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arcsinh(z)

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3-d plot

Inverse hyperbolic functions:

If z=sinh(w) then w=arcsinh(z). This function is the inverse hyperbolic sine. Like all inverse hyperbolic functions, the inverse hyperbolic sine is multivalued and has branch lines appearing as discontinuities in the colors. The image shows only the principal part, which is obtained by restricting the arguments of z to the interval from 0 to 2Pi. The principal part is characterized by property that arcsinh(0)=0. Since the hyperbolic sine sinh(z) maps the real numbers onto the real numbers in a one-to-one fashion, the inverse hyperbolic sine is well-defined and real-valued on the whole real axis.