#extract tx min from equation numerically
import numpy as np

x = np.linspace(47,49,10000)

x0              = 48.0333          
gamma           = 6.90003          
Lambda_p        = 0.0467036       
phi             = 0.129305         

lorentz_dis_beta = (1 + (4*Lambda_p**2 
	- 4*Lambda_p*np.cos(phi))*gamma**2/ ( 4*(x0-x)**2 + (gamma)**2 ) 
	- ( 8*Lambda_p*np.sin(phi)*(x-x0)*gamma ) / ( 4*(x0-x)**2 + gamma**2 ) ) 

print 1-lorentz_dis_beta.min()