# Addition is to Integration as Multiplication is to ______

Addition is to Integration as Multiplication is to ______ ? Everyone knows that definite integration is "a way to sum continuum-many terms" in a rough sense. Can we "multiply continuum-many factors" in a similar sense?

• How about $e^{\int f}$? Jul 13, 2015 at 15:53
• @ajotatxe Its True! But i don't think OP was expecting this :) Jul 13, 2015 at 15:58
• Related (duplicate?): Is there a “continuous product”? Jul 14, 2015 at 10:37

If $f\colon [a,b] \to (0,\infty)$, then the closest analogue to a multiplicative integral of $f$ is $${\prod}_a^b \,f(x)^{dx} \;\underset{\mathrm{def}}{=}\; \exp\left(\int_a^b \ln f(x)\,dx\right).$$ This has the following properties:

1. If $a<r<b$, then $\displaystyle{\prod}_a^b\,f(x)^{dx} = \left({\prod}_a^r\,f(x)^{dx}\right)\left({\prod}_r^b\,f(x)^{dx}\right)$.

2. If $f(x)$ is a constant funtion $C$, then $\displaystyle{\prod}_a^b\,f(x)^{dx} = C^{b-a}$.

3. $\displaystyle{\prod}_a^a \,f(x)^{dx} = 1$ for any function $f$.

4. $\displaystyle {\prod}_a^b\,\bigl(f(x)g(x)\bigr)^{dx} \;=\; \left({\prod}_a^b\,f(x)^{dx}\right)\left({\prod}_a^b\,g(x)^{dx}\right)$

5. $\displaystyle\left({\prod}_a^b \,f(x)^{dx}\right)^k \;=\; {\prod}_a^b \,\bigl(f(x)^k\bigr)^{dx}$ for any constant $k$.

6. The (Monte Carlo) geometric mean of $f(x)$ on $[a,b]$ is $\displaystyle\left({\prod}_a^b \,f(x)^{dx}\right)^{1/(b-a)}$.

7. If $f$ is continuous, then $\displaystyle{\prod}_a^b \,f(x)^{dx}$ is the limit of a "Riemann product": $$\displaystyle{\prod}_a^b \,f(x)^{dx} \;=\; \lim_{\max \Delta x_i\to 0}\, \prod_{i=1}^n f(x_i^*)^{\Delta x_i}.$$

8. If $\displaystyle F(t) = {\prod}_a^t f(x)^{dx}$, then $e^{F'(t)/F(t)} = f(t)$. Similarly, for any $C^1$ function $f\colon [a,b]\to\mathbb{R}$, we have $${\prod}_a^b \bigl(e^{f'(x)/f(x)}\bigr)^{dx} = \frac{f(b)}{f(a)}$$

• did you just come up with this? I haven't seen this before, does it have a name or are you still thinking about it :)? Jul 13, 2015 at 17:07
• @user190080 I've known about it for years, but I'm not sure whether I figured it out on my own or heard about it from somebody. There is a Wikipedia article for it under the name "product integral". Jul 13, 2015 at 17:17
• ah, nice, thanks for the reference! Jul 13, 2015 at 17:19
• Maybe the following explanation is also helpful: Basically, $\exp$ transforms addition into multiplication (i.e. $\exp : (\Bbb{R}, +) \to ( (0,\infty), \cdot)$ is an isomorphism). In particular, we have $x \cdot y = \exp(\ln x + \ln y)$. Now, we basically just replace addition on the right hand side by integration (after all, we are asking "addition is to integration as multiplication is to ...") and obtain the "product integral". Jul 13, 2015 at 17:56

Convolution is analogous to multiplication.

• don't know why, but that would be my answer too :)
– john
Jul 13, 2015 at 15:57
• This is a pretty good answer, I think. Convolution definitely captures the mathematical structure of multiplication. Jul 13, 2015 at 17:05

Product Integral. Math With Bad Drawings has an article that mentions it.

I think the one presented by Jim Belk is only Type I. Here is Type II.