Integral of exponential of Shannon Entropy Function

Asked May 12 '13 at 08:39

Active Oct 02 '22 at 22:26

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Here I am going to ask a similar question as rde asked , that is what is the integral of exponential of entropy function. That is what is the value of

$$F[H(x)]=\int_{-1}^{+1} e^{ikH(f(x^2))} dx$$

where $H(x)=-x\operatorname{Log}_2(x)-(1-x)\operatorname{Log}_2(1-x)$ and $f_0$ and $f$ are linear function of type $ax+b$

More generally, what I want is the integral if $x=\cos(\theta)$ then

$\int_{-1}^{1} e^{ikf_0(x^2)H(f(x^2))}d(x)$ where $f_0\neq f(x)$

edited Oct 02 '22 at 22:26

Another User

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asked May 12 '13 at 08:39

Ars3nous

What interval is that integral over? – Qiaochu Yuan May 12 '13 at 08:45
I have edited the question a bit @QiaochuYuan, with updated integrals – Ars3nous May 12 '13 at 09:10
Your question looks to me a little localized, poorly motivated and with some bad or poor notation (you state properties of $f_0$ before using it, $x = \cos \theta$ is out of context and irrelevant, etc). – leonbloy May 13 '13 at 19:19

Integral of exponential of Shannon Entropy Function

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