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

# ignore warnings
import warnings
warnings.filterwarnings("ignore")

def brownian_motion(N, T, h, seed=42):

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import numpy as np

def drifted_brownian_motion(mu, sigma, N, T, seed=42):
    """Simulates a Brownian Motion with drift.
    
    :param float mu: drift coefficient
    :param float sigma: volatility coefficient
    :param int N : number of discrete steps
    :param int T: number of continuous time steps
    :param int seed: initial seed of the random generator

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def geometric_brownian_motion(G0, mu, sigma, N, T):
    """Simulates a Geometric Brownian Motion.
    
    :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
    :return list: the geometric Brownian Motion
    """

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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

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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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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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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 )