 2

 

 

 

arccos--+ arccos(x)

 

 

 

 

3

 

 

 

Command:

ACOS2S(ACOS(2/3)+ACOS(X))

 

 

Result:

π/2-ASIN(2/3)+π/2-ASIN(X)

 

 

See also:

ASIN2C, ASIN2T, ATAN2S

 

 

ACOSH

Analytic Function

 

 

Type:

 

 

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-valuedinverse 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

(Ñis 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

 

 

3-8 Full Command and Function Reference