Introduction to the Cosecant Function

Defining the cosecant function

The cosecant function is an old mathematical function. It was mentioned in the works of G. J. von Lauchen Rheticus (1596) and E. Gunter (around 1620). It was widely used by L. Euler (1748) and T. Olivier, Wait, and Jones (1881).

The classical definition of the cosecant function for real arguments is: "the cosecant of an angle in a right‐angle triangle is the ratio of the length of the hypotenuse to the length of the opposite leg." This description of is valid for when this triangle is nondegenerate. This approach to the cosecant can be expanded to arbitrary real values of if the arbitrary point in the ,‐Cartesian plane is considered and is defined as the ratio assuming that α is the value of the angle between the positive direction of the ‐axis and the direction from the origin to the point .

Comparing the classical definition with the definition of the sine function shows that the following formula can also be used as a definition of the cosecant function:

A quick look at the cosecant function

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

Representation through more general functions

The cosecant function can be represented using more general mathematical functions. As the ratio of one divided by the sine function that is a particular case of the generalized hypergeometric, Bessel, Struve, and Mathieu functions, the cosecant function can also be represented as ratios of one and 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 useful to write the cosecant function as particular cases of one special function. That can be done using doubly periodic Jacobi elliptic functions that degenerate into the cosecant function when their second parameter is equal to or :

Definition of the cosecant function for a complex argument

In the complex ‐plane, the function is defined using or the exponential function in the points and through the formula:

In the points , where is zero, 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 cosecant function over the complex plane.

The best-known properties and formulas for the cosecant function

Values in points

Using the connection between the sine and cosecant functions, the following table of cosecant function values for angles between 0 and can be derived:

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 integers , , 1, or 2:

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 the real 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 the following Laurent series expansion at the origin that converges for all finite values with :

where are the Bernoulli numbers.

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

Integral representation

The function has 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 cosecant function:

Limit representation

The cosecant function has the following limit representation:

Indefinite integration

Indefinite integrals of expressions that contain the 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 cosecant function are sometimes simple. For example, the famous Catalan constant can be defined as the value of the following integral:

This constant also appears in the following integral:

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

Finite summation

The following finite sums that contain the cosecant function have simple values:

Infinite summation

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

Finite products

The following finite product from the cosecant can also be represented using the cosecant function:

Addition formulas

The cosecant of a sum and the cosecant of a difference can be represented by the formulas that follow from corresponding formulas for the sine of a sum and the sine of a difference:

Multiple arguments

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

Half-angle formulas

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

To make this formula correct for all complex , a 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 cosecant functions can be described by the following formulas:

Products involving the direct function

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

Inequalities

Some inequalities for the cosecant function can be easily derived from the corresponding inequalities for the sine function:

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 vertical 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 trigonometric functions

Cosecant and secant functions are connected by a very simple formula that contains the linear function in the argument:

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

Representations through hyperbolic functions

The cosecant function has representations using the hyperbolic functions:

Applications

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