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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?

MT_
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  • I've only very rarely seen $F$ denote an antiderivative of $f$ without it being explicitly mentioned it is so. So if you're gonna declare variables for a first antiderivative, you might as well do it for antiderivatives of all orders. – Git Gud Apr 25 '14 at 20:17
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    Let $D=\frac{d}{dx}$ then $\int f(x)dx=\left(D^{-1}f\right)(x)=f^{(-1)}(x)$ – user5402 Apr 25 '14 at 20:20
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    I' ve seen $ f^{(-1)} (x) $ and $ f^{(-2)} (x)$. – Américo Tavares Apr 25 '14 at 20:21
  • Note that antiderivatives are only defined up to a constant and, more generally, the $n^{th}$ antiderivative is only defined up to a polynomial of degree $n-1$. So such notation won't uniquely determine a function, unlike the case of derivatives. – Qiaochu Yuan Apr 26 '14 at 05:38

4 Answers4

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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.

Umberto P.
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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

Old Earth
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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 $$D^{0}:= f \\D^{-n-1}\{f\}:=D^{-1}\{D^{-n}\{f\}\}$$

MphLee
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I've never come across this before, but a conventional notation could be:

For the first antiderivative $F_1(x)$ and for the second $F_2(x)$

Ellya
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