Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

If $D_x$ is the differential operator. eg. $D_x x^3=3 x^2$.

How can I find out what the operator $Q_x=(1+(k D_x)^2)^{(-1/2)}$ does to a (differentiable) function $f(x)$? ($k$ is a real number)

For instance what is $Q_x x^3$?

share|cite|improve this question
Whenever in want for $f(D)$ performed on a specific function, where $D$ is an operator, I think it's generally a good idea to expand $f$ as a power series in $D$. For instance, $$Qx^3=x^3-3k^2x$$ (with aid from W|A). – anon Sep 27 '11 at 11:26
Fractional powers of differential operators have been studied since the beginnings of calculus. One possible interpretation of your operator $Q_x$ could be rooted using fractional calculus: – Bill Cook Sep 27 '11 at 19:00
up vote 3 down vote accepted

$1+kD^2$ has positive spectrum but this is not enough for the existence of a square root operator on the same space of functions. This is because derivative is an unbounded operator (so that the expansion of $Q_x$ as a power series in $D$ may not converge), and the square root function is multi-valued.

The restriction to functions that are polynomials does give a well-defined $Q$ using the power series.

share|cite|improve this answer

It probably means this: Expand the expression $(1+(kt)^2)^{-1/2}$ as a power series in $t$, getting $$ a_0 + a_1 t + a_2 t^2 + a_3 t^3 + \cdots, $$ and then put $D_x$ where $t$ was: $$ a_0 + a_1 D_x + a_2 D_x^2 + a_3 D_x^3 + \cdots $$ and then apply that operator to $x^3$: $$ a_0 x^3 + a_1 D_x x^3 + a_2 D_x^2 x^3 + a_3 D_x^3 x^3 + \cdots. $$ All of the terms beyond the ones shown here will vanish, so you won't have an infinite series.

share|cite|improve this answer

Dirac won a Nobel Prize for finding the square root of a pesky differential operator while working on the relativistic Schrodinger equation. <--- That video gives a clear overview of the hacks Dirac used.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.