Physics Python Formula

Question

I am a college student. I need to compute this formula for my final year Physics project. I am putting my own version which is having a math domain error.

FORMULA : https://i.stack.imgur.com/wGhG0.jpg

The symbol shown is finite products which works same as summation sign but instead of adding, multiplication takes place. And instead of infinity I have to take values from 0 to 100.*

def funq(q, x):
    return (math.sqrt(1 - q) / 2 * math.pi * math.sqrt(4 - (1 - q) * (x ** 2))) * nprod(lambda i: (1 - q ** (i + 1))*((1 + q ** i) ** 2) - (1 - q)*(q ** i)*(x ** 2), [0, 100])

I have values for x and q, and here I have to take i from 0 to 100.

xs = [-1.9786357294030505, -1.8729683005605666, -1.8690079734953249, -1.7893930922097478, -1.7786223508087187, -1.7330418476321756, -1.7218150017905753, -1.7053215143651612, -1.6859860461491698, -1.6440572818865058, -1.6284242718994277, -1.594340198236287, -1.583713661880595, -1.5662028064425124, -1.511369193363463, -1.4959328789222448, -1.4756020410385255, -1.4362949249640893, -1.423699326374759, -1.3920503186588522, -1.3666704786700927, -1.3392600432337527, -1.3188944726286376, -1.300630969628771, -1.284216913905989, -1.2671896486219765, -1.2368473426034123, -1.235421991642431, -1.1970513167149672, -1.163387360445743, -1.1524460267982806, -1.13212800790605, -1.11682277889514, -1.0932584182021468, -1.0719251220099668, -1.0678870348976006, -1.0210296076133556, -1.0130488003608291, -0.9975858444089151, -0.9796452267641969, -0.9530125782990486, -0.9464384291595466, -0.9191313448853328, -0.9038898666356475, -0.8902743815564222, -0.8674005928510681, -0.8508650887302857, 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xs = np.array(xs)
xs.sort()
fig, ax = plt.subplots(figsize=(12, 5))
q = 0.015
ys = [funq(q, x) for x in xs]
ax.plot(xs, ys)
ax.set_ylim(ymin=0)
ax.set_xlabel('x')
ax.set_ylabel('funq')
plt.show()

ERROR

Traceback (most recent call last):
  File "C:\Users\Parvathy\PycharmProjects\Hermite Polynomial\gaussian2p0.py", line 78, in <module>
ys = [funq(q, x) for x in xs]
  File "C:\Users\Parvathy\PycharmProjects\Hermite Polynomial\gaussian2p0.py", line 78, in <listcomp>
ys = [funq(q, x) for x in xs]
File "C:\Users\Parvathy\PycharmProjects\Hermite Polynomial\gaussian2p0.py", line 57, in funq
return (math.sqrt(1 - q) / 2 * math.pi * math.sqrt(4 - (1 - q) * (x ** 2))) * nprod(
ValueError: math domain error

Answer 1

You copied the formula, but you forgot to explain the part which describes the domain. From how the formula looks like, q should certainly be <= 1 (which would directly give an error for trying to square root a negative number), maybe it supposed to be between 0 and 1, with x between -2 and 2?

Also note that you forgot to put the multiplication signs. Without these, Python supposes you are doing function calls, leading to strange error messages.

To be able to call your function, it best to write it, well, as a function:

def funq(q, x):
    return (math.sqrt(1 - q) / 2 * math.pi * math.sqrt(4 - (1 - q) * (x ** 2))) * nprod(
        lambda i: (1 - q ** (i + 1))*((1 + q ** i) ** 2) - (1 - q)*(q ** i)*(x ** 2), [0, 100])

If the function wouldn't be using the multiprecision mpmath, it could be called with a complete numpy array of x values in one go. But, when mpmath functions are included, the function needs to be called on each x value, one by one. Beware that this can be extremely slow.

Here is some code that applies the function to 10 different q values, and shows them as a curve. As the x values aren't in order (there seem to be 3 leaps of 200 values), the values are sorted to prevent the plot going highwire.

import matplotlib.pyplot as plt
import math
from mpmath import nprod
import numpy as np

def funq(q, x):
    return (math.sqrt(1 - q) / 2 * math.pi * math.sqrt(4 - (1 - q) * (x ** 2))) * nprod(
        lambda i: (1 - q ** (i + 1))*((1 + q ** i) ** 2) - (1 - q)*(q ** i)*(x ** 2), [0, 100])

xs = 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xs = np.array(xs)
xs.sort()
fig, ax = plt.subplots(figsize=(12, 5))
for q in np.arange(0, 1, 0.1):
    ys = [funq(q, x) for x in xs]
    ax.plot(xs, ys, label=f'q={q:.1f}')
ax.set_ylim(ymin=0)
ax.set_xlabel('x')
ax.set_ylabel('funq')
ax.legend()
plt.show()

PS: For a single value of q, you could write:

q = 0.37
ys = [funq(q, x) for x in xs]
ax.plot(xs, ys)

Here xs is a list of x values, and ys a list of corresponding y values. Matplotlib will connect these values one-by-one to show a curve. Numpy is Python's superpower library to easily and fastly work with lists (or matrices or n-D arrays) of numbers. For an introduction to numpy, maybe this tutorial could be a start.

The label= parameter to plot are meant for the label to put into the legend. f'q={q:.1f}' is Python's new way to format strings.

As you seem to be very new to Python, a tutorial (e.g. this one) could be a start.