import math

# y=x在[-pi,pi]上的傅里叶级数

def get_fourier1(n: int):

    def f(x: float):

        a = []
        b = []

        for i in range(1, n + 1):
            a.append(0)
            b.append(2 * (((-1) ** (i + 1)) / i))
        
        res = a[0] / 2

        for i in range(1, n + 1):
            res += a[i - 1] * math.cos(i * x)
            res += b[i - 1] * math.sin(i * x)

        return res

    return f

# y=x^2

def get_fourier2(n: int):

    def f(x: float):

        a = []
        b = []

        for i in range(1, n + 1):
            a.append((4 * ((-1) ** i)) / (i ** 2))
            b.append(0)
        
        res = (2 * math.pi * math.pi / 3) / 2

        for i in range(1, n + 1):
            res += a[i - 1] * math.cos(i * x)
            res += b[i - 1] * math.sin(i * x)

        return res

    return f

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-3, 3, 1000)

plt.plot(x, x, '-', label=r'$y=x$')

p = get_fourier1(10)
p_f = np.vectorize(p)
y_1 = p_f(x)
plt.plot(x, y_1, '-', label=r'$Fourier, n=10$')

plt.title('Fourier Expand')
plt.xlabel('X')
plt.ylabel('Y')
plt.legend()
plt.grid()
plt.show()

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-4, 4, 100)
y = x * x

plt.plot(x, y, '-', label=r'$y=x^2$')

# p = get_fourier2(2)
# p_f = np.vectorize(p)
# y_1 = p_f(x)
# plt.plot(x, y_1, '-', label=r'$Fourier, n=2$')

# p = get_fourier2(3)
# p_f = np.vectorize(p)
# y_1 = p_f(x)
# plt.plot(x, y_1, '-', label=r'$Fourier, n=3$')

p = get_fourier2(11)
p_f = np.vectorize(p)
y_1 = p_f(x)
plt.plot(x, y_1, '-', label=r'$Fourier, n=4$')

plt.title('Fourier Expand')
plt.xlabel('X')
plt.ylabel('Y')
plt.legend()
plt.grid()
plt.show()