import matplotlib.pyplot as plt
import numpy as np

L = [1]
for k in range(9):
    L.append(L[k] + 1j*L[k]/abs(L[k]))

def g(t):
    return sum([np.arctan(1/np.sqrt(1+k))-np.arctan(1/np.sqrt(t+k)) for k in range(9999)])

def x(t):
    return np.sqrt(t)*np.cos(g(t))

def y(t):
    return np.sqrt(t)*np.sin(g(t))

T = np.arange(1,10,0.1)
X = x(T)
Y = y(T)
plt.axis('equal')
plt.plot(X,Y)
plt.plot([z.real for z in L],[z.imag for z in L])
plt.show()