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arctanh(z)

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3-d plot

Inverse hyperbolic functions:

If z=tanh(w) then w=arctanh(z) is called the inverse hyperbolic tangent. It is a multivalued function and the image above shows only the principal part. It has the property arctanh(0)=0 and shows branch lines extending from the points +1 and -1 to infinity. The function has singularities at z=1 and -1; here the function value becomes infinity. This can also be seen from the following formula, which describes the relation with the natural logarithm:

arctanh(z) = 1/2 ln( (1+z)/(1-z) )