Python：在渐近区域附近求解

Question

For E=0.46732451 and t=1.07589765 I am trying to solve for the upper limit of the integral t= \\int_{0}^{z} 1/sqrt(2*(0.46732451-z**2)), I plotted this function and it looks like this .

Around t=1 it kind of asymptotes.

I have the following code

import numpy as np
from scipy import integrate
from scipy.optimize import fsolve

def fg(z_up,t,E):
      def h(z,E):
           return 1/(np.sqrt(2*(E-z**2)))

      b, err = integrate.quad(h, 0, z_up,args=(E)) 
      return b-t 



x0 = 0.1
print fsolve(fg, x0, args=(1.07589765, 0.46732451))[0]

But this code just outputs the guess value, no matter what I put, so I am guessing it has something to do with the fact that the curve asymptotes there. I should note that this code works for other values of t which are away from the asymptotic region.

Can anyone help me resolve this?

Thanks

EDIT After playing around for a while, I solved the problem, but it's kind of a patchwork, it only works for similar problems not in general (or does it?)

I made the following changes: the maximum value that z can attain is sqrt(0.46732451) , so I set x0=0.5*np.sqrt(0.46732451) and set factor anywhere between 0.1 to 1 , and out pops the correct answer. I don't have an explanation for this, perhaps someone who is an expert in this matter can help?

Answer 1

You should use bisect instead, as it handles nan without problems:

print bisect(fg, 0.4, 0.7, args=(1.07589765, 0.46732451))

Here 0.4 and 0.7 are taken as an example but you can generalize this for almost any diverging integral by using 0 and let's say 1e12 as the limits.

However, I'm not sure I understand what you really want to do... if you want to find the limit at which the integral diverges, cf. your

I am trying to solve for the upper limit of the integral

then it's simply for z_up -> \\sqrt{E} \\approx 0,683611374 ... So to find the (approximate) numerical value of the integral you just have to decrease z_up from that value until quad stops giving a nan ...