Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1993 • Question 9

Question

Math 6410 • ODE Fall 1993 • Question 9

Let \[ f=x_+^\lambda = \begin {cases} 0, & x\le 0,\\ x^\lambda , & x>0, \end {cases} \] where \(-1<\lambda <0\). Then \(f\in L^1_{\operatorname {loc}}(\mathbb {R})\), hence \(f\) may be thought of as a distribution.

Compute \(\partial f\). Note that \(\partial f/\partial x\notin L^1_{\operatorname {loc}}(\mathbb {R})\).

Raw LaTeX

Let
\[
f=x_+^\lambda=
\begin{cases}
0, & x\le0,\\
x^\lambda, & x>0,
\end{cases}
\]
where \(-1<\lambda<0\). Then \(f\in L^1_{\operatorname{loc}}(\R)\), hence \(f\) may be
thought of as a distribution.

Compute \(\partial f\). Note that \(\partial f/\partial x\notin
L^1_{\operatorname{loc}}(\R)\).

Topic Tags

Distributions and Weak Solutions