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The hyperbolic cosecant is the algebraic inverse of the hyperbolic sine. Solving for the exponential gives

cschw=2ew -ew =z ze2w -2ew -z =0 ew =1 ±z2+1 z w=csch1z =ln(1 ±z2+1 z)

Applying the behavior of the logarithm, the inverse hyperbolic cosecant on an arbitrary branch is

csch1z =ln(1 ±z2+1 z) +2πni

The individual branches look like this:

The real part of this function retains the same numerical value between branches, while the imaginary part moves up and down in value. Visualize the imaginary part of several branches simultaneously: