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.
