How to calculate the inverse hyperbolic functions of a complex number?

To calculate the inverse hyperbolic functions of a complex number, use the formulas for the inverse hyperbolic functions.

The inverse hyperbolic functions are defined as follows:

arcsinh(z) = ln(z + sqrt(z^2 + 1))
arccosh(z) = ln(z + sqrt(z^2 - 1))
arctanh(z) = 1/2 ln((1 + z)/(1 - z))

To calculate the inverse hyperbolic functions of a complex number, first express the complex number in terms of its real and imaginary parts, z = x + iy. Then substitute x and y into the appropriate formula for the inverse hyperbolic function.

For example, to find arcsinh(2 + 3i), we have:

arcsinh(2 + 3i) = ln(2 + 3i + sqrt((2 + 3i)^2 + 1))
= ln(2 + 3i + sqrt(-8 + 12i))
= ln(2 + 3i + 2i sqrt(2 - 3i))
= ln(2 + 2i sqrt(2 - 3i) + i sqrt(2 - 3i))

Similarly, to find arccosh(2 + 3i), we have:

arccosh(2 + 3i) = ln(2 + 3i + sqrt((2 + 3i)^2 - 1))
= ln(2 + 3i + sqrt(-6 + 12i))
= ln(2 + 3i + 2i sqrt(3 - 2i))
= ln(2 + 2i sqrt(3 - 2i) + i sqrt(3 - 2i))

And to find arctanh(2 + 3i), we have:

arctanh(2 + 3i) = 1/2 ln((1 + 2 + 3i)/(1 - 2 - 3i))
= 1/2 ln((-1 + 3i)/(-1 - 3i))
= 1/2 ln((8 - 6i)/10)
= 1/2 ln(4 - 3i)