Introduction to the Hyperbolic Tangent Function

Defining the hyperbolic tangent function

The hyperbolic tangent function is an old mathematical function. It was first used in the work by L'Abbe Sauri (1774).

This function is easily defined as the ratio between the hyperbolic sine and the cosine functions (or expanded, as the ratio of the half‐difference and half‐sum of two exponential functions in the points and ):

After comparison with the famous Euler formulas for the sine and cosine functions, and , it is easy to derive the following representation of the hyperbolic tangent through the circular tangent function:

This formula allows the derivation of all the properties and formulas for the hyperbolic tangent from the corresponding properties and formulas for the circular tangent.

A quick look at the hyperbolic tangent function

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

Representation through more general functions

The hyperbolic tangent function can be represented using more general mathematical functions. As the ratio of the hyperbolic sine and cosine functions that are particular cases of the generalized hypergeometric, Bessel, Struve, and Mathieu functions, the hyperbolic tangent function can also be represented as ratios of those special functions. But these representations are not very useful. It is more useful to write the hyperbolic tangent function as particular cases of one special function. That can be done using doubly periodic Jacobi elliptic functions that degenerate into the hyperbolic tangent function when their second parameter is equal to or :

Definition of the hyperbolic tangent function for a complex argument

In the complex ‐plane, the function is defined by the same formula that is 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 tangent function over the complex plane.

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

Values in points

The values of the hyperbolic tangent for special values of its argument can be easily derived from corresponding values of the circular tangent in the 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 , 0, or ⅈ:

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 1. (b) is an essential singular point.

It is a periodic function with period :

The function is an odd 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 function satisfies the following first‐order nonlinear differential equation:

Series representation

The function has a simple series expansion at the origin that converges for all finite values with :

where are the Bernoulli numbers.

Integral representation

The function has a well-known integral representation through the following definite integral along the positive part of the real axis:

Continued fraction representations

The function has the following simple continued fraction representations:

Indefinite integration

Indefinite integrals of expressions involving the hyperbolic tangent 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 contains the hyperbolic tangent function are sometimes simple. For example, the famous Catalan constant can be defined through the following integral:

Some special functions can be used to evaluate more complicated definite integrals. For example, the hypergeometric function is needed to express the following integral:

Finite summation

The following finite sum that contains the hyperbolic tangent function can be expressed using hyperbolic cotangent functions:

Addition formulas

The hyperbolic tangent of a sum can be represented by the rule: "the hyperbolic tangent of a sum is equal to the sum of the hyperbolic tangents divided by one plus the product of the hyperbolic tangents". A similar rule is valid for the hyperbolic tangent of the difference:

Multiple arguments

In the case of multiple arguments , , …, the function can be represented as the ratio of the finite sums that includes powers of hyperbolic tangents:

Half-angle formulas

The hyperbolic tangent of a half‐angle can be represented using two hyperbolic functions by the following simple formulas:

The hyperbolic sine function in the last formula can be replaced by the hyperbolic cosine function. But it leads to a more complicated representation that is valid in a horizontal strip:

The last restrictions can be removed by slightly modifying the formula (now the identity is valid for all complex ):

Sums of two direct functions

The sum of two hyperbolic tangent functions can be described by the rule: "the sum of hyperbolic tangents is equal to the hyperbolic sine of the sum multiplied by the hyperbolic secants". A similar rule is valid for the difference of two hyperbolic tangents:

Products involving the direct function

The product of two hyperbolic tangent functions and the product of the hyperbolic tangent and cotangent have the following representations:

Inequalities

The most famous inequality for the hyperbolic tangent 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 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 tangent and cotangent functions are connected by a very simple formula that contains the linear function in the argument:

The hyperbolic tangent function can also be represented through other hyperbolic functions by the following formulas:

Representations through trigonometric functions

The hyperbolic tangent function has representations that use the trigonometric functions:

Applications

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