import numpy as np
import sympy as sym
from IPython.display import display,Math

print('Multiplication with complex numbers')
display(Math('z = (a\\quad bi)'))
display(Math('w = (c\\quad di)'))
display(Math('z\\times w = (a\\quad bi) \\times (c\\quad di)'))
display(Math('\\quad \\quad \\quad= ac + adi + cbi + bdi^2'))
display(Math('\\quad \\quad \\quad= ac + adi + cbi - bd'))

print('Complex number: 複素数')
display(Math('z'))
display(Math('z = a + bi'))
display(Math('w = a - bi'))
print('Complex conjugate: 複素共役')
display(Math('z^*'))
display(Math('z^* = a - bi'))
display(Math('w^* = a + bi'))

print('Multiplication with complex conjugate')
display(Math('z\\times z^* = (a\\quad bi) \\times (a- bi)'))
display(Math('\\quad \\quad \\quad= a^2 - abi + abi - bi^2'))
display(Math('\\quad \\quad \\quad= a^2 + b^2'))

Multiplication with complex numbers

$\displaystyle z = (a\quad bi)$

$\displaystyle w = (c\quad di)$

$\displaystyle z\times w = (a\quad bi) \times (c\quad di)$

$\displaystyle \quad \quad \quad= ac + adi + cbi + bdi^2$

$\displaystyle \quad \quad \quad= ac + adi + cbi - bd$

Complex number: 複素数

$\displaystyle z$

$\displaystyle z = a + bi$

$\displaystyle w = a - bi$

Complex conjugate: 複素共役

$\displaystyle z^*$

$\displaystyle z^* = a - bi$

$\displaystyle w^* = a + bi$

Multiplication with complex conjugate

$\displaystyle z\times z^* = (a\quad bi) \times (a- bi)$

$\displaystyle \quad \quad \quad= a^2 - abi + abi - bi^2$

$\displaystyle \quad \quad \quad= a^2 + b^2$

z1 = np.complex(4,5)
z2 = np.complex(6,-2)

w = np.real(z1)*np.real(z2) + np.real(z1)*np.imag(z2)*1j + np.imag(z1)*1j*np.real(z2) + np.imag(z1)*1j*np.imag(z2)*1j

print(w)
print(z1*z2)
print(sym.sympify(w))
print(sym.sympify(z1*z2))

(34+22j)
(34+22j)
34.0 + 22.0*I
34.0 + 22.0*I

np.conj(z1)

(4-5j)

np.conj(z2)

(6+2j)

a,b = sym.symbols('a,b',real=True)

z = a + b*sym.I

print(z)
sym.conjugate(z)

a + I*b

a - I*b

z * sym.conjugate(z)

(a - I*b)*(a + I*b)

sym.expand(z * sym.conjugate(z))

a**2 + b**2

display(Math('z\\times x^* = %s' %sym.latex(sym.expand(z*sym.conjugate(z)))))

$\displaystyle z\times x^* = a^{2} + b^{2}$