Intereting Posts

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What is

$$\int_{K} e^{a \cdot x+ b \cdot y} \mu(x,y)$$

where $K$ is the Koch curve and $\mu(x,y)$ is a uniform measure look here.

**Attempt:**

I can evaluate the integral numerically and I have derived a method to integrate $e^x$ over some cantor sets, look here. When I tried using that method to integrate the Koch Curve, I end up unable to express the integral in direct terms of its self. Here’s a proof that integration can be done over the Koch Curve…

**Information:** I’d like a symbolic answer if its available, but infinite series/products for this integral are great too. If there’s a reference that actually handles **this** specific function over fractals and derives a symbolic result, that’s good to. Also feel free to change $K$ to any other (non-trivial of course ðŸ˜‰ ) variant of the Koch curve if that makes it easier to compute. I warn only that because the goal is to integrate over *any* fractal rather than just one or two special examples, you shouldn’t pick needlessly trivial examples…

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**Motivation:** The derivation of this result allows for integration over a fractal, however the actual reason this is useful, is because of the usefulness of the exponential function. For instance, the concept of average temperature over a fractal is a very interesting concept. $e^x$ type functions allow for rudimentary temperature fields to be constructed and theoretically integrated over fractals. $e^x$ type functions are useful for many kinds of problems, but they seem to be difficult to integrate over fractals. In addition, developing a theory for integrals over fractals, requires a large library of results, and $e^x$ should definitely be included in that list of integrable functions.

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not an answer yet, just some thoughts.

Say our Koch curve $K$ starts at $(0,0)$, ends at $(1,0)$ and the midpoint is at $(1/2, 1/(2\sqrt{3}\;))$. Mark seems to have used this, since his computation with $a=b=1$ agrees with mine.

Self-similarity is described by two maps of the plane to itself:

$$

L(x,y) = \left(\frac{x}{2}+\frac{y}{2\sqrt{3}},\frac{x}{2\sqrt{3}}-\frac{y}{2}\right),

\\

R(x,y) = \left(\frac{x}{2}-\frac{y}{2\sqrt{3}}+\frac{1}{2},-\frac{x}{2\sqrt{3}}-\frac{y}{2}+\frac{1}{2\sqrt{3}}\right),

$$

So $L(K)$ is the left half and $R(K)$ is the right half. Set $K$ is the unique nonempty compact set with $K = L(K) \cup R(K)$. Map $L$ shrinks by factor $1/\sqrt{3}$, reflects in the $x$-axis, rotates by $\pi/6$, and fixes the point $(0,1)$. Map $R$ shrinks by factor $1/\sqrt{3}$, reflects in the $x$-axis, rotates by $-\pi/6$ and fixes the point $(1,0)$.

The measure $\mu$ on $K$ is made up of two parts, which

are images of $\mu$ under $L, R$, respectively, with half the measure. That is, for integrable $f$ we have

$$

\int_K f\,d\mu = \int_{L(K)} f\,d\mu+\int_{R(K)} f\,d\mu

= \frac{1}{2}\int_K f\circ L\,d\mu + \frac{1}{2}\int_K f\circ R\,d\mu

$$

Now if we write

$$

q(a,b) := \int_K e^{ax+by}d\mu(x,y)

$$

the self-similarity shows

$$

q(a,b) = \frac{1}{2}q\left(\frac{a}{2}+\frac{b}{2\sqrt{3}}, \frac{a}{2\sqrt{3}}-\frac{b}{2}\right)+\frac{1}{2}\exp\left(\frac{a}{2}+\frac{b}{2\sqrt{3}}\right)q\left(\frac{1}{2}-\frac{b}{2\sqrt{3}},-\frac{a}{2\sqrt{3}}-\frac{b}{2}\right)

$$

We could use this recursively to evaluate $q(a,b)$ numerically. At each iteration, the point $(a,b)$ where $q$ should be evaluated moves closer to the origin by factor $1/\sqrt{3}$. We stop when we are “close enough” to $(0,0)$, since we know $q(0,0)=1$. But, of course, at each iteration the number of exponentials we have to evaluate doubles, so it is a slow method.

This “answer” is in response to your comment that you’d be interested in seeing series/product solutions. As I’m sure you know, it’s not difficult (in principle) to compute the integral of $x^p$ or $y^p$ with respect to a self-similar measure. (I have Mathematica code that automates the procedure.) Thus, we can get an approximation by simply writing

$$

e^{ax+by}=e^{ax}e^{by},

$$

replacing the exponential expressions with a finite sum approximation, and then integrating. The result is:

$$

\left(1+\frac{a}{2}+\frac{19a^2}{120}+\frac{3 a^3}{80}+\frac{92983 a^4}{13023360}+\frac{5935 a^5}{5209344}+\frac{618497323 a^6}{3948161817600}+\cdots\right)\times \\

\left(1+\frac{b}{6 \sqrt{3}}+\frac{b^2}{120}+\frac{b^3}{1008\sqrt{3}}+\frac{83b^4}{2604672}+\frac{601 b^5}{234420480 \sqrt{3}}+\frac{2095657 b^6}{35533456358400}+\cdots\right)

$$

Unfortunately, I see no significant simplification beyond this. In particular, I am not able to find closed form expressions for the integrals of the power functions – only exact expressions for specific integers.

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