gives the inverse hyperbolic secant of the complex number .
- Mathematical function, suitable for both symbolic and numerical manipulation.
- For certain special arguments, ArcSech automatically evaluates to exact values.
- ArcSech can be evaluated to arbitrary numerical precision.
- ArcSech automatically threads over lists.
- ArcSech[z] has branch cut discontinuities in the complex plane running from to and to .
- ArcSech can be used with Interval and CenteredInterval objects. »
Background & Context
- ArcSech is the inverse hyperbolic secant function. For a real number , ArcSec[x] represents the hyperbolic angle measure such that .
- ArcSech automatically threads over lists. For certain special arguments, ArcSech automatically evaluates to exact values. When given exact numeric expressions as arguments, ArcSech may be evaluated to arbitrary numeric precision. Operations useful for manipulation of symbolic expressions involving ArcSech include FunctionExpand, TrigToExp, TrigExpand, Simplify, and FullSimplify.
- ArcSech is defined for complex argument by . ArcSech[z] has branch cut discontinuities in the complex plane.
- Related mathematical functions include Sech, ArcCsch, and ArcSec.
Examplesopen allclose all
Basic Examples (6)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Asymptotic expansion at Infinity:
Numerical Evaluation (6)
The precision of the output tracks the precision of the input:
Evaluate for complex arguments:
Evaluate ArcSech efficiently at high precision:
ArcSech threads elementwise over lists and matrices:
ArcSech can be used with Interval and CenteredInterval objects:
Specific Values (4)
Plot the ArcSech function:
Function Properties (10)
ArcSech is defined for all real values from the interval :
ArcSech achieves all real values greater than or equal to 0:
Function range for arguments from the complex domain:
ArcSech is not an analytic function:
ArcSech is decreasing over its real domain:
ArcSech is injective:
ArcSech is not surjective:
ArcSech is non-negative over its real domain:
It has both singularity and discontinuity in (-∞,0] and [1,∞):
ArcSech is neither convex nor concave:
Series Expansions (3)
Function Identities and Simplifications (3)
Function Representations (5)
Represent using ArcCosh:
Representation through inverse Jacobi functions:
Represent using Hypergeometric2F1:
ArcSech can be represented in terms of MeijerG:
ArcSech can be represented as a DifferentialRoot:
Properties & Relations (2)
Compose with inverse functions:
Use PowerExpand to disregard multivaluedness of the ArcSech:
Alternatively, evaluate under additional assumptions:
Use TrigToExp to express in terms of logarithm:
Wolfram Research (1988), ArcSech, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcSech.html (updated 2021).
Wolfram Language. 1988. "ArcSech." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/ArcSech.html.
Wolfram Language. (1988). ArcSech. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcSech.html