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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 .
Evaluate numerically:
Series expansion:
Evaluate numerically:
Click for copyable input
Click for copyable input
Click for copyable input
Series expansion:
Click for copyable input
Evaluate for complex arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Simple exact purely imaginary values are generated automatically:
ArcCosh threads element-wise over lists:
Convert to logarithm:
TraditionalForm formatting:
ArcCosh can deal with real-valued intervals from :
Infinite arguments give symbolic results:
ArcCosh can be applied to power series:
Find the rapidity of a boost that makes the energy of a body twice its rest energy:
Fraction of the speed of light required:
Width at the base of the inverted catenary arch of Gateway Arch in St. Louis, Missouri, in feet:
Plot the real and imaginary part of ArcCosh:
Compositions with the inverse function might need PowerExpand to simplify to an identity:
Alternatively, use additional assumptions:
This shows the branch cuts of the ArcCosh function:
Solve an inverse trigonometric equation:
Solve for zeros:
Solve the differential equation satisfied by ArcCosh:
Verify it is satisfied by ArcCosh:
Generically :
On branch cuts, machine-precision inputs can give numerically wrong answers:
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