How to proof that $ \frac{1}{2}(t^{2H} + s^{2H} -|t-s|^{2H})= H(2H-1) \int_0^t \int_0^s |u-v|^{2H-2} \, du \, dv $ I have trying to usee derivative in the right hand but I have a doubt as to the born of integration.
Integral with absolute value
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definite-integrals
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0Please see http://meta.math.stackexchange.com/questions/3399/why-should-we-accept-answers and http://meta.math.stackexchange.com/questions/3286/how-do-i-accept-an-answer? – 2012-10-16
1 Answers
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Hint: You may assume $t \le s$. Then write the inner integral as \begin{align*} \int_0^s |u-v|^{2H-2} du &= \int_0^v (v-u)^{2H-2} \, du + \int_v^s (u-v)^{2H-2}\, du \end{align*} If $H$ is an integer, you can (that seems to be even simpler) use \[ |v-u|^{2H-2} = (v-u)^{2H-2} \]
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0My warm thanks for the clarification. – 2012-10-16