Lately I’ve been thinking about how to prove the existence and uniqueness of solutions to differential equations. This is a very hard problem in general; it’s solved in the case where the equation is first-order by the Picard-Lindelöf theorem, but that theorem involves more analysis than I’m familiar with at the moment. Instead, I’ve just been looking at a very particular case where proving existence and uniqueness turns out to be very easy.
Suppose is a positive integer and is an entire complex-valued function such that for every complex number , , i.e. is a solution of the homogeneous linear differential equation with constant coefficients whose characteristic equation is . You might therefore call an th differential root of unity. This is my own term, I don’t know if any actual mathematicians use it.
Like an th root of unity, has the property that for every pair of non-negative integers and such that , , because by the definition of congruence, there is an integer such that , so
if you accept that , which is obvious and can be proven by induction. Because of this property, we can define for every integer modulo as the common value of for integers in the congruence class associated with .
The key insight which allows all the th differential roots of unity to be found is that the Taylor series of at 0 can be written as a sum of series, where each series in the sum consists of the terms whose indices in the original series are in a particular congruence class modulo . So for every complex number ,
This is assuming that for every integer such that , the series (which will be denoted by ) converges, but this can be easily proven by the ratio test:
since approaches as approaches .
So, is a linear combination of , … and . Therefore, the set of all such linear combinations definitely includes all the differential th roots of unity; however, it may also include other functions. In order to ensure that this is not the case, we only need to show that is a differential th root of unity for every integer such that , because any linear combination of differential th roots of unity is a differential th root of unity too (since the associated differential equation is linear). Well, for every complex number , using the rule for differentiation of power series, we have
Therefore the general solution to the differential equation is
where for every integer such that , is an arbitrary complex number. Furthermore, will always have the property that . We can use this property to discover some properties of the functions , , … and . For convenience, let us define and for every integer modulo as and , where is the unique integer such that which is in the congruence class associated with . This makes these properties simpler to express.
(1) For every integer modulo , we have
and for every integer modulo , . But clearly if , otherwise ; therefore, if , otherwise . In particular, but if is not 0.
(2) For every pair of integers and modulo , is a differential th root of unity, because (since for every integer in the congruence class associated with , ). Therefore we have
and for every integer modulo , . Now, , which we know from (1) is 1 if (i.e. ), otherwise 0. Therefore .
(3) For every integer modulo and every complex number , is a differential th root of unity, because it is the horizontal translation of units to the left, so its th derivative is simply the horizontal translation of units to the left. Therefore, for every complex number ,
and for every integer modulo , is the th derivative of at 0, which is simply , i.e. (by (2)). Therefore
which can be generalised by induction to
from which it follows in particular that for every positive integer ,
I’m going to stop here, but there are lots of questions to ask. First of all, when this formula shows that , and it can be generalised to hold when is rational, which allows to be used to extend the exponential functions to . Can it also be generalised like this when ? Also, we know that when , and , and when ; do any similar relations hold in general between the differential roots of unity?