Casio fx-CG 100 Python: Infinite Series and Fresnel Integrals
Introduction
Many functions, such as the Fresnel Integrals, the Zeta function, and the error function, can be calculated using infinite polynomials by the sum:
f(x) = Σ( g(x, n) from n = 0 to n = ∞) or
f(x) = Σ( g(x, n) from n = 1 to n = ∞)
where:
f(x): the function to be calculated
g(x, n): the infinite series representative of (x)
g(x, n) often includes the factorial function n!, where n starts at either 0 or 1 to infinity. While I was programming on fx-CG 100, I did not find a factorial function in the math module that is included in the MicroPython. So I included a definition of the factorial function as:
# factorial function
def fact(n):
f=1
if n<=1:
return 1
else:
for i in range(2,n+1):
f*=i
return f
While this function does not check for negative integers, it gives the default values 0! = 1! = 1.
The main program sets up a sum, ask for initial values, and set up a check variable w = g(x,n), with a high initial value to “fail” the loop test. The loop template is set up as:
w=initial value
n=0 (start value of n)
while abs(w)>=1e-20: (1e-20 is a tolerance value, which can be adjusted)
w=g(x,n)
s+=w
n+=1
An Example: Cosine of x Squared
Radians mode is used and is the default angle mode in Python.
cos(x²) = 1 – x^4 ÷ 2 + x^8 ÷ 24 – x^12 ÷ 720 + x^16 ÷ 40320 - …
= Σ( x^(4*n) * (-1)^n ÷ (2n)!, n = 0 to n = ∞)
In this case, g(x,n) = x^(4*n) * (-1)^n ÷ (2 * n)!
I set accuracy tolerance of 1 * 10^-20 but formatted the answer to 12 decimal places.
Casio fx-CG 100 Micropython: cossq.py
# cosine of x squared by series
# template of infinite series
# math module so pi is allowed
from math import *
# factorial function
def fact(n):
f=1
if n<=1:
return 1
else:
for i in range(2,n+1):
f*=i
return f
# main program
# setup sum
s=0
# ask for x
print("cos(x**2) by series")
x=eval(input("x radians: "))
# series
# set term artificially high
w=100
# set counter at beginning
n=0
# series loop
while abs(w)>=1e-20:
w=x**(4*n)*(-1)**(n)/fact(2*n)
s+=w
n+=1
# answer
print("{0:.12f}".format(s))
Examples:
Example 1: Input: x = 0.5, Result: 0.968912421711
Example 2: Input: x = 1.7, Result: -0.968517164228
Fresnel Integrals
There are two Fresnel integrals.
Cosine Fresnel Integral:
C(x) = ∫( cos( t^2) dt, t = 0 to t = x)
This integral is represented by the infinite series:
C(x) = Σ( x^(4*n + 1) * (-1)^n ÷ ((2* n)! * (4 * n + 1)), n = 0 to n = ∞)
Sine Fresnel Integral:
S(x) = ∫( sin( t^2) dt, t = 0 to t = x)
This integral is represented by the infinite series:
S(x) = Σ( x^(4*n + 3) * (-1)^n ÷ ((2* n + 1)! * (4 * n + 3)), n = 0 to n = ∞)
Casio fx-CG 100 Micropython: fresnel.py
# Fresnel Integrals
# template of infinite series
# Eddie W. Shore, 11/23/2025
from math import *
# factorial function
def fact(n):
f=1
if n<=1:
return 1
else:
for i in range(2,n+1):
f*=i
return f
# main program
# c: cosine, s=sine
c=0
s=0
# ask for x
x=eval(input("x radians: "))
# series
# set term artificially high
w=100
# set counter at beginning
n=0
# series loop
while abs(w)>=1e-20:
a=(-1)**n*x**(4*n+1)/(fact(2*n)*(4*n+1))
b=(-1)**n*x**(4*n+3)/(fact(2*n+1)*(4*n+3))
w=max(a,b)
c+=a
s+=b
n+=1
# answer
print("x:{0:.12f}".format(x))
print("Fresnel Cosine:\n{0:.12f}".format(c))
print("Fresnel Sine:\n{0:.12f}".format(s))
Examples:
Example 1: x = 0.4,
C(0.4) ≈ 0.398977212913, S(0.4) ≈ 0.021294355570
Example 2: x = 3.1,
C(3.1) ≈ 0.605132097879, S(3.1) ≈ 0.785491219284
Eddie
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