Example 7

Find e^{-2 + iπ/3}.

In Chapter 8, the hyperbolic cosine and hyperbolic sine were defined in terms of e^x by the equations

Euler's Formula leads to similar equations for the cosine and sine.

In the next section we will make use of complex valued functions, that is, functions f(t) that assign a complex number z = f(t) to each real number t. The derivative of a complex valued function is obtained by differentiating the real and complex parts separately. Thus, if h(t) = f(t) + ig(t), where g and h are real functions, then

h'(t)= f'(t) + ig'(t).

For example, if h(t) = e^rt, where r = a + ib is a complex constant, then

Summing up, the usual rule (e^rt)' = re^rt still holds when r is a complex constant. We can also consider complex valued differential equations. The example we shall need is the homogeneous linear differential equation

z' + rz = 0,

where r is a complex constant. The general solution of this equation is

z(t) = Ce^-rt,

where C is a complex constant. This solution can be checked by differentiation as before.