Tanh [z]
gives the hyperbolic tangent of z.
Tanh
Tanh [z]
gives the hyperbolic tangent of z.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- Sinh [z]/Cosh [z] is automatically converted to Tanh [z]. TrigFactorList [expr] does decomposition.
- For certain special arguments, Tanh automatically evaluates to exact values.
- Tanh can be evaluated to arbitrary numerical precision.
- Tanh automatically threads over lists. »
- Tanh can be used with Interval and CenteredInterval objects. »
Background & Context
- Tanh is the hyperbolic tangent function, which is the hyperbolic analogue of the Tan circular function used throughout trigonometry. Tanh [α] is defined as the ratio of the corresponding hyperbolic sine and hyperbolic cosine functions via . Tanh may also be defined as , where is the base of the natural logarithm Log .
- Tanh automatically evaluates to exact values when its argument is the (natural) logarithm of a rational number. When given exact numeric expressions as arguments, Tanh may be evaluated to arbitrary numeric precision. TrigFactorList can be used to factor expressions involving Tanh into terms containing Sinh , Cosh , Sin , and Cos . Other operations useful for manipulation of symbolic expressions involving Tanh include TrigToExp , TrigExpand , Simplify , and FullSimplify .
- Tanh threads element-wise over lists and matrices. In contrast, MatrixFunction can be used to give the hyperbolic tangent of a square matrix (i.e. the power series for the hyperbolic tangent function with ordinary powers replaced by matrix powers) as opposed to the hyperbolic tangents of the individual matrix elements.
- Tanh [x] approaches for small negative x and for large positive x. Tanh satisfies an identity similar to the Pythagorean identity satisfied by Tan , namely . The definition of the hyperbolic tangent function is extended to complex arguments by way of the identities and . Tanh has poles at values for an integer and evaluates to ComplexInfinity at these points. Tanh [z] has series expansion sum_(k=0)^infty(2^(2 k)(2^(2k)-1) TemplateBox[{{2, , k}}, BernoulliB] )/((2 k)!)z^(2 k-1) about the origin that may be expressed in terms of the Bernoulli numbers BernoulliB .
- The inverse function of Tanh is ArcTanh . Additional related mathematical functions include Sinh , Coth , and Tan .
Examples
open all close allBasic Examples (5)
Evaluate numerically:
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at 0:
Asymptotic expansion at a singular point:
Scope (46)
Numerical Evaluation (5)
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Tanh can take complex number inputs:
Evaluate Tanh efficiently at high precision:
Compute the elementwise values of an array using automatic threading:
Or compute the matrix Tanh function using MatrixFunction :
Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:
Or compute average-case statistical intervals using Around :
Specific Values (4)
Values of Tanh at fixed purely imaginary points:
Values at infinity:
Zero of Tanh :
Find the zero using Solve :
Substitute in the value:
Visualize the result:
Simple exact values are generated automatically:
More complicated cases require explicit use of FunctionExpand :
Visualization (3)
Plot the Tanh function:
Plot the real part of :
Plot the imaginary part of :
Plot the real part of :
Function Properties (12)
Tanh is defined for all real values:
Complex domain:
Tanh achieves all real values from the open interval :
Tanh is an odd function:
Tanh has the mirror property tanh(TemplateBox[{z}, Conjugate])=TemplateBox[{{tanh, (, z, )}}, Conjugate]:
Tanh is an analytic function of over the reals:
While it is not analytic on the complex plane, it is meromorphic:
Tanh is monotonic:
Tanh is injective:
Tanh is not surjective:
Tanh is neither non-negative nor non-positive:
Tanh has no singularities or discontinuities:
Tanh is neither convex nor concave:
TraditionalForm formatting:
Differentiation (3)
First derivative:
Higher derivatives:
Formula for the ^(th) derivative:
Integration (3)
Indefinite integral of Tanh :
Definite integral of an odd integrand over an interval centered at the origin is 0:
More integrals:
Series Expansions (4)
Integral Transforms (2)
Compute the Laplace transform using LaplaceTransform :
Function Identities and Simplifications (6)
Function Representations (4)
Representation through Tan :
Representation through Bessel functions:
Representation through Jacobi functions:
Representation through Mathieu functions:
Applications (4)
Plot a tractrix pursuit curve:
Plot a pseudosphere:
Calculate the finite area of the surface extending to infinity:
Velocity of a relativistic body in a constant force field:
Solution of the Burgers' equation using the tanh method:
Properties & Relations (13)
Basic parity and periodicity properties of Tanh are automatically applied:
Expressions containing hyperbolic functions do not automatically simplify:
Use Refine , Simplify , and FullSimplify to simplify expressions containing Tanh :
Use FunctionExpand to express special values in radicals:
Compose with inverse functions:
Solve a hyperbolic equation:
Numerically find a root of a transcendental equation:
Reduce a hyperbolic equation:
Integrals:
Integral transforms:
Obtain Tanh from sums and integrals:
Tanh appears in special cases of special functions:
Tanh is a numeric function:
Possible Issues (4)
Machine-precision input is insufficient to give a correct answer:
With exact input, the answer is correct:
A larger setting for $MaxExtraPrecision can be needed:
No power series exists at infinity, where Tanh has an essential singularity:
In TraditionalForm , parentheses are needed around the argument:
Neat Examples (1)
Continued fraction expansion:
Tech Notes
Related Guides
Related Links
History
Introduced in 1988 (1.0) | Updated in 1996 (3.0) ▪ 2021 (13.0)
Text
Wolfram Research (1988), Tanh, Wolfram Language function, https://reference.wolfram.com/language/ref/Tanh.html (updated 2021).
CMS
Wolfram Language. 1988. "Tanh." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/Tanh.html.
APA
Wolfram Language. (1988). Tanh. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Tanh.html
BibTeX
@misc{reference.wolfram_2025_tanh, author="Wolfram Research", title="{Tanh}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Tanh.html}", note=[Accessed: 04-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_tanh, organization={Wolfram Research}, title={Tanh}, year={2021}, url={https://reference.wolfram.com/language/ref/Tanh.html}, note=[Accessed: 04-January-2026]}