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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