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Elementary Functions > Csc[z] > Introduction to the Cosecant Function

The best-known properties and formulas for the cosecant function

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

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:

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 .

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

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:

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

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

The cosecant function has the following limit representation:

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 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:

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

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

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

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:

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:

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.

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

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

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

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:

The cosecant function has representations using the hyperbolic functions:

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