Introduction to the Hyperbolic Cosine Function

Defining the hyperbolic cosine function

The hyperbolic cosine function is an old mathematical function. It was used in the works of V. Riccati (1757), D. Foncenex (1759), and J. H. Lambert (1768).

This function is easily defined as the half‐sum of two exponential functions in the points and :

After comparison with the famous Euler formula for cosine: , it is easy to derive the following representation of the hyperbolic cosine through the circular cosine:

The previous formula allows the derivation of all properties and formulas for hyperbolic cosine from corresponding properties and formulas for the circular cosine.

The following formula can sometimes be used as an equivalent definition of the hyperbolic cosine function:

This series converges for all finite numbers .

A quick look at the hyperbolic cosine function

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

Representation using more general functions

The function is a particular case of more complicated mathematical functions. For example, it is a special case of the generalized hypergeometric function with the parameter at :

It is also a particular case of the modified Bessel function with the parameter , multiplied by :

Other Bessel functions can also be expressed through hyperbolic cosine functions for similar values of the parameter:

Struve functions can also degenerate into the hyperbolic cosine function for a similar value of the parameter:

But the function is also a degenerate case of the doubly periodic Jacobi elliptic functions when their second parameter is equal to or :

Finally, the function is the particular case of one more class of functions—the Mathieu functions:

Definition of the hyperbolic cosine for a complex argument

In the complex ‐plane, the function is defined by the same formula that is used for real values:

Here are two graphics showing the real and imaginary parts of the hyperbolic cosine function over the complex plane.

The best-known properties and formulas for the hyperbolic cosine function

Values in points

The values of the hyperbolic cosine function for special values of its argument can be easily derived from the corresponding values of the circular cosine 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 even be rational numbers, 0, or 1. Here are some examples:

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 entire analytical function of that is defined over the whole complex ‐plane and does not have branch cuts and branch points. It has an essential singular point at . It is a periodic function with the period :

The function is an even function with mirror symmetry:

Differentiation

The derivatives of have simple representations using either the function or the function:

Ordinary differential equation

The function satisfies the simplest possible linear differential equation with constant coefficients:

The complete solution of this equation can be represented as a linear combination of and with arbitrary constant coefficients and :

The function also satisfies first‐order nonlinear differential equations:

Series representation

The function has a simple series expansion at the origin that converges in the whole complex ‐plane:

Product representation

The following famous infinite product representation for clearly illustrates that at :

Indefinite integration

Indefinite integrals of expressions that contain the hyperbolic cosine 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:

The last integral cannot be evaluated in closed form using the known classical special functions for arbitrary values of parameters and .

Definite integration

Definite integrals that contain the hyperbolic cosine are sometimes simple, as shown the following example:

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

Integral transforms

Numerous formulas for integral transforms from circular cosine functions cannot be easily converted into corresponding formulas with a hyperbolic cosine function because the hyperbolic cosine grows exponentially at infinity. This applies for Fourier cosine and sine transforms, and for Mellin, Hilbert, Hankel, and other transforms.

An exceptional case is the Laplace transform that itself includes the exponential function in the kernel:

Finite summation

The following finite sums of the hyperbolic cosine can be expressed using the hyperbolic functions:

Infinite summation

The following infinite sums can be expressed using elementary functions:

Finite products

The following finite products from the hyperbolic cosine can be expressed using elementary functions:

Infinite products

The following infinite product that contains the hyperbolic cosine function can be expressed using the hyperbolic sine function:

Addition formulas

The hyperbolic cosine of a sum can be represented by the rule: "the hyperbolic cosine of a sum is equal to the sum of the product of the hyperbolic cosines and the product of the hyperbolic sines." A similar rule is valid for the hyperbolic cosine of the difference:

Multiple arguments

In the case of multiple arguments , , …, the function can be represented as the finite sum that contains powers of the hyperbolic sine and cosine:

The function can also be represented as the finite sum that contains only the hyperbolic cosine of :

Half-angle formulas

The hyperbolic cosine of the 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 floor functions.

Sums of two direct functions

The sum of two hyperbolic cosine functions can be described by the rule: "the sum of the hyperbolic cosines is equal to the doubled hyperbolic cosine of the half‐difference multiplied by the hyperbolic cosine of the half‐sum." A similar rule is valid for the difference of two hyperbolic cosines:

Products involving the direct function

The product of two hyperbolic cosine functions or the product of the hyperbolic cosine and hyperbolic sine have the following representations:

Powers of the direct function

The integer powers of the hyperbolic cosine functions can be expanded as finite sums of hyperbolic cosine functions with multiple arguments. These sums contain binomial coefficients:

Inequalities

The best-known inequalities for the hyperbolic cosine function are 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 right half of the horizontal strip . It can be generalized to the full horizontal strip by changing to in its right side:

For the whole complex plane, a much more complicated relation (that contains the unit step, real part, imaginary part, and the floor functions) holds:

Representations through other hyperbolic functions

Hyperbolic cosine and sine functions are connected by a very simple formula that contains the linear function in the argument:

Another famous formula, connecting and , is expressed in the analog of the well‐known Pythagorean theorem:

The restriction on can be removed, but the formula will get a complicated coefficient that contains the unit step, real part, imaginary part, and the floor functions and c(z) ⩵ 1:

The hyperbolic cosine function can also be represented using other hyperbolic functions by the following formulas:

Representation through trigonometric functions

The hyperbolic cosine function has similar representations using related trigonometric functions and formulas:

Applications

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