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def derivative(f, a, h=0.01):
    '''Approximates the derivative of the function f in a
    
    :param function f: function to differentiate
    :param float a: the point of differentiation
    :param float h: step size
    :return float: the derivative of f in a
    '''
    return (f(a + h) - f(a - h))/(2*h)

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import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation

x_start, y_start = -2, -2  # an interesting region starts here
width, height = 4, 4  # for 4 units up and right
density_per_unit = 200  # how many pixles per unit

# real and imaginary axis
re = np.linspace(x_start, x_start + width, width * density_per_unit )

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def julia_quadratic(zx, zy, cx, cy, threshold):
    """Calculates whether the number z[0] = zx + i*zy with a constant c = x + i*y
    belongs to the Julia set. In order to belong, the sequence 
    z[i + 1] = z[i]**2 + c, must not diverge after 'threshold' number of steps.
    The sequence diverges if the absolute value of z[i+1] is greater than 4.
    
    :param float zx: the x component of z[0]
    :param float zy: the y component of z[0]
    :param float cx: the x component of the constant c
    :param float cy: the y component of the constant c

IlievskiV

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation

x_start, y_start = -2, -1.5  # an interesting region starts here
width, height = 3, 3  # for 3 units up and right
density_per_unit = 250  # how many pixles per unit

# real and imaginary axis
re = np.linspace(x_start, x_start + width, width * density_per_unit )

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def mandelbrot(x, y, threshold):
    """Calculates whether the number c = x + i*y belongs to the 
    Mandelbrot set. In order to belong, the sequence z[i + 1] = z[i]**2 + c
    must not diverge after 'threshold' number of steps. The sequence diverges
    if the absolute value of z[i+1] is greater than 4.
    
    :param float x: the x component of the initial complex number
    :param float y: the y component of the initial complex number
    :param int threshold: the number of iterations to considered it converged
    """

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def calculate_integral(f, a, b, n):
    '''Calculates the integral based on the composite trapezoidal rule
    relying on the Riemann Sums.

    :param function f: the integrand function
    :param int a: lower bound of the integral
    :param int b: upper bound of theintergal
    :param int n: number of trapezoids of equal width
    :return float: the integral of the function f between a and b
    '''

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import numpy as np
def gbm_mean(G0, mu, sigma, N, T):
    """Simulates the mean of the Geometric Brownian Motion, which is:
        E(t) = G0*e^{(mu + sigma^{2}/2)*t}
        
    :param float G0: initial value
    :param float mu: drift coefficient
    :param float sigma: diffusion coefficient
    :param int N: number of discrete steps
    :param int T: number of continuous time steps