# Opt3

Given that the power spectrum and structure function are related by

$\phi _{n}(\kappa )={\frac {1}{4\pi ^{2}\kappa ^{2}}}\int _{0}^{\infty }{\frac {\sin(\kappa R)}{\kappa R}}{\frac {\partial }{\partial R}}\left[R^{2}{\frac {\partial D_{n}(R)}{\partial R}}\right]\,dR\,$ ,

use the structure function $D_{n}(R)=C_{n}^{2}R^{2/3}\,$ and the integral formula

$\int _{0}^{\infty }x^{\alpha }\sin(x)\,dx=2^{\alpha }{\sqrt {\pi }}\,{\frac {\Gamma ({\frac {\alpha }{2}}+1)}{\Gamma ({\frac {1}{2}}-{\frac {\alpha }{2}})}}\,$ to obtain the Kolmogorov power law spectrum:

$\phi _{n}(\kappa )=0.033\,C_{n}^{2}\kappa ^{-11/3},\,\,\,{\frac {1}{L_{0}}}<\!\!<\kappa <\!\!<{\frac {1}{l_{0}}}\,$ .

Substituting the structure function into the power spectrum above,

$\phi _{n}(\kappa )={\frac {1}{4\pi ^{2}\kappa ^{2}}}\int _{0}^{\infty }{\frac {\sin(\kappa R)}{\kappa R}}C_{n}^{2}{\frac {\partial }{\partial R}}\left[R^{2}{\frac {2}{3}}R^{-1/3}\right]\,dR\,$ $={\frac {1}{4\pi ^{2}\kappa ^{2}}}C_{n}^{2}{\frac {2}{3}}\int _{0}^{\infty }{\frac {\sin(\kappa R)}{\kappa R}}{\frac {5}{3}}R^{2/3}\,dR\,$ $={\frac {2}{3}}{\frac {5}{3}}{\frac {1}{4\pi ^{2}\kappa ^{2}}}\kappa ^{-1}C_{n}^{2}\int _{0}^{\infty }\sin(\kappa R)R^{-1/3}\,dR\,$ Now to use the integral formula, let $x=\kappa R,\,\,\,R={\frac {x}{\kappa }},\,\,\,dR={\frac {dx}{\kappa }}\,$ $={\frac {10}{36}}{\frac {1}{\pi ^{2}\kappa ^{3}}}C_{n}^{2}\int _{0}^{\infty }\sin(x)\left({\frac {x}{\kappa }}\right)^{-1/3}{\frac {dx}{\kappa }}\,$ $={\frac {10}{36}}{\frac {1}{\pi ^{2}\kappa ^{11/3}}}C_{n}^{2}\int _{0}^{\infty }x^{-1/3}\sin(x)\,dx\,$ $={\frac {10}{36}}{\frac {1}{\pi ^{2}\kappa ^{11/3}}}C_{n}^{2}2^{-1/3}{\sqrt {\pi }}\,{\frac {\Gamma ({\frac {-1}{6}}+1)}{\Gamma ({\frac {1}{2}}+{\frac {1}{6}})}}\,$ $=.033\,C_{n}^{2}\kappa ^{-11/3}\,$ 