gives the arc secant of the complex number .
- Mathematical function, suitable for both symbolic and numerical manipulation.
- All results are given in radians.
- For real outside the interval to , the results are always in the range to , excluding .
- For certain special arguments, ArcSec automatically evaluates to exact values.
- ArcSec can be evaluated to arbitrary numerical precision.
- ArcSec automatically threads over lists.
- ArcSec[z] has a branch cut discontinuity in the complex plane running from to .
- ArcSec can be used with Interval and CenteredInterval objects. »
Background & Context
- ArcSec is the inverse secant function. For a real number , ArcSec[x] represents the radian angle measure, , , such that .
- ArcSec automatically threads over lists. For certain special arguments, ArcSec automatically evaluates to exact values. When given exact numeric expressions as arguments, ArcSec may be evaluated to arbitrary numeric precision. Operations useful for manipulation of symbolic expressions involving ArcSec include FunctionExpand, TrigToExp, TrigExpand, Simplify, and FullSimplify.
- ArcSec is defined for complex argument via . ArcSec[z] has a branch cut discontinuity in the complex plane.
- Related mathematical functions include Sec, ArcCsc, and ArcSech.
Examplesopen allclose all
Basic Examples (5)
Numerical Evaluation (6)
The precision of the output tracks the precision of the input:
Evaluate for complex arguments:
Evaluate ArcSec efficiently at high precision:
ArcSec threads elementwise over lists and matrices:
ArcSec can be used with Interval and CenteredInterval objects:
Specific Values (5)
Plot the ArcSec function:
Function Properties (10)
ArcSec is defined for all real values except from the interval :
ArcSec lies between and :
Function range for arguments from the complex domain:
ArcSec is not an analytic function:
ArcSec is monotonic in a specific range:
ArcSec is injective:
ArcSec is not surjective:
ArcSec is non-negative on its real domain:
It has both singularity and discontinuity for x in [-1,1]:
ArcSec is neither convex nor concave:
Indefinite integral of ArcSec:
Definite integral over the interval :
Series Expansions (3)
Function Identities and Simplifications (3)
Function Representations (5)
Represent using ArcCos:
Representation through inverse Jacobi functions:
Represent using Hypergeometric2F1:
ArcSec can be represented in terms of MeijerG:
ArcSec can be represented as a DifferentialRoot:
Branch cut of ArcSec runs along the real axis:
Solve a differential equation:
Visualize multiple complex trigonometric functions using Parallelize to speed up computations:
Properties & Relations (4)
Compose with inverse functions:
Use PowerExpand to disregard multivaluedness of the ArcSec:
Alternatively, evaluate under additional assumptions:
Use TrigToExp to express in terms of logarithm:
Use ExpToTrig to convert back:
Use FunctionExpand to convert trigs of arctrigs into an algebraic function:
Wolfram Research (1988), ArcSec, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcSec.html (updated 2021).
Wolfram Language. 1988. "ArcSec." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/ArcSec.html.
Wolfram Language. (1988). ArcSec. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcSec.html