persistence of differential equations

The persistence of analytic relations has important consequencesfor the theory of differential equations in the complex plane.Suppose that a function $f$ satisfies a differential equation$F(x,f(x),f^{\prime }(x),\mathrm{\dots },f^{(n)})=0$ where $F$ is a polynomial.This equation may be viewed as a polynomial relation between the$n+2$ functions $\mathrm{id},f,f^{\prime },\mathrm{\dots },f^{(n)}$ hence, by thepersistence of analytic relations, it will also hold for the analyticcontinuations of these functions. In other words, if analgebraic differential equation holds for a function in someregion, it will still hold when that function is analyticallycontinued to a larger region.

An interesting special case is that of the homogeneous lineardifferential equation with polynomial coefficients. In thatcase, we have the principle of superposition which guaranteesthat a linear combination of solutions is also a solution.Hence, if we start with a basis of solutions to our equationabout some point and analytically continue them back to ourstarting point, we obtain linear combinations of thosesolutions. This observation plays a very important role in thetheory of differential equations in the complex plane and isthe foundation for the notion of monodromy group and Riemann’sglobal characterization of the hypergeometric function.

For a less exalted illustrative example, we can consider thecomplex logarithm. The differential equation

$$x{y}^{\prime \prime }+y^{\prime }=0$$

has as solutions $y=1$ and $y=\log x$. While the formeris as singly valued as functions get, the latter is multiplyvalued. Hence upon performong analytic continuation, we expectthat the second solution will continue to a linear combinationof the two solutions. This, of course is exactly what happens;upon analytic continuation, the second solution becomes thesolution $y=\log x+n\pi i$ where $n$ is an integer whosevalue depends on how we carry out the analytic continuation.