import numpy as np
import matplotlib.pyplot as plt
from IPython.display import display,Math

print("Euler's formula")
display(Math('e^{ik} = \\cos(k) + i\\sin(k)'))
display(Math('me^{ik} = m(\\cos(k) + i\\sin(k))')) # m: magnitude, k: angle
display(Math('e^{i0} = 1 + 0i'))
display(Math('e^{i\\pi/2} = 0 + 1i'))
display(Math('e^{i\\pi} = -1 + 0i'))
display(Math('\\quad \\longrightarrow \\quad e^{i\\pi} + 1 = 0')) # the most elegant equation

Euler's formula

$\displaystyle e^{ik} = \cos(k) + i\sin(k)$

$\displaystyle me^{ik} = m(\cos(k) + i\sin(k))$

$\displaystyle e^{i0} = 1 + 0i$

$\displaystyle e^{i\pi/2} = 0 + 1i$

$\displaystyle e^{i\pi} = -1 + 0i$

$\displaystyle \quad \longrightarrow \quad e^{i\pi} + 1 = 0$

k = np.pi/6
m = 2.3

eul = m*np.exp(1j*k)
cis = m*(np.cos(k) + 1j*np.sin(k)) #cosine imagenary sine

print(cis)
print(eul)

(1.991858428704209+1.1499999999999997j)
(1.991858428704209+1.1499999999999997j)

mag = np.abs(eul)
ang = np.angle(eul)

print(m,mag)
print(k,ang)

plt.polar([0,ang],[0,mag],'r')
plt.polar(k,m,'bo')

plt.show()

2.3 2.3
0.5235987755982988 0.5235987755982987

Exercise

def eulerFromCosSine():
    re = eval(input('cosine part: '))
    im = eval(input('sine part: '))
    
    m = np.sqrt(re**2 + im**2)
    k = np.arctan2(im,re)
    
    plt.polar([0,k],[0,m])
    plt.title('me$^{i\\phi}$, m=%g, $\\phi$= %g' %(m,k))
    plt.thetagrids([0,45,130,200,222])
    plt.show()

eulerFromCosSine()

cosine part: 1
sine part: 1

np.pi/4

0.7853981633974483