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:
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:
The derivatives of have simple representations using either the function or the function:
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:
The function has a simple series expansion at the origin that converges in the whole complex ‐plane:
The following famous infinite product representation for clearly illustrates that at :
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 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:
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:
The following finite sums of the hyperbolic cosine can be expressed using the hyperbolic functions:
The following infinite sums can be expressed using elementary functions:
The following finite products from the hyperbolic cosine can be expressed using elementary functions:
The following infinite product that contains the hyperbolic cosine function can be expressed using the hyperbolic sine function:
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:
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 :
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.
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:
The product of two hyperbolic cosine functions or the product of the hyperbolic cosine and hyperbolic sine have the following representations:
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:
The bestknown inequalities for the hyperbolic cosine function are the following:
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:
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:
The hyperbolic cosine function has similar representations using related trigonometric functions and formulas:
The hyperbolic cosine function is used throughout mathematics, the exact sciences, and engineering.
