19 Analytic functions [1N4]

All definitions and theorems needed to solve the following exercises, may be found in Chap. 6 of [ 2 ] or Chap. 8 of [ 22 ] .

E411

[1N5]Prerequisites:4.

Verify that the function $𝜑 : ℝ \to ℝ$

𝜑 (x) = {\begin{cases} e^{- 1 / x} & s e x > 0 \\ 0 & s e x \leq 0 \end{cases}

(also seen in 4) is not analytic.

Hidden solution: [UNACCESSIBLE UUID ’1N6’]

E411

[1N7] Note:Exercise 2, written exam March 2010.

Let $I \subseteq ℝ$ be a not-empty open interval. Let $f : I \to ℝ$ be of class $C^{\infty}$ , and such that $\forall x \in I, \forall k \geq 0$ we have $f^{(k)} (x) \geq 0$ . Prove that $f$ is analytic.

[UNACCESSIBLE UUID ’1N8’]

Hidden solution: [UNACCESSIBLE UUID ’1N9’] See also the exercise 12.

[UNACCESSIBLE UUID ’1NB’] [1NC]Prerequisites:1.

Show that $f (x) = \frac{1}{1 + x^{2}}$ is analytic on all $ℝ$ , but the radius of convergence of the Taylor seried centered in $x_{0}$ is $\sqrt{1 + x_{0}^{2}}$ .

Hidden solution: [UNACCESSIBLE UUID ’1ND’][UNACCESSIBLE UUID ’1NF’]

Study similarly $f (x) = \sqrt{x^{2} + 1}$ or $f (x) = e^{1 / (x^{2} + 1)}$ . [1NG] Let $f : ℝ \to ℝ$ be a $C^{\infty}$ class function; fix $x_{0} \in ℝ$ and define

g (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (x_{0})}{n!} (x - x_{0})^{n}

using the Taylor series; suppose $g$ has radius of convergence $R > 0$ : So $g : J \to ℝ$ is a well-defined function, where $J = (x_{0} - R, x_{0} + R)$ . Can it happen that $f (x) \neq g (x)$ for a point $x \in J$ ?

And if $f$ is analytic? ¹

Hidden solution: [UNACCESSIBLE UUID ’1NH’] [1NJ]Let $I \subseteq ℝ$ be a nonempty open interval. Let $f : I \to ℝ$ be a $C^{\infty}$ class function. Let

b_{n} = sup_{x \in I} | f^{(n)} (x) | = ‖ f^{(n)} ‖_{\infty};

\underset{n \to \infty}{lim sup} \frac{1}{n} \sqrt[n]{b_{n}} < \infty

then $f$ is analytic.

Show with a simple example that the request is not necessary.

Hidden solution: [UNACCESSIBLE UUID ’1NK’][UNACCESSIBLE UUID ’1NM’] [1NN]Note:Exercise 1, written exam, June 30th, 2017.

Let $f$ be a continuous function on the interval $[0, 1]$ . Prove that the function

F (t) = \int_{0}^{1} f (x) e^{t x} d x

is analytic on $ℝ$ .

Hidden solution: [UNACCESSIBLE UUID ’1NP’] [1NQ] Let $I = (0, 1)$ , find an example of an analytic function $f : I \to ℝ$ not identically zero, but such that $A = {x \in I : f (x) = 0}$ has an accumulation point in $ℝ$ . Compare this example with Prop. 6.8.4 in the notes [ 2 ] ; and with the example 8.

Hidden solution: [UNACCESSIBLE UUID ’1NR’]

[UNACCESSIBLE UUID ’1NS’]