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Sech
✖
Sech

✖
Sech[z]

gives the hyperbolic secant of z.

Details

  • 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. »
  • Sech can be used with Interval and CenteredInterval objects. »

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 sum_(k=0)^infty(TemplateBox[{{2, , k}}, EulerE])/((2 k)!)z^(2 k) 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.

Examples

open allclose all

Basic Examples  (5)Summary of the most common use cases

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-d3freo
Out[1]=1

Plot over a subset of the reals:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-ftxzrs
Out[1]=1

Plot over a subset of the complexes:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-kiedlx
Out[1]=1

Series expansion at the origin:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-d84plt
Out[1]=1

Asymptotic expansion at a singular point:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-qrwggr
Out[1]=1

Scope  (47)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-l274ju
Out[1]=1

Evaluate to high precision:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-q9hex5
Out[1]=1

The precision of the output tracks the precision of the input:

In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-cgotwz
Out[2]=2

Sech can take complex number inputs:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-ko71k6
Out[1]=1

Evaluate Sech efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-di5gcr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-bq2c6r
Out[2]=2

Compute the elementwise values of an array using automatic threading:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-thgd2
Out[1]=1

Or compute the matrix Sech function using MatrixFunction:

In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-o5jpo
Out[2]=2

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-crg1wt
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-lmyeh7
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0mq4j6-f4gayw
Out[3]=3

Or compute average-case statistical intervals using Around:

In[4]:=4
✖
https://wolfram.com/xid/0mq4j6-cw18bq
Out[4]=4

Specific Values  (4)

Values of Sech at fixed purely imaginary points:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-nww7l
Out[1]=1

Values at infinity:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-do145d
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-bxpy4
Out[2]=2

Maximum of Sech:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-cw39qs
Out[1]=1

Find the maximum as a root of :

In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-bgtr3o
Out[2]=2

Substitute in the value:

In[3]:=3
✖
https://wolfram.com/xid/0mq4j6-oiazhe
Out[3]=3

Visualize the result:

In[4]:=4
✖
https://wolfram.com/xid/0mq4j6-bp76w5
Out[4]=4

Simple exact values are generated automatically:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-eyjevv
Out[1]=1

More complicated cases require explicit use of FunctionExpand:

In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-mnn
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0mq4j6-p3j
Out[3]=3

Visualization  (3)

Plot the Sech function:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-ecj8m7
Out[1]=1

Plot the real part of :

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-bo5grg
Out[1]=1

Plot the imaginary part of :

In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-d4yw4c
Out[2]=2

Polar plot with :

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-epb4bn
Out[1]=1

Function Properties  (12)

Sech is defined for all real values:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-cl7ele
Out[1]=1

Complex domain:

In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-de3irc
Out[2]=2

Sech achieves all real values from the interval :

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-evf2yr
Out[1]=1

Sech is an even function:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-dnla5q
Out[1]=1

Sech has the mirror property sech(TemplateBox[{z}, Conjugate])=TemplateBox[{{sech, (, z, )}}, Conjugate]:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-heoddu
Out[1]=1

Sech is an analytic function of on the reals:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-h5x4l2
Out[1]=1

While it is not analytic on the complex plane, it is meromorphic:

In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-xl0tdy
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0mq4j6-e434t9
Out[3]=3

Sech is neither non-decreasing nor non-increasing:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-g6kynf
Out[1]=1

Sech is not injective:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-gi38d7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-ctca0g
Out[2]=2

Sech is not surjective:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-hkqec4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-b1r9xi
Out[2]=2

Sech is non-negative:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-84dui
Out[1]=1

Sech has no singularities or discontinuities:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-mdtl3h
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-mn5jws
Out[2]=2

Sech is neither convex nor concave:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-kdss3
Out[1]=1

TraditionalForm formatting:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-b3991r

Differentiation  (3)

First derivative:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-mmas49
Out[1]=1

Higher derivatives:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-nfbe0l
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-ehb25t
Out[2]=2

Formula for the ^(th) derivative:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-odmgl1
Out[1]=1

Integration  (3)

Indefinite integral of Sech:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-bponid
Out[1]=1

Definite integral of an even function over the interval centered at the origin:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-b9jw7l
Out[1]=1

This is twice the integral over half the interval:

In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-fnnzyy
Out[2]=2

More integrals:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-cz1s1n
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-f6pje1
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0mq4j6-ca0slx
Out[3]=3

Series Expansions  (4)

Find the Taylor expansion using Series:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-ewr1h8
Out[1]=1

Plot the first three approximations for Sech around :

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-binhar
Out[2]=2

General term in the series expansion of Sech:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-kevxtm
Out[1]=1

The first term of Fourier series for Sech:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-f64drv
Out[1]=1

Sech can be applied to power series:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-m3hn12
Out[1]=1

Integral Transforms  (2)

Compute the Laplace transform using LaplaceTransform:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-d3nogi
Out[1]=1

FourierTransform:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-ecytiv
Out[1]=1

Function Identities and Simplifications  (6)

Sech of a double angle:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-mjplp7
Out[1]=1

Sech of a sum:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-nfe4y
Out[1]=1

Convert multiple‐angle expressions:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-bbbslh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-kq05k
Out[2]=2

Convert sums of hyperbolic functions to products:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-f7kbnv
Out[1]=1

Expand assuming real variables and :

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-dfb1e1
Out[1]=1

Convert to exponentials:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-j5k3nt
Out[1]=1

Function Representations  (4)

Representation through Cos:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-df304y
Out[1]=1

Representation through Bessel functions:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-ewa69v
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-9y9s4
Out[2]=2

Representation through Jacobi functions:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-b622gi
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-hst6yl
Out[2]=2

Representation in terms of MeijerG:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-hry552
Out[1]=1

Applications  (7)Sample problems that can be solved with this function

Plot a tractrix pursuit curve:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-pwz39
Out[1]=1

Plot a pseudosphere:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-c5789q
Out[1]=1

Calculate the finite area of the surface extending to infinity:

In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-b0l9w2
Out[2]=2

A soliton in the Korteweg–de Vries equation:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-cle9bq
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-jw5dcq
Out[2]=2

A Schrödinger equation with a zero energy solution:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-jiszow
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-jzrotj
Out[2]=2

Calculate the CDF of the hyperbolic secant PDF:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-bvxz5r
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-chai88
Out[2]=2

Plot the PDF and CDF:

In[3]:=3
✖
https://wolfram.com/xid/0mq4j6-djrfr0
Out[3]=3

Solve a differential equation:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-looaq
Out[1]=1

Compute nonlinear Schrödinger equation: a soliton profile perturbed by a periodic potential:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-ebw3sx
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-eq8dkv
Out[2]=2

Properties & Relations  (11)Properties of the function, and connections to other functions

Basic parity and periodicity properties of Sech get automatically applied:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-caywmk
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-f7wju8
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0mq4j6-cs2kii
Out[3]=3

Expressions containing hyperbolic functions do not automatically simplify:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-pc0o9
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-ehd33x
Out[2]=2

Use Refine, Simplify, and FullSimplify to simplify expressions containing Sech:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-brx6g5
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-bqs961
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0mq4j6-o9izg
Out[3]=3

Use FunctionExpand to express special values in radicals:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-c9tx1v
Out[1]=1

Compose with inverse functions:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-hldu3c
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-cet7tq
Out[2]=2

Solve a hyperbolic equation:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-c5h0de
Out[1]=1

Numerically find a root of a transcendental equation:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-bctqgm
Out[1]=1

Reduce a hyperbolic equation:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-fxve3q
Out[1]=1

Obtain Sech from sums, products, and integrals:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-eitopx
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-e3vehy
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0mq4j6-g6ngu
Out[3]=3

Sech appears in special cases of special functions:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-j5ud1h
Out[1]=1

Sech is a numeric function:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-jk9o7t
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-gho5cl
Out[2]=2

Possible Issues  (5)Common pitfalls and unexpected behavior

Machine-precision input is insufficient to give a correct answer:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-c6gpss
Out[1]=1

With exact input, the answer is correct:

In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-prvh3
Out[2]=2

A larger setting for $MaxExtraPrecision can be needed:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-iyk1qv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-kpy4c
Out[2]=2

The inverse of Sech evaluates to Cosh:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-hmn3gv
Out[1]=1

No power series exists at infinity, where Sech has an essential singularity:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-ifiand
Out[1]=1

In TraditionalForm, parentheses are needed around the argument:

In[1]:=1
✖
https://wolfram.com/xid/0mq4j6-c4azs6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0mq4j6-fv4ia
Out[2]=2
Wolfram Research (1988), Sech, Wolfram Language function, https://reference.wolfram.com/language/ref/Sech.html (updated 2021).
✖
Wolfram Research (1988), Sech, Wolfram Language function, https://reference.wolfram.com/language/ref/Sech.html (updated 2021).

Text

Wolfram Research (1988), Sech, Wolfram Language function, https://reference.wolfram.com/language/ref/Sech.html (updated 2021).

✖
Wolfram Research (1988), Sech, Wolfram Language function, https://reference.wolfram.com/language/ref/Sech.html (updated 2021).

CMS

Wolfram Language. 1988. "Sech." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/Sech.html.

✖
Wolfram Language. 1988. "Sech." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/Sech.html.

APA

Wolfram Language. (1988). Sech. Wolfram Language & System Documentation Center. Retrieved from 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

BibTeX

@misc{reference.wolfram_2025_sech, author="Wolfram Research", title="{Sech}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Sech.html}", note=[Accessed: 13-May-2025 ]}

✖
@misc{reference.wolfram_2025_sech, author="Wolfram Research", title="{Sech}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Sech.html}", note=[Accessed: 13-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_sech, organization={Wolfram Research}, title={Sech}, year={2021}, url={https://reference.wolfram.com/language/ref/Sech.html}, note=[Accessed: 13-May-2025 ]}

✖
@online{reference.wolfram_2025_sech, organization={Wolfram Research}, title={Sech}, year={2021}, url={https://reference.wolfram.com/language/ref/Sech.html}, note=[Accessed: 13-May-2025 ]}