Calculating the inverse hyperbolic sine of a complex number

Excel does not provide any functions for calculating the inverse hyperbolic trig functions for complex numbers (z).  Here, and in future posts, I provide formulas to make these calculations.  First, for the inverse hyperbolic sin function, which we label SINH-1z, the formula would be:

ln(z+√(𝑧^2+1))

Using Excel’s functions this would become:

=IMLN(IMSUM(A1,IMSQRT(IMSUM(IMPOWER(A1,2),1))))

If A1 contained 1+2i this formula would return 1.46935174436819+1.06344002357775i.

Since these functions are multi-valued this formula returns the primary or principal value.  In Excel the imaginary unit can only be written as lower-case i or j.  Most of the “IM” functions were introduced with the Analysis ToolPak in Excel 4.  Of the functions used in the above formula, only IMSUM supported arrays prior to 2007.