Introduction to the Hyperbolic Cotangent Function

Defining the hyperbolic cotangent function

The hyperbolic cotangent function is an old mathematical function. It was first used in the articles by L'Abbe Sauri (1774).

This function is easily defined as the ratio of the hyperbolic sine and cosine functions (or expanded, as the ratio of the half‐sum and half‐difference of two exponential functions in the points and ):

This function can also be defined as reciprocal to the hyperbolic tangent function:

After comparison with the famous Euler formulas for the cosine and sine functions, and , it is easy to derive the following representation of the hyperbolic cotangent through the circular cotangent:

This formula allows for the derivation of all properties and formulas for the hyperbolic cotangent from the corresponding properties and formulas for circular cotangent function.

A quick look at the hyperbolic cotangent function

Here is a graphic of the hyperbolic cotangent function for real values of its argument .

Representation through more general functions

The hyperbolic cotangent function can be represented using more general mathematical functions. As the ratio of the hyperbolic cosine and sine functions that are particular cases of the generalized hypergeometric, Bessel, Struve, and Mathieu functions, the hyperbolic cotangent function can also be represented as ratios of those special functions. But these representations are not very useful. It is more useful to write the hyperbolic cotangent function as particular cases of one special function. This can be done using doubly periodic Jacobi elliptic functions that degenerate into the hyperbolic cotangent function when their second parameter is equal to or :

Definition of the hyperbolic cotangent for a complex argument

In the complex ‐plane, the function is defined by the same formula as for real values.

In the points , where has zeros, the denominator of the last formula equals zero and has singularities (poles of the first order).

Here are two graphics showing the real and imaginary parts of the hyperbolic cotangent function over the complex plane:

The best-known properties and formulas for the hyperbolic cotangent function

Values in points

The values of the hyperbolic cotangent for special values of its argument can be easily derived from the corresponding values of the circular cotangent function in the special points of the circle:

The values at infinity can be expressed by the following formulas:

General characteristics

For real values of argument , the values of are real.

In the points , the values of are algebraic. In several cases, they can be , 0, or ⅈ:

The values of can be expressed using only square roots if and is a product of a power of 2 and distinct Fermat primes {3, 5, 17, 257, …}.

The function is an analytical function of that is defined over the whole complex ‐plane and does not have branch cuts and branch points. It has an infinite set of singular points:

(a) are the simple poles with residues 1. (b) is an essential singular point.

It is a periodic function with period :

The function is an odd function with mirror symmetry:

Differentiation

The first derivative of has simple representations using either the function or the function:

The derivative of has much more complicated representations than symbolic derivatives for and :

where is the Kronecker delta symbol: and .

Ordinary differential equation

The function satisfies the following first‐order nonlinear differential equation:

Series representation

The function has a simple Loran series expansion that converges for all finite values with :

where the are the Bernoulli numbers.

Integral representation

The function has a well-known integral representation through the following definite integral along the positive part of the real axis:

Continued fraction representations

The function has the following continued fraction representation:

Indefinite integration

Indefinite integrals of expressions involving the hyperbolic cotangent function can sometimes be expressed using elementary functions. However, special functions are frequently needed to express the results even when the integrands have a simple form (if they can be evaluated in closed form). Here are some examples:

Definite integration

Definite integrals that contain the hyperbolic cotangent function are sometimes simple:

Some special functions can be used to evaluate more complicated definite integrals. For example, the polylogarithm function is needed to express the following integral:

Finite summation

The following finite sum that contains the hyperbolic cotangent function can be expressed using the hyperbolic cotangent functions:

Addition formulas

The hyperbolic cotangent of a sum can be represented by the rule: "the hyperbolic cotangent of a sum is equal to the product of the hyperbolic cotangents plus one divided by the sum of the hyperbolic cotangents." A similar rule is valid for the hyperbolic cotangent of the difference:

Multiple arguments

In the case of multiple arguments , , …, the function can be represented as the ratio of the finite sums containing powers of hyperbolic cotangents:

Half-angle formulas

The hyperbolic cotangent of a half‐angle can be represented using two hyperbolic functions by the following simple formulas:

The hyperbolic sine function in the last formula can be replaced by a hyperbolic cosine function. But it leads to a more complicated representation that is valid in a horizontal strip:

The last restrictions can be removed by modifying the last identity (now the identity is valid for all complex ):

Sums of two direct functions

The sum of two hyperbolic cotangent functions can be described by rule: "the sum of the hyperbolic cotangents is equal to the hyperbolic sine of the sum multiplied by the hyperbolic cosecants." A similar rule is valid for the difference of two hyperbolic cotangents:

Products involving the direct function

The product of two hyperbolic cotangents and the product of the hyperbolic cotangent and tangent have the following representations:

Inequalities

The most famous inequality for the hyperbolic cotangent function is the following:

Relations with its inverse function

There are simple relations between the function and its inverse function :

The second formula is valid at least in the horizontal strip . Outside of this strip, a much more complicated relation (containing the unit step, real part, and the floor functions) holds:

Representations through other hyperbolic functions

The hyperbolic cotangent and tangent functions are connected by a very simple formula that contains the linear function in the argument:

The hyperbolic cotangent function can also be represented through other hyperbolic functions by the following formulas:

Representations through trigonometric functions

The hyperbolic cotangent function has similar representations using related trigonometric functions by the following formulas:

Applications

The hyperbolic cotangent function is used throughout mathematics, the exact sciences, and engineering.