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Inverse hyperbolic sine

The inverse hyperbolic sine can be expresses with logarithms as

 


Explanation

We start with the hyperbolic sine to which applies

Substitution of eθ = k gives

We continue only with the positive solution and substitute k = eθ so that

Now we take the logarithm on both sides

After substitution of θ = arsinh x we get

 


Example 1

You can see that sinh−1(0) = 0, as

arsinh (0) = ln (0 + √1)  = ln 1 = 0

 


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