Are there real analytic functions whose derivative is not the sum of the derivatives of the terms in its Taylor expansion?

Question

In complex analysis we know that for every complex analytic function $f(z)$, that $f(z)$ has (by definition) a Taylor expansion around every point $z_0$: $\sum_{n=0}^{\infty} a_n (z-z_0)^{n}$ with radius of convergence $R>0$. Furthermore we know that $\frac{d}{dz}f(z)=\sum_{n=1}^{\infty} a_n n (z-z_0)^{n-1}$, and this series has again radius of convergence $R$.

I couldn’t, however, find a similar theorem for analytic functions on $\mathbb{R}$. Clarification: For $a,b \in \mathbb{R}$ and $f \in C^{\infty}((a,b))$, we say that $f$ is analytic if for every point $x_0 \in (a,b) \ \exists R >0$ s.t. f has a Taylor series around $x_0$ with radius of convergence $R$ that agrees with $f$ on $(x_0-R,x_0+R) \cap (a,b)$.

So my question is, are there real analytic functions s.t. $\frac{d}{dx}f(x) \neq \sum_{n=1}^{\infty} a_n n (x-x_0)^{n-1} $?

Answer 1

You can always differentiate a power series termwise. The proof is fairly straightforward.

Recall the following theorem.

Let $f_1, f_2, \ldots$ be a sequence of differentiable functions $[a, b] \to \mathbb{R}$. Suppose that $f_1', f_2', \ldots$ converges uniformly, and suppose there is an $x_0 \in [a, b]$ such that $\lim\limits_{n \to \infty} f_n(x)$ exists. Then $f_1, f_2, \ldots$ converges uniformly to some $f$, and $f'$ is the limit of the $f_i'$s.

With this theorem in hand, consider $f_i = \sum\limits_{n = 0}^i a_n (z - z_0)^n$. Note that the radius of convergence for the differential partial sums $f_i' = \sum\limits_{n = 0}^i n a_n (z - z_0)^{n - 1}$ is also $R$ by the root test. In fact, the sequence of $f_i'$s converges uniformly for any radius $< R$. So we can apply the above theorem.

Answer 2

If $f(x) = \sum_{n = 0}^{\infty}{a_n (x - x_0)^n}$ (with domain $(a,b)$) where the sum has radius of convergence $R > 0$, then you can define

$$\begin{split} g : \{ |z - x_0| < R \} &\to \mathbb{C} \\ z &\mapsto g(z) = \sum_{n=0}^{\infty}{a_n (z - x_0)^n } \end{split}$$

Whenever $x \in (x_0 - R, x_0 + R) \cap (a,b)$ you have $g(x) = f(x)$, also $g$ is holomorphic so for every $n$ $g^{(n)}(x_0)$ exists and clearly

$$f^{(n)}(x_0) = g^{(n)}(x_0) = a_n$$

Therefore if the theorem holds for complex analytic functions it holds also for real analytic functions!

Answer 3

Some calculus book and most books on real analysis should have a proof of termwise differentiation of power series. As evidence, here is where I found the property proved in some calculus and analysis books taken off my shelf.

Apostol, Calculus, Volume 1, first theorem of section 9.21.

Apostol, Mathematical Analysis, Theorem 13-24.

Bers, Calculus, Lemma 6 in Section 8.5.

Lang, A First Course in Calculus, 5th edition, Theorem 7.2.

Rudin, Principles of Mathematical Analysis, 2nd edition, Theorem 8.1.

Strichartz, The Way of Analysis, Theorem 7.4.2.

Kolmogorov and Sinai, Introductory Real Analysis: no treatment of power series.

I think the best way to see at a glance all the basic properties of real-analytic functions once you are familiar with complex-analytic functions is to take advantage of your knowledge of analytic functions in complex analysis. The starting point is that for a power series $\sum_{n \geq 0} a_nx^n$ with real coefficients (we may center it at the origin without loss of generality), Hadamard's radius-of-convergence formula for this series centered at $0$ on $\mathbf R$ and centered at $0$ on $\mathbf C$ are the same formula: the maximal open interval of convergence of that series in $\mathbf R$ is $(-R,R)$ and the maximal open disc of convergence of that series in $\mathbf C$ is $\{z : |z| < R\}$ for the same value of $R$, where $1/R = \varlimsup \sqrt[n]{|a_n|}$. Allowing a general center now, a function expressible as a power series on an open interval $I$ in $\mathbf R$ is an analytic function on the open disc in $\mathbf C$ having $I$ as a diameter. So if you know that termwise differentiation is valid for complex power series (perhaps proved using the Cauchy integral formula or some other technique of complex analysis), then you immediately get termwise differentiation of real power series since the real derivative (using a limit as $h \to 0$ in $\mathbf R$) is a restricted version of the complex derivative (using a limit as $h \to 0$ in $\mathbf C$) if the complex derivative exists.