Introduction to the Secant Function

Defining the secant function

The word "secant" was introduced by Alhabaš Alhãsib around 800. Later on Th. Fincke (1583) used the word "secans" in Latin for notation of the corresponding function. The secant function also appeared in the works of A. Magini (1592) and B. Cavalieri (1643). J. Kresa (1720) used the symbol "sec" that was later widely used by L. Euler (1748).

The classical definition of the secant function for real arguments is: "the secant of an angle in a right‐angle triangle is the ratio of the length of the hypotenuse to the adjacent leg." This description of is valid for when this triangle is nondegenerate. This approach to the secant can be expanded to arbitrary real values of if consideration is given to the arbitrary point in the ,‐Cartesian plane 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 definition with the definition of the cosine function shows that the following formula can also be used as a definition of the secant function:

A quick look at the secant function

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

Representation through more general functions

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

Definition of the secant 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 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 secant function over the complex plane.

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.