On-line Math 21

On-line Math 21

  Inverse hyperbolic functions

Inverses of hyperbolic functions can be written easily in terms of functions we already know, such as:
sinh-1x = ln(x+
Ö
 
x2+1
 
 ),
which follows from the definition.

Why is that true? Well, set y = sinh-1(x) , so
x
=
sinh(y)
=
1
2
 ( ey-e-y) ,
or, solving for ey by first multiplying both sides by ey , which gives a quadratic in ey ,
2eyx
=
( ey) 2-1,
0
=
( ey) 2-2eyx-1,
which by the quadratic formula gives
ey
=
2x±
Ö
4x2+4
2
=
x±
Ö
 
x2+1
 
 .
However, the `` - '' of the ± can't possibly be correct, since that would be a negative number on the right (why? And why is that a problem?), so
ey = x+
Ö
 
x2+1
 
 .
Taking natural logarithms of both sides gives
y = ln æ
è 		x+
Ö
 
x2+1
 
 ö
ø 		.

Copyright (c) 2000 by David L. Johnson.

File translated from TEX by TTH, version 2.61.
On 24 May 2000, 04:50.