Modified hyperbolic tangent

The modified hyperbolic tangent (mtanh or mth) is a special S-shaped function based on the hyperbolic tangent, given by

$\operatorname{mtanh}x = \frac {e^{ax} - e^{-bx}} {e^{cx} + e^{-dx}}$

This function was proposed by Elena Soboleva (of Ukraine) as a utility function for multi-objective optimization and choice modelling in decision-making.^[1][2][3]

In other papers it is used for approximation of current-voltage characteristics of field-effect transistors^[4] and light-emitting diodes.

With parameters a = b = c = d = 1 the modified hyperbolic tangent function reduces to the conventional tanh(x) function, whereas for a = b = 1 and c = d = 0, the term becomes equal to sinh(x).

See also

• e (mathematical constant)
• Equal incircles theorem, based on sinh
• Inverse hyperbolic functions
• List of integrals of hyperbolic functions
• Poinsot's spirals
• Sigmoid function

References

1. Soboleva, Elena. V.; Beskorovainyi, V. V. (2008). The utility function in problems of structural optimization of distributed objects (in Russian). Kharkiv University of Air Force. p. 121.
2. Soboleva, Elena V. (2009). The S-shaped utility function of individual criteria for multi-objective decision-making in design (in Russian). Kharkiv National University of Radioelectronics. p. 247.
3. Beskorovainyi, V. V.; Soboleva, Elena V. (2010). ИДЕНТИФИКАЦИЯ ЧАСТНОй ПОлЕЗНОСТИ МНОГОФАКТОРНЫХ АлЬТЕРНАТИВ С ПОМОЩЬЮ S-ОБРАЗНЫХ ФУНКЦИй [Identification of utility functions in multi-objective choice modelling by using S-shaped functions] (PDF) (in Russian). Kharkiv National University of Radioelectronics. pp. 50–54. УДК 519.688: 004.896; БИОНИКА ИНТЕЛЛЕКТА 2010. No 1 (72). line feed character in |title= at position 61 (help)
4. Tuev, Vasily I.; Uzhanin, Maxim V. (2009). ПРИМЕНЕНИЕ МОДИФИЦИРОВАННОЙ ФУНКЦИИ ГИПЕРБОЛИЧЕСКОГО ТАНГЕНСА ДЛЯ АППРОКСИМАЦИИ ВОЛЬТАМПЕРНЫХ ХАРАКТЕРИСТИК ПОЛЕВЫХ ТРАНЗИСТОРОВ [Using modified hyperbolic tangent function to approximate the current-voltage characteristics of field-effect transistors] (in Russian). Tomsk Politehnic University. pp. 135–138. 4/314/2009. Retrieved 2015-11-05.