How to calculate the hyperbolic functions of a complex number?

To calculate the hyperbolic functions of a complex number, use the definitions of sinh, cosh, and tanh.

The hyperbolic functions of a complex number z = x + iy are defined as:

sinh(z) = (e^z - e^-z)/2
cosh(z) = (e^z + e^-z)/2
tanh(z) = sinh(z)/cosh(z)

To calculate these functions, first express z in terms of its real and imaginary parts:

z = x + iy

Then, use the definitions of the exponential function and trigonometric functions to evaluate e^z, e^-z, sinh(z), cosh(z), and tanh(z):

e^z = e^x * e^iy = e^x * (cos(y) + i sin(y))
e^-z = e^-x * e^-iy = e^-x * (cos(y) - i sin(y))

sinh(z) = (e^z - e^-z)/2 = (e^x * (cos(y) + i sin(y)) - e^-x * (cos(y) - i sin(y)))/2
= (e^x - e^-x)/2 * cos(y) + i (e^x + e^-x)/2 * sin(y)

cosh(z) = (e^z + e^-z)/2 = (e^x * (cos(y) + i sin(y)) + e^-x * (cos(y) - i sin(y)))/2
= (e^x + e^-x)/2 * cos(y) + i (e^x - e^-x)/2 * sin(y)

tanh(z) = sinh(z)/cosh(z) = [(e^x - e^-x)/2 * cos(y) + i (e^x + e^-x)/2 * sin(y)] / [(e^x + e^-x)/2 * cos(y) + i (e^x - e^-x)/2 * sin(y)]
= sinh(x)/cosh(x) + i tan(y)