The best-known properties and formulas for the secant function

Values in points

Using the connection between the cosine and secant functions gives the following table of values of the secant function for angles between 0 and 2 π:

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 a real period :

The function is an even 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 series expansion at the origin that converges for all finite values with :

where are the Euler numbers.

The secant function can also be presented 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 secant function:

Limit representation

The secant function has the following limit representation:

Indefinite integration

Indefinite integrals of expressions involving the secant 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 secant function are sometimes simple and their values can be expressed through elementary functions. Here is one example:

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

Finite summation

Finite sums that contain the secant function have the following simple values:

Infinite summation

The evaluation limit of the last formula in the previous subsubsection for gives the following value for the corresponding infinite sum:

Finite products

The following finite product from the secant can be represented through the cosecant function:

Infinite products

The following infinite product from the secant can be represented through the cosecant function:

Addition formulas

The secants of a sum and a difference can be represented by the following formulas that are derived from the cosines of a sum and a difference:

Multiple arguments

In the case of multiple arguments , , , …, the function can be represented as a rational function including powers of a secant. Here are two examples:

Half-angle formulas

The secant 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 secant functions can be described by the following formulas:

Products involving the direct function

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

Inequalities

One of the most famous inequalities for a secant 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 (containing the unit step, real part, and the floor functions) holds:

Representations through other trigonometric functions

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

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

Representations through hyperbolic functions

The secant function has representations using the hyperbolic functions:

Applications

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