gives the hyperbolic secant of z.
- Mathematical function, suitable for both symbolic and numerical manipulation.
- 1/Cosh[z] is automatically converted to Sech[z]. TrigFactorList[expr] does decomposition.
- For certain special arguments, Sech automatically evaluates to exact values.
- Sech can be evaluated to arbitrary numerical precision.
- Sech automatically threads over lists.
Background & Context
- Sech is the hyperbolic secant function, which is the hyperbolic analogue of the Sec circular function used throughout trigonometry. It is defined as the reciprocal of the hyperbolic cosine function as . It is defined for real numbers by letting be twice the area between the axis and a ray through the origin intersecting the unit hyperbola . Sech[α] then represents the reciprocal of the horizontal coordinate of the intersection point. The equivalent definition of hyperbolic secant is , where is the base of the natural logarithm Log.
- Sech automatically evaluates to exact values when its argument is the (natural) logarithm of a rational number. When given exact numeric expressions as arguments, Sech may be evaluated to arbitrary numeric precision. TrigFactorList can be used to factor expressions involving Sech into terms containing Sinh, Cosh, Sin, and Cos. Other operations useful for manipulation of symbolic expressions involving Sech include TrigToExp, TrigExpand, Simplify, and FullSimplify.
- Sech threads element-wise over lists and matrices. In contrast, MatrixFunction can be used to give the hyperbolic secant of a square matrix (i.e. the power series for the hyperbolic secant function with ordinary powers replaced by matrix powers) as opposed to the hyperbolic secants of the individual matrix elements.
- Sech[x] decreases exponentially as x approaches . Sech satisfies an identity similar to the Pythagorean identity satisfied by Sec, namely . The definition of the hyperbolic secant function is extended to complex arguments by way of the identity . Sech has poles at values for an integer and evaluates to ComplexInfinity at these points. Sech[z] has series expansion about the origin that may be expressed in terms of the Euler numbers EulerE.
- The inverse function of Sech is ArcSech. Related mathematical functions include Cosh and Csch.
Examplesopen allclose all
Basic Examples (5)
Numerical Evaluation (6)
Specific Values (4)
Plot the Sech function:
Function Properties (12)
Sech is defined for all real values:
Sech achieves all real values from the interval :
Sech is an even function:
Sech has the mirror property :
Sech is an analytic function of on the reals:
Sech is neither non-decreasing nor non-increasing:
Sech is not injective:
Sech is not surjective:
Sech is non-negative:
Sech has no singularities or discontinuities:
Sech is neither convex nor concave:
Indefinite integral of Sech:
Series Expansions (4)
Function Identities and Simplifications (6)
Properties & Relations (11)
Basic parity and periodicity properties of Sech get automatically applied:
Use FunctionExpand to express special values in radicals:
Obtain Sech from sums, products, and integrals:
Sech appears in special cases of special functions:
Sech is a numeric function:
Possible Issues (5)
A larger setting for $MaxExtraPrecision can be needed:
No power series exists at infinity, where Sech has an essential singularity:
In TraditionalForm, parentheses are needed around the argument:
Wolfram Research (1988), Sech, Wolfram Language function, https://reference.wolfram.com/language/ref/Sech.html (updated 1996).
Wolfram Language. 1988. "Sech." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 1996. https://reference.wolfram.com/language/ref/Sech.html.
Wolfram Language. (1988). Sech. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Sech.html