Introduction to the Hyperbolic Secant Function
Defining the hyperbolic secant function
The hyperbolic secant function is an old mathematical function.
This function is easily defined as the ratio of one and hyperbolic cosine functions:
After comparison with the famous Euler formula for the cosine function , it is easy to derive the following representation of the hyperbolic secant through the circular secant:
The previous formula allows establishment of all the properties and formulas for the hyperbolic secant from corresponding properties and formulas for the circular secant.
A quick look at the hyperbolic secant function Here is a graphic of the hyperbolic secant function for real values of its argument .
Representation through more general functions
The hyperbolic secant function can be represented using more general mathematical functions. As the reciprocal to the hyperbolic cosine function that is a particular case of the generalized hypergeometric, Bessel, Struve, and Mathieu functions, the hyperbolic secant function can also be represented as reciprocal to 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 hyperbolic secant function when their second parameter is equal to or :
Definition of the hyperbolic secant function for a complex argument In the complex ‐plane, the function is defined by the same formula used for real values: 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 hyperbolic secant function over the complex plane.
The bestknown properties and formulas for the hyperbolic secant function
Values in points
The values of the hyperbolic secant function for special values of its argument can be easily derived from the corresponding values of the circular secant function in special points of the circle:
The values at infinity can be expressed by the following formulas:
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 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 the symbolic derivatives for and :
where is the Kronecker delta symbol: and .
Ordinary differential equation
The functions 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 hyperbolic secant function can also be presented using other kinds of series with the following formulas:
Integral representation
The function has a wellknown 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 hyperbolic secant function:
Limit representation
The hyperbolic secant function has the following limit representation:
Indefinite integration
Indefinite integrals of expressions that contain the hyperbolic 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 hyperbolic 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, gamma functions, incomplete beta functions, and the Catalan constant are needed to express the following integrals:
Finite summation
The following finite sum that contain the hyperbolic secant has a simple value:
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 hyperbolic secant can be represented through the hyperbolic cosecant function:
Infinite products
The following infinite product from the hyperbolic secant can be represented through the hyperbolic cosecant function:
Addition formulas
The hyperbolic secant of a sum or difference can be represented in terms of hyperbolic sine and cosine as shown in the following formulas:
Multiple arguments
In the case of multiple arguments , , …, the function can be represented as a rational function that contains powers of the hyperbolic secant. Here are two examples:
Halfangle formulas
The hyperbolic secant of a half‐angle can be represented by the following simple formula that is valid in a horizontal 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 hyperbolic secant functions can be described by the following formulas:
Products involving the direct function
The product of two hyperbolic secants and the product of hyperbolic secant and cosecant have the following representations:
Inequalities
One of the most famous inequalities for the hyperbolic 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 horizontal half‐strip . This can be expanded to a full horizontal 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 hyperbolic functions
The hyperbolic secant and cosecant functions are connected by a very simple formula including the linear function in the argument:
The hyperbolic secant function can also be represented using other hyperbolic functions by the following formulas:
Representations through trigonometric functions
The hyperbolic secant function has the following representations using the trigonometric functions:
Applications
The hyperbolic secant function is used throughout mathematics, the exact sciences, and engineering.
