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Example 6

Find the square roots of i.

By the computation in Example 3, the polar form of i is i = cis (π/2).

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The two square roots of i are shown in Figure 14.5.4.

14_differential_equations-136.gif

Figure 14.5.4

We now turn to complex exponents, which are useful in the study of differential equations. In order to give a meaning to an exponent e2, we consider infinite series of complex numbers. The sum of an infinite series of complex numbers is defined by summing the real and imaginary parts separately. If zn = xn + iyn, and the series ∑xn and ∑yn both converge, the sum of the series ∑zn is defined by the formula

14_differential_equations-137.gif

In Chapter 9, we found that for real numbers z the exponent ez is given by the power series

14_differential_equations-138.gif

When z is a complex number, this formula is taken as the definition of ez. It can be shown that the power series converges for every z and that the exponential rule eu+z = eu ez holds for complex exponents. In the case that z is a purely imaginary number z = iy, the power series takes the form

14_differential_equations-139.gif

Using the power series for cos y and sin y, we obtain Euler's Formula:

eiy = cos y + i sin y = cis y.

When z is a complex number z = x + iy, the exponent ez is given by the formula gx + iy = ex eiy = ex(cos y + i sin y).


Last Update: 2006-11-16