gives the inverse hyperbolic cosine of the complex number .
- Mathematical function, suitable for both symbolic and numerical manipulation.
- For certain special arguments, ArcCosh automatically evaluates to exact values.
- ArcCosh can be evaluated to arbitrary numerical precision.
- ArcCosh automatically threads over lists.
- ArcCosh[z] has a branch cut discontinuity in the complex plane running from to .
Background & Context
- ArcCosh is the inverse hyperbolic cosine function. For a real number , ArcCosh[x] represents the hyperbolic angle measure such that .
- ArcCosh automatically threads over lists. For certain special arguments, ArcCosh automatically evaluates to exact values. When given exact numeric expressions as arguments, ArcCosh may be evaluated to arbitrary numeric precision. Operations useful for manipulation of symbolic expressions involving ArcCosh include FunctionExpand, TrigToExp, TrigExpand, Simplify, and FullSimplify.
- ArcCosh is defined for complex argument by . ArcCosh[z] has a branch cut discontinuity in the complex plane.
- Related mathematical functions include Cosh, ArcSinh, and ArcCos.
Examplesopen allclose all
Basic Examples (5)
Asymptotic expansion at Infinity:
Numerical Evaluation (6)
Specific Values (4)
Plot the ArcCosh function:
Function Properties (10)
ArcCosh is defined for all real values greater than or equal to 1:
ArcCosh achieves all real values greater than or equal to 0:
ArcCosh is not an analytic function:
ArcCosh is increasing over its real domain:
ArcCosh is injective:
ArcCosh is not surjective:
ArcCosh is non-negative over its real domain:
ArcCosh is concave over its real domain:
Series Expansions (4)
Function Identities and Simplifications (3)
Plot the real and imaginary part of ArcCosh:
Properties & Relations (5)
Compositions with the inverse function might need PowerExpand to simplify to an identity:
This shows the branch cuts of the ArcCosh function:
Solve the differential equation satisfied by ArcCosh:
Verify it is satisfied by ArcCosh:
Possible Issues (2)
Neat Examples (1)
Wolfram Research (1988), ArcCosh, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcCosh.html.
Wolfram Language. 1988. "ArcCosh." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ArcCosh.html.
Wolfram Language. (1988). ArcCosh. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcCosh.html