4.1 Power Series

In the preceding chapter we allowed the dependent variable (y or u) to be complex, but required the independent variable (x or t) to be real. We found that the solutions could be constructed in terms of elementary func-tions. In the present chapter, not only the dependent variable but also the independent variable is allowed to be complex. The derivatives are there-fore complex derivatives, so the functions differentiated must be analytic. We shall again be able to construct the solutions, although now they will take the form of power-series, and may represent new, unfamiliar functions. We provide in §4.1 statements, without proof, of the few facts regarding analytic functions that we require in the present chapter. However, in this and the next chapter, some familiarity with methods of complex analysis, as described in books on complex-function theory, is desirable