Anti derivative notation?
For derivatives, we use $f'(x), f''(x),$ etc. until that comes too unwieldy so we just use $f^{(n)} (x)$
What about for anti derivatives? I've seen using $F(x)$ to denote the first antiderivative of $f(x)$, but what would one do if you want to find the second anti derivative of $f(x)$? What would that be denoted by?
$\endgroup$ 44 Answers
$\begingroup$I'm not aware of a standard notation for this. If I were to make something up, $f^{(-1)}(x)$, $f^{(-2)}(x)$, and so on would be consistent with the existing notation.
One thing to remember is that unlike $f^{(n)}(x)$, the antiderivative $f^{(-n)}(x)$ will represent a family of functions, not a single function.
$\endgroup$ $\begingroup$I've never come across this before, but a conventional notation could be:
For the first antiderivative $F_1(x)$ and for the second $F_2(x)$
$\endgroup$ $\begingroup$As all consistent my choice of notation will be recognised as;
$\frac{D_y}{D_x}$ for the first derivative
$\frac{D^n_y}{D^n_x}$ for the nth derivative
$\frac{D^\star_y}{D^\star_x}$ for the first anti derivative
$\frac{D^{n\star}_y}{D^{n\star}_x}$ for the nth antiderivative
$\frac{\Phi_y}{\Phi_x}$ for the junctional derivative
$\endgroup$ $\begingroup$I don't know if common, but I personally find this intuitive: $$\int f:=D^{-1}\{f\},$$ which is a solution set (not a function) and where, if we let $\phi\in\int f$ then and $\int 0$ be the solution set of $Df=0$, we get an analogous relation between homogeneous and non-homogeneous solutions of the equation $D\phi=f$ $$\int f=\int 0+\phi$$From there I'd go with $\int\int f=D^{-2}\{f\}$, $\int\int\int f=D^{-3}\{f\}$ but for arbitrary $n$ in general use $$D^{0}:= f \\D^{-n-1}\{f\}:=D^{-1}\{D^{-n}\{f\}\}$$
$\endgroup$
