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Inverse hyperbolic functions help

The following shows how inverse sinh can be written as ln(x+sqrt(x2+1)) but I am confused by the line shown in red, can someone tell me the mathematical step involved and then proceed to explain it from thereon?

y = argsinh(x)
<=>
x = sinh(y)
<=>
x = (ey - e-y)/2
<=>
x = (e2y - 1)/(2.ey)
<=>
e2y - 2x ey - 1 = 0
<=>
ey is the positive root of previous quadratic equation
<=>
ey = x + sqrt(x2+1)
<=>
y = ln( x + sqrt(x2+1) )

Reply 1

ghostwalker

Original post by lee_vassallo

x = (ey - e-y)/2
<=>
x = (e2y - 1)/(2.ey)
<=>
e2y - 2x ey - 1 = 0
<=>
ey is the positive root of previous quadratic equation
<=>
ey = x + sqrt(x2+1)
<=>
y = ln( x + sqrt(x2+1) )

Multiply top and bottom of fraction by e^y.

Then muptiply by 2 and rearrange to get a quadratic in e^y.

Find roots. Since e^y is >0 you're only interested in the +ve root.

Take logs base e.

Reply 2

lee_vassallo

Original post by ghostwalker

Multiply top and bottom of fraction by e^y.

Then muptiply by 2 and rearrange to get a quadratic in e^y.

Find roots. Since e^y is >0 you're only interested in the +ve root.

Take logs base e.