20 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 ] .
- E412
-
Verify that the function \(𝜑:ℝ→ℝ\)
\[ 𝜑(x) = \begin{cases} e^{-1/x} & se ~ x{\gt}0 \\ 0 & se ~ x≤ 0 \end{cases} \](also seen in 4) is not analytic.
Hidden solution: [UNACCESSIBLE UUID ’1N6’]
- E412
[1N7] Note:Exercise 2, written exam March 2010.
Let \(I⊆ ℝ\) be a not-empty open interval. Let \(f:I→ℝ\) be of class \(C^∞\), and such that \(∀ x∈ I,∀ k≥ 0\) we have \(f^{(k)}(x)≥ 0\). Prove that \(f\) is analytic.
[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 \(f:ℝ→ℝ\) be a \(C^∞\) class function; fix \(x_ 0∈ ℝ\) and define
using the Taylor series; suppose \(g\) has radius of convergence \(R{\gt}0\): So \(g:J→ℝ\) is a well-defined function, where \(J=(x_ 0-R,x_ 0+R)\). Can it happen that \(f(x)≠ g(x)\) for a point \(x∈ J\)?
And if \(f\) is analytic? 1
Hidden solution: [UNACCESSIBLE UUID ’1NH’] [1NJ]Let \(I⊆ ℝ\) be a nonempty open interval. Let \(f:I→ ℝ\) be a \(C^∞\) class function. Let
if
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
is analytic on \(ℝ\).
Hidden solution: [UNACCESSIBLE UUID ’1NP’] [1NQ] Let \(I=(0,1)\), find an example of an analytic function \(f:I→ℝ\) not identically zero, but such that \(A=\{ x∈ I:f(x)=0\} \) 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’]