Hyperbolic functions

Hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. They are defined in terms of the exponential function and are important in many areas of mathematics, including algebra, geometry, calculus, and complex analysis. Hyperbolic functions are closely related to the exponential function, as evident from their definitions. The basic hyperbolic functions are:

Definition of Hyperbolic functions

1. Hyperbolic Sine ($\sinh$):
   $$
   \sinh(x) = \frac{e^x – e^{-x}}{2}
   $$
   This function is an odd function, similar to the sine function in trigonometry.
2. Hyperbolic Cosine ($\cosh$):
   $$
   \cosh(x) = \frac{e^x + e^{-x}}{2}
   $$
   This function is an even function, similar to the cosine function in trigonometry.
3. Hyperbolic Tangent ($\tanh$):
   $$
   \tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x – e^{-x}}{e^x + e^{-x}}
   $$
   This function is similar to the tangent function in trigonometry.
4. Hyperbolic Cotangent ($\coth$):
   $$
   \coth(x) = \frac{\cosh(x)}{\sinh(x)} = \frac{e^x + e^{-x}}{e^x – e^{-x}}
   $$
   This is the reciprocal of the hyperbolic tangent.
5. Hyperbolic Secant ($sech$):
   $$
   sech(x) = \frac{1}{\cosh(x)} = \frac{2}{e^x + e^{-x}}
   $$
   This is the reciprocal of the hyperbolic cosine.
6. Hyperbolic Cosecant ($csch$):
   $$
   csch(x) = \frac{1}{\sinh(x)} = \frac{2}{e^x – e^{-x}}
   $$
   This is the reciprocal of the hyperbolic sine.

Odd and Even Hyperbolic functions

Like trigonometric functions, hyperbolic functions have properties of being odd or even. $\sinh(x)$ is odd, and $\cosh(x)$ is even.
We know
$\sin(x) = – \sin(-x)$
Similary
$$
\sinh(x) = \frac{e^x – e^{-x}}{2}
$$
$$
\sinh(-x) = \frac{e^{-x} – e^{x}}{2}= – \frac{e^x – e^{-x}}{2}= – \sinh(x)
$$

We know
$\cos(x) = \cos(-x)$
Similary
$$
\cosh(x) = \frac{e^x + e^{-x}}{2}
$$
$$
\cosh(-x) = \frac{e^{-x} + e^{x}}{2}= \cosh(x)
$$

Hyperbolic Identities

There are several identities involving hyperbolic functions that are analogous to trigonometric identities. For example, $\cosh^2(x) – \sinh^2(x) = 1$, which is similar to the Pythagorean identity in trigonometry.

(I) $\cosh^2(x) – \sinh^2(x) = 1$
(II) $\tanh ^2(x) + sech^2(x) = 1$
(III) $\coth^2(x) – cosech^2(x) = 1$

Proof of I

$\cosh^2(x) – \sinh^2(x) $
$= [\frac{e^x + e^{-x}}{2}]^2 – [\frac{e^x – e^{-x}}{2}]^2$
$= \frac {e^{2x} + e^{-2x} + 2}{4} – \frac {e^{2x} + e^{-2x} – 2}{4}$
$=1$

Proof of II

Recall the definitions of $\tanh(x)$ and $sech(x)$:

1. $\tanh(x) = \frac{\sinh(x)}{\cosh(x)}$
2. $sech(x) = \frac{1}{\cosh(x)}$

Substituting these values we get

$$
\tanh^2(x) + sech^2(x) = \frac{\sinh^2(x) + 1}{\cosh^2(x)}
$$

Use the identity $\cosh^2(x) – \sinh^2(x) = 1$, which can be rearranged to $\sinh^2(x) + 1 = \cosh^2(x)$:

$$
\tanh^2(x) + sech^2(x) = \frac{\cosh^2(x)}{\cosh^2(x)} = 1
$$

Proof of III

The hyperbolic cotangent and cosecant functions are defined as follows:

1. $\coth(x) = \frac{\cosh(x)}{\sinh(x)}$
2. $csch(x) = \frac{1}{\sinh(x)}$

Substituting these values we get

$$
\coth^2(x) – csch^2(x) = \frac{\cosh^2(x) – 1}{\sinh^2(x)}
$$

Use the identity $\cosh^2(x) – \sinh^2(x) = 1$, which can be rearranged to $\cosh^2(x) = 1 + \sinh^2(x)$:

$$
\coth^2(x) – csch^2(x) = \frac{1 + \sinh^2(x) – 1}{\sinh^2(x)}
$$

$$
\coth^2(x) – csch^2(x) = \frac{\sinh^2(x)}{\sinh^2(x)} = 1
$$

Inverse Hyperbolic Functions

They are the inverses of the hyperbolic functions. Just as the inverse trigonometric functions (like arcsin, arccos, and arctan) are used to find the angle corresponding to a given trigonometric ratio, inverse hyperbolic functions are used to find the hyperbolic angle corresponding to a hyperbolic function value. The main inverse hyperbolic functions are:

1. Inverse Hyperbolic Sine ($\text{arsinh}$ or $\sinh^{-1}$):
   $$
   \text{arsinh}(x) = \ln\left(x + \sqrt{x^2 + 1}\right)
   $$
   This function gives the value whose hyperbolic sine is $x$.
2. Inverse Hyperbolic Cosine ($\text{arcosh}$ or $\cosh^{-1}$):
   $$
   \text{arcosh}(x) = \ln\left(x + \sqrt{x^2 – 1}\right)
   $$
   This is defined for $x \geq 1$. It gives the value whose hyperbolic cosine is $x$.
3. Inverse Hyperbolic Tangent ($\text{artanh}$ or $\tanh^{-1}$):
   $$
   \text{artanh}(x) = \frac{1}{2} \ln\left(\frac{1 + x}{1 – x}\right)
   $$
   This is defined for $-1 < x < 1$. It gives the value whose hyperbolic tangent is $x$.
4. Inverse Hyperbolic Cotangent ($\text{arcoth}$ or $\coth^{-1}$):
   $$
   \text{arcoth}(x) = \frac{1}{2} \ln\left(\frac{x + 1}{x – 1}\right)
   $$
   This is defined for $|x| > 1$. It gives the value whose hyperbolic cotangent is $x$.
5. Inverse Hyperbolic Secant ($\text{arsech}$ or $sech^{-1}$):
   $$
   \text{arsech}(x) = \ln\left(\frac{1}{x} + \sqrt{\frac{1}{x^2} – 1}\right)
   $$
   This is defined for $0 < x \leq 1$. It gives the value whose hyperbolic secant is $x$.
6. Inverse Hyperbolic Cosecant ($\text{arcsch}$ or $csch^{-1}$):
   $$
   \text{arcsch}(x) = \ln\left(\frac{1}{x} + \sqrt{\frac{1}{x^2} + 1}\right)
   $$
   This function gives the value whose hyperbolic cosecant is $x$.