To compute the hyperbolic functions of a complex number, we utilize the definitions of the hyperbolic sine, cosine, and tangent functions.

Let a complex number be defined as:

where ( x ) and ( y ) are the real and imaginary parts, respectively. The hyperbolic functions of ( z ) are defined as follows:

To evaluate these functions, we first express ( e^z ) and ( e^{-z} ) using Euler’s formula:

Now we can substitute these expressions into the definitions of the hyperbolic functions.

Calculating ( \sinh(z) ):

[ \sinh(z) = \frac{e^z - e^{-z}}{2} = \frac{e^x \cdot (\cos(y) + i \sin(y)) - e^{-x} \cdot (\cos(y) - i \sin(y))}{2} ]

This simplifies to:

[ \sinh(z) = \frac{(e^x - e^{-x})}{2} \cos(y) + i \frac{(e^x + e^{-x})}{2} \sin(y) ]

Calculating ( \cosh(z) ):

[ \cosh(z) = \frac{e^z + e^{-z}}{2} = \frac{e^x \cdot (\cos(y) + i \sin(y)) + e^{-x} \cdot (\cos(y) - i \sin(y))}{2} ]

This simplifies to:

[ \cosh(z) = \frac{(e^x + e^{-x})}{2} \cos(y) + i \frac{(e^x - e^{-x})}{2} \sin(y) ]

Calculating ( \tanh(z) ):

Finally, we find ( \tanh(z) ):

[ \tanh(z) = \frac{\sinh(z)}{\cosh(z)} = \frac{\left[ \frac{(e^x - e^{-x})}{2} \cos(y) + i \frac{(e^x + e^{-x})}{2} \sin(y) \right]}{\left[ \frac{(e^x + e^{-x})}{2} \cos(y) + i \frac{(e^x - e^{-x})}{2} \sin(y) \right]} ]

This can be further simplified to:

In summary, the hyperbolic functions for a complex number ( z = x + iy ) can be expressed in terms of its real and imaginary components, allowing for straightforward calculations in complex analysis.