I have a function:

delt=1 #trial
def f(z):
    return ((1-2*z)*np.exp(-delt/z))/(((1-z)**(2+delt))*(z**(2-delt)))

I also have a variable:

import scipy.integrate as integrate
var=integrate.quad(f,0,0.5)[0]  # equals 0.040353419593637516

Now I am trying to find the value p such that

integrate.quad(f,0.5,p)= var

and manually I can check that it is around 0.605

I define the following function to be used in optimization:

def integral(p):
    return integrate.quad(f,0.5, p)[0]-var

However, i get the following results:

import scipy.optimize as op
In[26]: op.root(integral,0.61)
    fjac: array([[-1.]])
     fun: -0.040353420516861596
 message: 'The iteration is not making good progress, as measured by the \n  improvement from the last ten iterations.'
    nfev: 18
     qtf: array([ 0.04035342])
       r: array([ 0.00072888])
  status: 5
 success: False
       x: array([ 0.50002065])
In[27]: op.fsolve(integral,0.61)
Out[27]: array([ 0.50002065])

Any idea why both root and fsolve might be failing?

  • 1
    $\begingroup$ integrate.quad(f, 0.5, 0.605) is approximately 0 and not close to var as you claim. Also, a plot of integral(p) for p>0.5 suggests that it is less than zero $\endgroup$
    – Stelios
    Commented Mar 9, 2017 at 9:06

1 Answer 1


As @Stelios mentioned, It seems like the integral from 0.5 to ~0.605 is close to 0, and then it turns negative.The antiderivative of your function is given by

$$\int f(z; t=1)\, dz = \frac{e^{-\frac{1}{z} - 1}\left[e\, (1-2z)z - e^{1/z}(z-1)^2\operatorname{Ei}\left(\frac{z-1}{z}\right)\right]}{2(z-1)^2}\, ,$$

keep in mind that it does not work for $z=0$, indeed, your original function is not defined at $z=0$, and the limit does not even exist. If you make the plot of the antiderivative you can see these features

enter image description here


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