Plotting functions with singularities

P3.1.1Plotting functions with singularities

Plot the functions

\begin{align*} f_1(x) &= \ln \left( \frac{1}{\sin^2x} \right)\\ f_2(x) &= \ln \left( \frac{1}{\cos^2x} \right) \end{align*}

on 1000 points across the range $-20 \le x \le 20$ . What happens to these functions at $x = n\pi/2$ for $n=0,\pm 1, \pm 2,\cdots$ ? What happens in your plot of them?

Show solution

If you run the code below the output should be something like the graph shown here.

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-20, 20, 1000)
def f1(x):
    return np.log(np.sin(x)**-2)
def f2(x):
    return np.log(np.cos(x)**-2)

plt.plot(x, f1(x))
plt.plot(x, f2(x))
plt.show()

Plotting functions with singularities with Matplotlib

Consider $$f_1(x)$$ . This function is infinite for $x = n\pi$ where $n = 0, 1, 2, \cdots$ as Python will tell you:

>>> f1(0)
__main__:2: RuntimeWarning: divide by zero encountered in double_scalars
inf

However, the discrete grid of $$x$$ values we have chosen does not actually include any of these points, and so the peaks appear as maxima which vary in height according to how close to $x = n\pi$ the points in x become. For example, in the vicinity of $$x=0$$ :

>>> x[498:502]
array([-0.06006006, -0.02002002,  0.02002002,  0.06006006])
>>> f1(x[498:502])
array([ 5.62602298,  7.82217861,  7.82217861,  5.62602298])

but near $x = \pi$ ,

>>> x[576:580]
array([ 3.06306306,  3.1031031 ,  3.14314314,  3.18318318])
>>> f1(x[576:580])
array([  5.09061561,   6.51523082,  12.93836992,   6.36034221])