ Note that the derivative of $\cosh$ is $\sinh$ (with no sign change), unlike the derivative of $\cos$ which is $-\sin$. ## Integrals The basic integrals follow directly from the derivatives: $$

\int \sinh x , dx = \cosh x + C

\int \cosh x , dx = \sinh x + C

\int \tanh x , dx = \ln(\cosh x) + C

$$ ## Inverse Hyperbolic Functions The inverse hyperbolic functions can be expressed in terms of logarithms: $$

\text{arsinh } x = \ln(x + \sqrt{x^2 + 1})

\text{arcosh } x = \ln(x + \sqrt{x^2 - 1}), \quad x \geq 1

\text{artanh } x = \frac{1}{2}\ln\left(\frac{1 + x}{1 - x}\right), \quad |x| < 1

$$ These formulas are derived by solving the definitions for $x$. The derivatives of inverse hyperbolic functions appear in many integration formulas. ## Addition Formulas Hyperbolic functions satisfy addition formulas analogous to trigonometric ones: $$

\sinh(x + y) = \sinh x \cosh y + \cosh x \sinh y

\cosh(x + y) = \cosh x \cosh y + \sinh x \sinh y

\tanh(x + y) = \frac{\tanh x + \tanh y}{1 + \tanh x \tanh y}

$$ Notice that $\cosh(x + y)$ has plus signs throughout, unlike $\cos(x + y)$ which has a minus sign. ## Connection to Trigonometric Functions Using Euler's formula and its hyperbolic analog, we can relate the two families: $$

\cosh(ix) = \cos x, \quad \sinh(ix) = i\sin x

\cos(ix) = \cosh x, \quad \sin(ix) = i\sinh x

$$

This shows that hyperbolic functions are trigonometric functions evaluated at imaginary arguments.

Applications

The Catenary

A flexible chain or cable hanging under its own weight forms a catenary curve, described by y=a*cosh(x/a) This is not a parabola, though it looks similar. The Gateway Arch in St. Louis follows an inverted catenary.

Special Relativity

In special relativity, hyperbolic functions describe rapidity, an alternative to velocity. The addition formula for tanh() corresponds to relativistic velocity addition. Lorentz transformations can be written using cosh() and sinh() of rapidity.

Differential Equations

Hyperbolic functions appear in solutions to differential equations like y″−y=0 where the general solution is y=A*cosh(x)+B*sinh(x) This contrasts with y″+y=0 solved by trigonometric functions.

Machine Learning

The tanh() function is widely used as an activation function in neural networks. Its output range of (−1,1) and its smooth gradient make it useful for learning.

Geometric Interpretation

Just as circular functions parametrize the unit circle x2+y2=1 hyperbolic functions parametrize the unit hyperbola x2−y2=1 The point (cosh(t),sinh(t)) lies on the right branch of this hyperbola.

The parameter t represents twice the signed area between the xaxis, the hyperbola, and the line from the origin to (cosh(t),sinh(t)) analogous to how the angle in radians relates to arc length on a circle.

Hyperbolic functions bridge exponential and trigonometric functions, appearing throughout mathematics, physics, and engineering wherever growth, decay, or hyperbolic geometry arise.