Category. Mathematics.
Abstract. Hyperbolic tangent: definition, plot, properties and identities.
Reference. This article is a part of Librow scientific formula calculator project.
Hyperbolic tangent is defined as
tanhx ≡ (ex − e−x) /(ex + e−x)Hyperbolic tangent is antisymmetric function defined everywhere on real axis. Its plot is depicted below — fig. 1.
[画像:Fig. 1. Plot of the hyperbolic tangent function y = tanh x.] Fig. 1. Plot of the hyperbolic tangent function y = tanhx.Function codomain is limited to the range (−1, 1).
Base:
tanh2x + sech2x = 1By definition:
tanhx ≡ sinhx /coshx ≡ 1 /cothxProperty of antisymmetry:
tanh−x = −tanhxHalf-argument:
tanh(x/2) = (coshx − 1) /sinhxDouble argument:
tanh(2x) = 2 tanhx /(tanh2x + 1)Triple argument:
tanh(3x) = (tanh3x + 3 tanhx) /(3 tanh2x + 1)Quadruple argument:
tanh(4x) = (4 tanh3x + 4 tanhx) /(tanh4x + 6 tanh2x + 1)Power reduction:
tanh2x = (cosh(2x) − 1) /(cosh(2x) + 1)Sum and difference of arguments:
tanh(x + y) = (tanhx + tanhy) /(1 + tanhx tanhy)Product:
tanhx tanhy = [cosh(x + y) − cosh(x − y)] /[cosh(x + y) + cosh(x − y)]Sum:
tanhx + tanhy = sinh(x + y) /(coshx coshy)Hyperbolic tangent function tanh or th of the real argument is supported by free version of the Librow calculator.
Hyperbolic tangent function tanh or th of the complex argument is supported by professional version of the Librow calculator.
To calculate hyperbolic tangent of the number:
tanh(-1);To calculate hyperbolic tangent of the current result:
tanh(rslt);To calculate hyperbolic tangent of the angle x in memory:
tanh(mem[x]);