The inverse hyperbolic sine sinh-1 x is also called area hyperbolic sine. This is a multivalve function and its name varies as arcsinh x or arsinh x.
Multivalve or multiple-valued Functions are the functions that provide two or more distinct values in The Range that is followed by function for at least one Point that exists in its Domain.
Inverse Hyperbolic Functions refers the area hyperbolic functions and area hyperbolic functions refer the area of unit Hyperbola x2 – y2 = 1 and inverse Trigonometric Functions gives the Arc Length of the unit Circle x2 + y2 = 1.
The inverse hyperbolic sine is expressed as:
Sinh-1 x = [ log ( x + √ ( 1 + x2)],
The first derivative of the inverse hyperbolic function can be expressed in the form
d (sin-1 x) / dx = 1 / √ (1 + x2),
And the indefinite integral of inverse hyperbolic sine is:
∫ sinh-1 x dx = x sinh-1 x - √ ( 1 + x2) + c,
The Maclaurin series for inverse hyperbolic sine is:
Sinh-1 x = x – (1/6) x3 + (3/40) x5 – (5/112) x7 + (35 / 1152) x9 …
Inverse hyperbolic sine has the branch cuts (-i ∞, -i) and (i, i ∞),
A branch cut is a curve in the complex plane across which an analytic multivalued function is discontinuous. The range and domain of this inverse hyperbolic sine function both are same i.e (-∞, ∞) on the real line.
Complex plane defined by inverse hyperbolic function i.e. arcsinh x is:
Sinh-1 x = log [ x + √ ( x2 + 1 )],
The graph y = sinh-1 x is the mirror image of y = sinh x in the line y = x.
The addition of two arcsinh functions is as follows:
arcsinh u ± arcsinh v = arcsinh [ u√( 1 + v2) ± v√(1 + u2)].