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:
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:
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 .
The function satisfies the following first‐order nonlinear differential equation:
The function has a simple series expansion at the origin that converges for all finite values with :
where are the Bernoulli numbers.
The function has a wellknown integral representation through the following definite integral along the positive part of the real axis:
The function has the following simple continued fraction representations:
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 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:
The following finite sum that contains the hyperbolic tangent function can be expressed using hyperbolic cotangent functions:
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:
In the case of multiple arguments , , …, the function can be represented as the ratio of the finite sums that includes powers of hyperbolic tangents:
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 ):
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:
The product of two hyperbolic tangent functions and the product of the hyperbolic tangent and cotangent have the following representations:
The most famous inequality for the hyperbolic tangent function is the following:
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:
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:
The hyperbolic tangent function has representations that use the trigonometric functions:
The hyperbolic tangent function is used throughout mathematics, the exact sciences, and engineering.
