Computing $\langle\sin(\gamma_i)\rangle= \int_{(S^2)^N} \sin(\gamma_i)p(\Theta)dS$

Question

I'm trying to evaluate the following integral, which I know must be zero, $$\langle\sin(\gamma_i)\rangle= \int_{(S^2)^N} \sin(\gamma_i)p(\Gamma)dS$$ Where, $$\langle \vec{a}(\vec{r_1},...,\vec{r_N})\rangle= \int_{(\mathbb{R})^N} \vec{a}(\vec{r_1},...,\vec{r_N})p(\vec{r_1},...,\vec{r_N})dV$$ and $p(\vec{r_1},...,\vec{r_N})$ is the pdf of the configuration $(\vec{r_1},...,\vec{r_N})$. There are a lot of definitions here which can be found in the reference material, but they should not be needed. We also have $$p(\Gamma)=\frac1Z \exp(\delta\sum_{j=1}^N \cos(\gamma_j))$$ Where $Z=\int_{(S^2)^N} \exp(\delta\sum_{j=1}^N \cos(\gamma_j))dS$ and $\Gamma=\{\gamma_i\}$. Therefore $$\langle\sin(\gamma_i)\rangle=$$ $$ \frac1Z \int_0^{2\pi} \cdots\int_0^{2\pi} \int_0^{\pi} \cdots\int_0^{\pi} \sin(\gamma_i)\exp(\delta\sum_{j=1}^N \cos(\gamma_j))\sin(\gamma_1)d\gamma_1 \cdots \sin(\gamma_N)d\gamma_N d\phi_1\cdots d\phi_N$$

$$=\frac{(4\pi)^N}{Z} \prod_{j=1}^N \int_0^\pi \sin(\gamma_i) \sin(\gamma_j) \exp(\delta\cos( \gamma_j)) d\gamma_j$$

Now $$Z=(4\pi)^N \prod_{j=1}^N \int_0^\pi \sin(\gamma_j) \exp(\delta \cos(\gamma_j)) d\gamma_j$$ Therefore $$\langle\sin(\gamma_i)\rangle=\frac{\int_0^\pi \sin^2(\gamma_i) \exp(\delta\cos( \gamma_i)) d\gamma_i}{\int_0^\pi \sin(\gamma_i) \exp(\delta \cos(\gamma_i)) d\gamma_i}\neq0 $$ (http://www.wolframalpha.com/input/?i=int_0%5Epi+sin%28x%29%5E2+e%5E%28cos%28x%29%29+dx)

I can't for the life of me see what's going wrong here, any help would be greatly appreciated.

EDIT: With reference to the pdf file, here is the proof as to why I know this integral must be zero,

On page 10 there is the proof of: $w_n=ww_{n-1}$

$w_n=\langle \vec{t}_i \cdot \vec{t}_{i+n}\rangle=\langle \cos(\gamma_i+...+γ_{i+n-1})\rangle$

We then expand the cos term into: $w_n=\langle \cos(γ_i)\rangle \langle\ \cos(\gamma_{i+1}+...+γ_{i+n-1})\rangle-\langle \sin(γ_i)\rangle \langle \sin(\gamma_{i+1}+...+γ_{i+n-1})\rangle$

$\Rightarrow w_n=ww_{n-1}+\langle \sin(γ_i)\rangle \langle\cdots\rangle$

$\Rightarrow \langle \sin(γ_i)\rangle=0$

Answer 1

I don't understand how you get from: $$ \frac1Z \int_0^{2\pi} \cdots\int_0^{2\pi} \int_0^{\pi} \cdots\int_0^{\pi} \sin(\gamma_i)\exp(\delta\sum_{j=1}^N \cos(\gamma_j))\sin(\gamma_1)d\gamma_1 \cdots \sin(\gamma_N)d\gamma_N d\phi_1\cdots d\phi_N$$ to $$ \frac{(4\pi)^N}{Z} \prod_{j=1}^N \int_0^\pi \sin(\gamma_i) \sin(\gamma_j) \exp(\delta\cos( \gamma_j)) d\gamma_j$$ In the first expression, $\sin(\gamma_i)$ appears once. In the right hand side it appears $N$ times. Shouldn't you get $$ \frac{(4\pi)^N}{Z} \prod_{j=1}^N \int_0^\pi \sin^{1+\delta_{ij}}(\gamma_j) \exp(\delta\cos( \gamma_j)) d\gamma_j ?$$

Answer 2

I don't know if I'm seeing this right, but when writing the multiple integrals you will have an integral of the type $$ \int_0^{2\pi} \sin(\gamma_i) \exp(\delta \cos(\gamma_i))d\gamma_i$$ which is zero (it is the integral of an odd periodic function over one period). Therefore the rest of the expression is zero.