How to notate higher anti-derivatives?

Question

We can represent the $nth$ derivative of $y$ with the following notation:

$$\frac{d^ny}{dx^n}$$

How can we represent the $nth$ anti-derivative of $y$?

Answer 1

To put a bit more context around this question, let's consider second derivatives. The notation $\frac{d^2y}{dx^2}$ is only one of at least four widely-used notations for representing such a derivative:

$$\frac{d^2 y}{dx^2},\quad y'',\quad \ddot y,\quad \mbox{or}\quad D^2 y.$$

The notations $y''$ and $\ddot y$ depend on an implicit understanding of what variable we are to differentiate over; for example, in many contexts $\ddot y$ is defined as a second derivative with respect to time. A variation on $D^2 y$ is to write $D_x^2 y$ so as to make that variable explicit. Perhaps you might consider $D_x^2 y$ a fifth notation.

For higher derivatives there is another variation, for example, one might write successive derivatives of $y$ as $y'$, $y''$, $y'''$, $y^{(4)}$, $y^{(5)}$, and so forth. You could say there is yet another type of notation (exemplified by $y^{(5)}$) in this list, since you might sometimes see $y^{(3)}$ instead of $y'''$ or $y''''$ instead of $y^{(4)}$, although one rarely sees a second derivative written $y^{(2)}$.

Only some of these notations generalize well to arbitrary $n$th derivatives: $$\frac{d^n y}{dx^n},\quad y^{(n)},\quad D^n y,\quad \mbox{or}\quad D_x^n y.$$

As suggested in one comment, for antiderivatives you could write $$\frac{d^{-n}y}{dx^{-n}},\quad y^{(-n)},\quad D^{-n}y,\quad \mbox{or}\quad D_x^{-n}y.$$

Alternatively, as suggested in another comment, make an operator $I$ corresponding to a kind of inverse of the operator $D$, and write $$I^n y\quad \mbox{or}\quad I_x^n y.$$

A wrinkle is it is not strictly correct to write $I_x D_x y = y.$ Instead, $I_x D_x y$ is a parameterized family of functions $y + a_0$, where $a_0$ is the parameter. It follows that $I_x^n D_x^n y = y + P(x)$ where $P$ is a polynomial of degree $n - 1$, that is, the $n$th antiderivative is a family of functions parameterized by the $n$ coefficients of $P$. On the other hand, $D_x^n I_x^n y = y$ without the need to introduce parameters.