Blancmange Curve - Integrating The Blancmange Curve

Integrating The Blancmange Curve

Given that the integral of from 0 to 1 is 1/2, the identity allows the integral over any interval to be computed by the following relation. The computation is recursive with computing time on the order of log of the accuracy required.

$\begin{align} I(x) &= \int_0^x{\rm blanc}(x)\,dx,\\ I(x) &=\begin{cases} 1/2+I(x-1) & \text{if }x \geq 1\\ 1/2-I(1-x) & \text{if }1/2 < x < 1 \\ I(2x)/4+x^2/2 & \text{if } 0 \leq x \leq 1/2 \\ -I(-x) & \text{if } x < 0 \end{cases} \\ \int_a^b{\rm blanc}(x)\,dx &= I(b) - I(a). \end{align}$

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