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By an odd coincidence, I stumbled upon a second question in as many weeks about power series, and once again the only way I know how to prove the result is by complex methods; once again, I am leaving it here as a challenge to any interested readers, and I would be particularly interested in knowing of a proof that was not based on complex analysis (or thinly disguised versions thereof), or for a reference to previous literature where something like this identity has occured. (I suspect for instance that something like this may have shown up before in free probability, based on the answer to part (ii) of the problem.)

Here is a purely algebraic form of the problem:

Problem 1 Let ${F = F(z)}$ be a formal function of one variable ${z}$. Suppose that ${G = G(z)}$ is the formal function defined by

$\displaystyle G := \sum_{n=1}^\infty \left( \frac{F^n}{n!} \right)^{(n-1)}$

$\displaystyle = F + \left(\frac{F^2}{2}\right)' + \left(\frac{F^3}{6}\right)'' + \dots$

$\displaystyle = F + FF' + (F (F')^2 + \frac{1}{2} F^2 F'') + \dots,$

where we use ${f^{(k)}}$ to denote the ${k}$-fold derivative of ${f}$ with respect to the variable ${z}$.

• (i) Show that ${F}$ can be formally recovered from ${G}$ by the formula

$\displaystyle F = \sum_{n=1}^\infty (-1)^{n-1} \left( \frac{G^n}{n!} \right)^{(n-1)}$

$\displaystyle = G - \left(\frac{G^2}{2}\right)' + \left(\frac{G^3}{6}\right)'' - \dots$

$\displaystyle = G - GG' + (G (G')^2 + \frac{1}{2} G^2 G'') - \dots.$

• (ii) There is a remarkable further formal identity relating ${F(z)}$ with ${G(z)}$ that does not explicitly involve any infinite summation. What is this identity?

To rigorously formulate part (i) of this problem, one could work in the commutative differential ring of formal infinite series generated by polynomial combinations of ${F}$ and its derivatives (with no constant term). Part (ii) is a bit trickier to formulate in this abstract ring; the identity in question is easier to state if ${F, G}$ are formal power series, or (even better) convergent power series, as it involves operations such as composition or inversion that can be more easily defined in those latter settings.

To illustrate Problem 1(i), let us compute up to third order in ${F}$, using ${{\mathcal O}(F^4)}$ to denote any quantity involving four or more factors of ${F}$ and its derivatives, and similarly for other exponents than ${4}$. Then we have

$\displaystyle G = F + FF' + (F (F')^2 + \frac{1}{2} F^2 F'') + {\mathcal O}(F^4)$

and hence

$\displaystyle G' = F' + (F')^2 + FF'' + {\mathcal O}(F^3)$

$\displaystyle G'' = F'' + {\mathcal O}(F^2);$

multiplying, we have

$\displaystyle GG' = FF' + F (F')^2 + F^2 F'' + F (F')^2 + {\mathcal O}(F^4)$

and

$\displaystyle G (G')^2 + \frac{1}{2} G^2 G'' = F (F')^2 + \frac{1}{2} F^2 F'' + {\mathcal O}(F^4)$

and hence after a lot of canceling

$\displaystyle G - GG' + (G (G')^2 + \frac{1}{2} G^2 G'') = F + {\mathcal O}(F^4).$

Thus Problem 1(i) holds up to errors of ${{\mathcal O}(F^4)}$ at least. In principle one can continue verifying Problem 1(i) to increasingly high order in ${F}$, but the computations rapidly become quite lengthy, and I do not know of a direct way to ensure that one always obtains the required cancellation at the end of the computation.

Problem 1(i) can also be posed in formal power series: if

$\displaystyle F(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots$

is a formal power series with no constant term with complex coefficients ${a_1, a_2, \dots}$ with ${|a_1|<1}$, then one can verify that the series

$\displaystyle G := \sum_{n=1}^\infty \left( \frac{F^n}{n!} \right)^{(n-1)}$

makes sense as a formal power series with no constant term, thus

$\displaystyle G(z) = b_1 z + b_2 z^2 + b_3 z^3 + \dots.$

For instance it is not difficult to show that ${b_1 = \frac{a_1}{1-a_1}}$. If one further has ${|b_1| < 1}$, then it turns out that

$\displaystyle F = \sum_{n=1}^\infty (-1)^{n-1} \left( \frac{G^n}{n!} \right)^{(n-1)}$

as formal power series. Currently the only way I know how to show this is by first proving the claim for power series with a positive radius of convergence using the Cauchy integral formula, but even this is a bit tricky unless one has managed to guess the identity in (ii) first. (In fact, the way I discovered this problem was by first trying to solve (a variant of) the identity in (ii) by Taylor expansion in the course of attacking another problem, and obtaining the transform in Problem 1 as a consequence.)

The transform that takes ${F}$ to ${G}$ resembles both the exponential function

$\displaystyle \exp(F) = \sum_{n=0}^\infty \frac{F^n}{n!}$

and Taylor’s formula

$\displaystyle F(z) = \sum_{n=0}^\infty \frac{F^{(n)}(0)}{n!} z^n$

but does not seem to be directly connected to either (this is more apparent once one knows the identity in (ii)).

My colleague Tom Liggett recently posed to me the following problem about power series in one real variable ${x}$. Observe that the power series

$\displaystyle \sum_{n=0}^\infty (-1)^n\frac{x^n}{n!}$

has very rapidly decaying coefficients (of order ${O(1/n!)}$), leading to an infinite radius of convergence; also, as the series converges to ${e^{-x}}$, the series decays very rapidly as ${x}$ approaches ${+\infty}$. The problem is whether this is essentially the only example of this type. More precisely:

Problem 1 Let ${a_0, a_1, \dots}$ be a bounded sequence of real numbers, and suppose that the power series

$\displaystyle f(x) := \sum_{n=0}^\infty a_n\frac{x^n}{n!}$

(which has an infinite radius of convergence) decays like ${O(e^{-x})}$ as ${x \rightarrow +\infty}$, in the sense that the function ${e^x f(x)}$ remains bounded as ${x \rightarrow +\infty}$. Must the sequence ${a_n}$ be of the form ${a_n = C (-1)^n}$ for some constant ${C}$?

As it turns out, the problem has a very nice solution using complex analysis methods, which by coincidence I happen to be teaching right now. I am therefore posing as a challenge to my complex analysis students and to other readers of this blog to answer the above problem by complex methods; feel free to post solutions in the comments below (and in particular, if you don’t want to be spoiled, you should probably refrain from reading the comments). In fact, the only way I know how to solve this problem currently is by complex methods; I would be interested in seeing a purely real-variable solution that is not simply a thinly disguised version of a complex-variable argument.

(To be fair to my students, the complex variable argument does require one additional tool that is not directly covered in my notes. That tool can be found here.)

At the core of almost any undergraduate real analysis course are the concepts of differentiation and integration, with these two basic operations being tied together by the fundamental theorem of calculus (and its higher dimensional generalisations, such as Stokes’ theorem). Similarly, the notion of the complex derivative and the complex line integral (that is to say, the contour integral) lie at the core of any introductory complex analysis course. Once again, they are tied to each other by the fundamental theorem of calculus; but in the complex case there is a further variant of the fundamental theorem, namely Cauchy’s theorem, which endows complex differentiable functions with many important and surprising properties that are often not shared by their real differentiable counterparts. We will give complex differentiable functions another name to emphasise this extra structure, by referring to such functions as holomorphic functions. (This term is also useful to distinguish these functions from the slightly less well-behaved meromorphic functions, which we will discuss in later notes.)

In this set of notes we will focus solely on the concept of complex differentiation, deferring the discussion of contour integration to the next set of notes. To begin with, the theory of complex differentiation will greatly resemble the theory of real differentiation; the definitions look almost identical, and well known laws of differential calculus such as the product rule, quotient rule, and chain rule carry over verbatim to the complex setting, and the theory of complex power series is similarly almost identical to the theory of real power series. However, when one compares the “one-dimensional” differentiation theory of the complex numbers with the “two-dimensional” differentiation theory of two real variables, we find that the dimensional discrepancy forces complex differentiable functions to obey a real-variable constraint, namely the Cauchy-Riemann equations. These equations make complex differentiable functions substantially more “rigid” than their real-variable counterparts; they imply for instance that the imaginary part of a complex differentiable function is essentially determined (up to constants) by the real part, and vice versa. Furthermore, even when considered separately, the real and imaginary components of complex differentiable functions are forced to obey the strong constraint of being harmonic. In later notes we will see these constraints manifest themselves in integral form, particularly through Cauchy’s theorem and the closely related Cauchy integral formula.

Despite all the constraints that holomorphic functions have to obey, a surprisingly large number of the functions of a complex variable that one actually encounters in applications turn out to be holomorphic. For instance, any polynomial ${z \mapsto P(z)}$ with complex coefficients will be holomorphic, as will the complex exponential ${z \mapsto \exp(z)}$. From this and the laws of differential calculus one can then generate many further holomorphic functions. Also, as we will show presently, complex power series will automatically be holomorphic inside their disk of convergence. On the other hand, there are certainly basic complex functions of interest that are not holomorphic, such as the complex conjugation function ${z \mapsto \overline{z}}$, the absolute value function ${z \mapsto |z|}$, or the real and imaginary part functions ${z \mapsto \mathrm{Re}(z), z \mapsto \mathrm{Im}(z)}$. We will also encounter functions that are only holomorphic at some portions of the complex plane, but not on others; for instance, rational functions will be holomorphic except at those few points where the denominator vanishes, and are prime examples of the meromorphic functions mentioned previously. Later on we will also consider functions such as branches of the logarithm or square root, which will be holomorphic outside of a branch cut corresponding to the choice of branch. It is a basic but important skill in complex analysis to be able to quickly recognise which functions are holomorphic and which ones are not, as many of useful theorems available to the former (such as Cauchy’s theorem) break down spectacularly for the latter. Indeed, in my experience, one of the most common “rookie errors” that beginning complex analysis students make is the error of attempting to apply a theorem about holomorphic functions to a function that is not at all holomorphic. This stands in contrast to the situation in real analysis, in which one can often obtain correct conclusions by formally applying the laws of differential or integral calculus to functions that might not actually be differentiable or integrable in a classical sense. (This latter phenomenon, by the way, can be largely explained using the theory of distributions, as covered for instance in this previous post, but this is beyond the scope of the current course.)

Remark 1 In this set of notes it will be convenient to impose some unnecessarily generous regularity hypotheses (e.g. continuous second differentiability) on the holomorphic functions one is studying in order to make the proofs simpler. In later notes, we will discover that these hypotheses are in fact redundant, due to the phenomenon of elliptic regularity that ensures that holomorphic functions are automatically smooth.