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import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from scipy.interpolate import interp1d
from scipy.optimize import curve_fit, fsolve


## Problem¶

Solve the following two equations in the two unknowns $c_L$ and $c_{\eta}$: $$\int_0^{\infty}E(\kappa)d\kappa = K,$$

$$\int_0^{\infty}2\nu\kappa^2E(\kappa)d\kappa = \epsilon.$$

Here, $\nu = 1.5\times 10^{-5}$, $K = 0.008$, and $\epsilon = 0.003$.

You can take $\infty\approx 100,000$.

We also have the following functions and relations: $$L = \frac{K^{3/2}}{\epsilon},$$ $$\eta = \left(\frac{\nu^3}{\epsilon}\right)^{1/4},$$

$$E(\kappa) = 1.5\epsilon^{2/3}\kappa^{-5/3}f_L(\kappa)f_{\eta}(\kappa),$$$$f_L(\kappa) = \left(\frac{\kappa L}{[(\kappa L)^2 + c_L]^{1/2}}\right)^{11/3},$$$$f_{\eta}(\kappa) = \exp(-5.2([(\kappa\eta)^4 + c_{\eta}^4]^{1/4}-c_{\eta})).$$
• Guesses: $c_L=4$, $c_{\eta}=0.5$.
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