The best-known properties and formulas for the cotangent function

Values in points

Students usually learn the following basic table of values of the cotangent function for special points of the circle:

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 , 0, or 1:

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 a simple Laurent series expansion at the origin that converges for all finite values with :

where 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 simple continued fraction representation:

Indefinite integration

Indefinite integrals of expressions that contain the 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 cotangent 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, to express the following integral, the Gauss hypergeometric function is needed:

Finite summation

The following finite sum that contains a cotangent function can be expressed in terms of a cotangent function:

Other finite sums that contain a cotangent function can be expressed in terms of a polynomial function:

Infinite summation

The following infinite sum that contains the cotangent function has a very simple value:

Finite products

The following finite product from the cotangent has a very simple value:

Addition formulas

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

Multiple arguments

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

Half-angle formulas

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

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

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

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

Sums of two direct functions

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

Products involving the direct function

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

Inequalities

The most famous inequality for the 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 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

Cotangent and tangent functions are connected by a very simple formula that contains the linear function in the following argument:

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

Representations through hyperbolic functions

The cotangent function has representations using the hyperbolic functions:

Applications

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