Introduction to the Hyperbolic Cosecant Function

Defining the hyperbolic cosecant function

The hyperbolic cosecant function is an old mathematical function.

This function has a simple definition as the reciprocal to hyperbolic sine functions:

After comparison with the famous Euler formula for the sine function , it is easy to derive the following representation of the hyperbolic cosecant through the circular cosecant:

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

A quick look at the hyperbolic cosecant function

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

Representation through more general functions

The hyperbolic cosecant function can be represented using more general mathematical functions. As the reciprocal to the hyperbolic sine function that is a particular case of the generalized hypergeometric, Bessel, Struve, and Mathieu functions, the hyperbolic cosecant function can also be represented as reciprocal to those special functions. Here are some examples:

But these representations are not very useful because they include complicated special functions in the denominators.

It is more interesting to write the hyperbolic cosecant function as particular cases of one special function. Such formulas take place for doubly periodic Jacobi elliptic functions that can degenerate into a hyperbolic cosecant function when their second parameter is equal to or :

Definition of the hyperbolic cosecant function for a complex argument

In the complex ‐plane, the function is defined by the same formula that was used to define 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 cosecant function over the complex plane.

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

Values in points

The values of the hyperbolic cosecant function for special values of its argument can be easily derived from the corresponding values of the circular cosecant 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 , , , or :

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 . (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 the symbolic derivatives for and :

where is the Kronecker delta symbol: and .

Ordinary differential equation

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

Series representation

The function has the following Laurent series expansion at the origin that converges for all finite values with :

where are the Bernoulli numbers.

The hyperbolic cosecant function can also be represented using other kinds of series by the following formulas:

Integral representation

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

Product representation

The famous infinite product representation for can be easily rewritten as the following product representation for the hyperbolic cosecant function:

Limit representation

The hyperbolic cosecant function has the following limit representation:

Indefinite integration

Indefinite integrals of expressions involving the hyperbolic cosecant 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 cosecant function are sometimes simple. Here is an example:

Some special functions can be used to evaluate more complicated definite integrals. For example, zeta and polylogarithmic functions are needed to express the following integrals:

Finite summation

The following finite sum that contains the hyperbolic cosecant has this simple value:

Infinite summation

The following infinite sum that contains the hyperbolic cosecant has this simple value:

where is the inverse elliptic nome.

Addition formulas

The hyperbolic cosecant of a sum or difference can be represented in terms of the hyperbolic sine and cosine as shown in the following formulas:

Multiple arguments

In the case of multiple arguments , , …, the function can be represented as a rational function that contains powers of the hyperbolic cosecants and secants. Here are two examples:

Half-angle formulas

The hyperbolic cosecant of a half‐angle can be represented by the following simple formula that is valid in a horizontal half‐strip:

This formula can be expanded to the full horizontal strip if an additional regulator of the sign is added:

To make this formula correct for all complex , a more complicated prefactor is needed:

where contains the unit step, real part, imaginary part, and the floor functions.

Sums of two direct functions

The sum and difference of two hyperbolic cosecant functions can be described by the following formulas:

Products involving the direct function

The product of two hyperbolic cosecants and the product of the hyperbolic cosecant and secant have the following representations:

Inequalities

One of the most famous inequalities for the hyperbolic cosecant 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 (that contains the unit step, real part, and the floor functions) holds:

Representations through other hyperbolic functions

The hyperbolic cosecant and secant functions are connected by a very simple formula that includes the linear function in the argument:

The hyperbolic cosecant function can also be represented using other hyperbolic functions by the following formulas:

Representations through trigonometric functions

The hyperbolic cosecant function has the following representations using the trigonometric functions:

Applications

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