Reference Guide
3-8 Full Command and Function Reference
arc
2
3
---
cos arc x( )cos+
Command:
ACOS2S(ACOS(2/3)+ACOS(X))
Result: π
/2-ASIN(2/3)+
π
/2-ASIN(X)
See also: ASIN2C, ASIN2T, ATAN2S
ACOSH
Type: Analytic Function
Description: Inverse Hyperbolic Cosine Analytic Function: Returns the inverse hyperbolic cosine of the
argument.
For real arguments x < 1, ACOSH returns the complex result obtained for the argument (x, 0).
The inverse of ACOSH is a relation, not a function, since COSH sends more than one argument to
the same result. The inverse relation for COSH is expressed by ISOL as the general solution:
s1*ACOSH(Z)+2*π*i*n1
The function ACOSH is the inverse of a part of COSH, a part defined by restricting the domain
of COSH such that:
•
each argument is sent to a distinct result, and
•
each possible result is achieved.
The points in this restricted domain of COSH are called the principal values of the inverse relation.
ACOSH in its entirety is called the principal branch of the inverse relation, and the points sent by
ACOSH to the boundary of the restricted domain of COSH form the branch cuts of ACOSH.
The principal branch used by the calculator for ACOSH was chosen because it is analytic in the
regions where the arguments of the real-valued inverse function are defined. The branch cut for the
complex-valued hyperbolic arc cosine function occurs where the corresponding real-valued
function is undefined. The principal branch also preserves most of the important symmetries.
The graphs below show the domain and range of ACOSH. The graph of the domain shows
where the branch cut occurs: the heavy solid line marks one side of the cut, while the feathered
lines mark the other side of the cut. The graph of the range shows where each side of the cut is
mapped under the function.
These graphs show the inverse relation s1*ACOSH(Z)+2*π*i*n1 for the case s1 = 1 and n1 = 0.
For other values of s1 and n1, the horizontal half-band in the lower graph is rotated to the left and
translated up and down. Taken together, the bands cover the whole complex plane, which is the
domain of COSH.
View these graphs with domain and range reversed to see how the domain of COSH is restricted
to make an inverse function possible. Consider the horizontal half-band in the lower graph as the
restricted domain Z = (x, y). COSH sends this domain onto the whole complex plane in the range
W = (u, v) = COSH(x, y) in the upper graph.
Access: …Ñ
HYPERBOLIC ACOSH
(Ñ is the right-shift of the 8key).
Flags: Principal Solution (–1), Numerical Results (–3)
Input/Output:
Level 1/Argument 1 Level 1/Item 1
z
→
acosh z
'symb'
→
'ACOSH(symb)'
See also: ASINH, ATANH, COSH, ISOL