Elementary Complex Analysis

Elementary Complex Analysis#

Revised

07 May 2023

Hide code cell source 
import numpy             as np
import matplotlib        as mpl
from   matplotlib        import cm
from   matplotlib.ticker import LinearLocator
import matplotlib.pyplot as plt
import matplotlib.image  as mpimg
plt.style.use('ggplot');
#mpl.projections.get_projection_names()
#plt.style.available

import scipy
from   scipy import integrate

import  math
import cmath

from IPython.display import display, Math

from   datetime import datetime as d
import locale                   as l
import platform                 as p
import sys                      as s

pad = 20
print(f"{'Executed'.upper():<{pad}}: {d.now()}")
print()
print(f"{'Platform'   :<{pad}}: "
      f"{p.mac_ver()[0]} | "
      f"{p.system()} | "
      f"{p.release()} | "
      f"{p.machine()}")
print(f"{''           :<{pad}}: {l.getpreferredencoding()}")
print()
print(f"{'Python'     :<{pad}}: {s.version}")
print(f"{''           :<{pad}}: {s.version_info}")
print(f"{''           :<{pad}}: {p.python_implementation()}")
print()
print(f"{'Matplotlib' :<{pad}}: {mpl.__version__}")
print(f"{'NumPy'      :<{pad}}: {np .__version__}")
print(f"{'SciPy'      :<{pad}}: {scipy.__version__}")

EXECUTED            : 2024-06-20 10:49:53.397333

Platform            : 14.5 | Darwin | 23.5.0 | arm64
                    : UTF-8

Python              : 3.11.9 | packaged by conda-forge | (main, Apr 19 2024, 18:34:54) [Clang 16.0.6 ]
                    : sys.version_info(major=3, minor=11, micro=9, releaselevel='final', serial=0)
                    : CPython

Matplotlib          : 3.8.4
NumPy               : 1.26.4
SciPy               : 1.13.1

def translate (z,z0=1+2j):
  return z0+z
def scale (z,r=0.5):
  return r*z
def rotate90 (z):
  "Multiply z by i to rotate by 90 degrees."
  return complex(-z.imag,z.real)
def rotate180 (z):
  "Multiply z by -1=i^2 to rotate by 180 degrees."
  return -z
def rotate270 (z):
  "Multiply z by -i=i^3 to rotate by 270 degrees."
  return complex(z.imag,-z.real)
def rotate (z,tau):
  "Rotate z by tau radians."
  return z*cmath.e**(tau*1j)
def uC (z):
  "Project a complex number onto the unit circle."
  return cmath.e**(1j*cmath.polar(z)[1])

def pC (S,xl=-10,yl=10,dpi=100,t='n',v='n'):
  "Plot a set of complex numbers in the complex plane."
  fig = plt.figure(dpi=dpi);
  ax = plt.subplot();
  ax.set_aspect(1);
  for s in S:
    for x,y in [np.array([[c.real,c.imag] for c in s]).T]:
      ax.scatter(x,y);
      if v=='y':
        ax.quiver([0]*len(x),[0]*len(y),x,y,scale_units='xy',scale=1,width=0.001);
  ax.add_patch(plt.Circle((0,0),1,facecolor='None',edgecolor='b'));
  ax.set_xticks(ticks=np.arange(xl,yl+1),labels=np.arange(xl,yl+1));
  ax.set_yticks(ticks=np.arange(xl,yl+1),labels=[f'{str(c)}i' for c in np.arange(xl,yl+1)]);
  ax.set_xlim(xl,yl);
  ax.set_ylim(xl,yl);
  ax.set_xlabel('Re');
  ax.set_ylabel('Im');

What is \(i\)?#

\( \boxed{ i\overset{\text{def}}=\sqrt{-1} } \)

\( \begin{aligned} i^0&=1\\ i^1&=\sqrt{-1}\\ i^2&=(\sqrt{-1})^2=-1\\ i^3&=ii^2=(i)(-1)=-i\\ i^4&=i^2i^2=(-1)(-1)=1=i^0\\ i^5&=ii^4=(i)(1)=i=i^1\\ i^6&=i^2i^4=(-1)(1)=-1=i^2\\ i^7&=ii^2i^4=(i)(-1)(1)=-i=i^3\\ i^8&=i^4i^4=(1)(1)=1=i^0\\ &...\\ i^{4n}&=1\\ i^{4n+1}&=i\\ i^{4n+2}&=-1\\ i^{4n+3}&=-i\\ \end{aligned} \)

where \(n={0,1,2,3,...}\)

def powers_of_i (ns):
  for n in range(ns):
    yield f"i^{n:>02} = {1j ** n}"

i = powers_of_i(25)
list(i)

['i^00 = (1+0j)',
 'i^01 = 1j',
 'i^02 = (-1+0j)',
 'i^03 = (-0-1j)',
 'i^04 = (1+0j)',
 'i^05 = 1j',
 'i^06 = (-1+0j)',
 'i^07 = (-0-1j)',
 'i^08 = (1+0j)',
 'i^09 = 1j',
 'i^10 = (-1+0j)',
 'i^11 = (-0-1j)',
 'i^12 = (1+0j)',
 'i^13 = 1j',
 'i^14 = (-1+0j)',
 'i^15 = (-0-1j)',
 'i^16 = (1+0j)',
 'i^17 = 1j',
 'i^18 = (-1+0j)',
 'i^19 = (-0-1j)',
 'i^20 = (1+0j)',
 'i^21 = 1j',
 'i^22 = (-1+0j)',
 'i^23 = (-0-1j)',
 'i^24 = (1+0j)']

Complex Numbers#

A complex number is an ordered pair \((a,b)\) where \(a,b\in\mathbb{R}\).

A complex number may take the form

\(z=a+bi\)

z = 3 + 2j
z

(3+2j)

type(z)

complex

The real part of a complex number#

Let \(z=a+bi\) where \(a,b\in\mathbb{R}\) be a complex number.

The real part of \(z\) is

\(\text{Re}(z)=a=a+0i\) where \(a,0\in\mathbb{R}\)

z.real

3.0

The imaginary part of a complex number#

Let \(z=a+bi\) where \(a,b\in\mathbb{R}\) be a complex number.

The imaginary part of \(z\) is

\(\text{Im}(z)=bi=0+bi\) where \(b,0\in\mathbb{R}\)

z.imag * 1j

2j

\( z=\text{Re}(z)+\text{Im}(z) \)

The set of complex numbers \(\mathbb{C}\)#

The set of complex numbers

\( \begin{aligned} \mathbb{C}=\{a+bi\mid a,b\in\mathbb{R}\} \end{aligned} \)

x is imaginary

\( \begin{aligned} x^2&=-1 \implies x=i \\ x^2&=-9 \implies x=3i \\ x^2&=-b \implies x=i\sqrt{b} \end{aligned} \)

x is complex

\( \begin{aligned} (x-1)^2&=-9 \implies x-1=3i \implies x=3i+1 \end{aligned} \)

Properties of complex numbers#

Additive Commutativity#

\( \alpha+\beta=\beta+\alpha \,\,\,\forall\alpha,\beta\in\mathbb{C} \)

PROOF

\(\blacksquare\)

a = (3 + 2j)
b = (5 + 4j)

print(a + b)
print(         b + a)
print(a + b == b + a)

(8+6j)
(8+6j)
True

Multiplicative Commutativity#

\( \alpha\beta=\beta\alpha \,\,\,\forall\alpha,\beta\in\mathbb{C} \)

PROOF

Let \(\alpha = a + bi\) and \(\beta = c + di\) where \(a, b, c, d \in \mathbb{R}\).

\( \begin{aligned} \alpha\beta &= (a + bi)(c + di) \\ &= ac + adi + bci + bdi^2 \\ &= (c + di)a + (c + di)bi \\ &= (c + di)(a + bi) \\ &= \beta\alpha \\ \end{aligned} \)

\(\blacksquare\)

a = (3 + 2j)
b = (5 + 4j)

print(a * b)
print(         b * a)
print(a * b == b * a)

(7+22j)
(7+22j)
True

Additive Associativity#

\( (\alpha+\beta)+\lambda=\alpha+(\beta+\lambda) \,\,\,\forall\alpha,\beta,\lambda\in\mathbb{C} \)

PROOF

\(\blacksquare\)

a = (3 + 2j)
b = (5 + 4j)
l = (7 + 6j)

print( a + b                    )
print(                    b + l )
print((a + b) + l               )
print(               a + (b + l))
print((a + b) + l == a + (b + l))

(8+6j)
(12+10j)
(15+12j)
(15+12j)
True

Multiplicative Associativity#

\( (\alpha\beta)\lambda=\alpha(\beta\lambda) \,\,\,\forall\alpha,\beta,\lambda\in\mathbb{C} \)

PROOF

\(\blacksquare\)

a = (3 + 2j)
b = (5 + 4j)
l = (7 + 6j)

print( a * b                    )
print(                    b * l )
print((a * b) * l               )
print(               a * (b * l))
print((a * b) * l == a * (b * l))

(7+22j)
(11+58j)
(-83+196j)
(-83+196j)
True

Existence of the additive identity element#

\( \lambda+0=\lambda \,\,\,\forall\lambda\in\mathbb{C},0\in\mathbb{R} \)

PROOF

\(\blacksquare\)

l = 3 + 2j
i = 0 + 0j

print(l         )
print(     l + i)
print(l == l + i)

(3+2j)
(3+2j)
True

Existence of the multiplicative identity element#

\( \lambda1=\lambda \,\,\,\forall\lambda\in\mathbb{C},1\in\mathbb{R} \)

PROOF

\(\blacksquare\)

l = 3 + 2j
i = 1 + 0j

print(l         )
print(     l * i)
print(l == l * i)

(3+2j)
(3+2j)
True

Existence of additive inverse elements#

\(\forall \alpha \in \mathbb{C}\), \(\exists \beta \in \mathbb{C}\) s.t. \(\alpha + \beta = 0\)
The additive inverse of an element \(\alpha\) is denoted \(-\alpha\).

PROOF

\(\blacksquare\)

Existence of multiplicative inverse elements#

\(\forall \alpha \in \mathbb{C}\) with \(\alpha \neq 0\), \(\exists \beta \in \mathbb{C}\) s.t. \(\alpha\beta = 1\)
The multiplicative inverse of an element \(\alpha \neq 0\) is denoted \(\alpha^{-1}\).

PROOF

\(\blacksquare\)

Distributivity#

\(\lambda(\alpha + \beta) = \lambda\alpha + \lambda\beta\)
\(\forall \lambda, \alpha, \beta \in \mathbb{C}\)

PROOF

\(\blacksquare\)

Operations on complex numbers#

Complex Addition#

\( (a+bi)+(c+di)=(a+c)+(b+d)i\,\,\,\text{where}\,a,b,c,d\in\mathbb{R} \)

PROOF

\( \begin{aligned} &(a+bi)+(c+di)\\ &=a+bi+c+di\\ &=a+c+bi+di\\ &=(a+c)+(bi+di)\\ &=(a+c)+(b+d)i\\ \end{aligned} \)

\(\blacksquare\)

a=1
b=2
c=3
d=4

display(
  complex(a,b) + complex(c,d),
  complex((a+c), (b+d)),
  complex(a,b) + complex(c,d) == complex((a+c), (b+d)),
)

(4+6j)

(4+6j)

True

Complex Multiplication#

\( (a+bi)(c+di)=(ac-bd)+(ad+bc)i\,\,\,\text{where}\,a,b,c,d\in\mathbb{R} \)

PROOF

\( \begin{aligned} &(a+bi)(c+di)\\ &=ac+adi+bci+bdi^2\\ &=ac+bdi^2+adi+bci\\ &=ac-bd+adi+bci\\ &=(ac-bd)+(adi+bci)\\ &=(ac-bd)+(ad+bc)i\\ \end{aligned} \)

\(\blacksquare\)

a=1
b=2j
c=3
d=4j

display(
  (a+b) * (c+d),
  complex((a*c - b.imag*d.imag),(a*d.imag + b.imag*c)),
  (a+b) * (c+d) == complex((a*c - b.imag*d.imag),(a*d.imag + b.imag*c)),
)

(-5+10j)

(-5+10j)

True

Complex Subtraction#

\( \beta-\alpha\equiv\beta+(-\alpha) \)

a=1
b=2
c=3
d=4

display(
  complex(a,b) -  complex(c,d),
  complex(a,b) + -complex(c,d),
  complex(a,b) -  complex(c,d) == complex(a,b) + -complex(c,d),
)

(-2-2j)

(-2-2j)

True

Complex Division#

\( \begin{aligned} \frac{\beta}{\alpha}\equiv\beta\alpha^{-1} \end{aligned} \)

Complex arithmetic obeys the same predence rules as real arithmetic.

1+2j*3

(1+6j)

(1+2j)*3

(3+6j)

4*3j**2

(-36+0j)

(4*3j)**2

(-144+0j)

Complex Conjugate of a complex number#

DEFINITION
Let \(z \in \mathbb{C}\).
The complex conjugate of \(z\) is \(\bar{z} \overset{\text{def}}= \,(\text{Re}\, z) - \,(\text{Im}\, z) i\).
If \(z = a + bi\), then \(\bar{z} = a - bi\).

(1+2j).conjugate()

(1-2j)

Absolute Value of a complex number#

DEFINITION
Let \(z \in \mathbb{C}\).
The absolute value of \(z\) is \(|z| \overset{\text{def}}= \sqrt{\,(\text{Re}\, z)^2 + \,(\text{Im}\, z)^2}\).
If \(z = a + bi\), then \(|z| = \sqrt{a^2 + b^2}\).

Claim

\(\forall z \in \mathbb{C}, |z| \ge 0\). (The absolute value of any complex number is nonnegative.)

Proof

This follows from the fact that the sum of squares of a collection of real numbers is nonnegative.

\(\blacksquare\)

Claim

If \(z \in \mathbb{R}\), then \(z = \bar{z}\). (The complex conjugate of a real number is itself.)

Proof

\(z = a + 0i = a = a - 0i = \bar{z}\)

\(\blacksquare\)

abs(1+2j) == np.sqrt(1**2 + 2**2)

True

Operations on a complex number and its conjugate#

Sum of \(z\) and \(\bar{z}\)#

Let \(z \in \mathbb{C}\).
\(z + \bar{z} = 2a\)

Proof

\( \begin{aligned} z + \bar{z} &= (a + bi) + (a - bi) \\ &= a + bi + a - bi \\ &= 2a + bi - bi \\ &= 2a \end{aligned} \)

\(\blacksquare\)

Difference of \(z\) and \(\bar{z}\)#

Let \(z \in \mathbb{C}\).
\(z - \bar{z} = 2bi\)

Proof

\( \begin{aligned} z - \bar{z} &= (a + bi) - (a - bi) \\ &= a + bi - a + bi \\ &= 2bi + a - a \\ &= 2bi \end{aligned} \)

\(\blacksquare\)

Product of \(z\) and \(\bar{z}\)#

Let \(z \in \mathbb{C}\).
\(z \bar{z} = |z|^2\)

Proof

\( \begin{aligned} z \bar{z} &= (a + bi) (a - bi) \\ &= a^2 - abi + abi - (bi)^2 \\ &= a^2 - b^2 i^2 \\ &= a^2 + b^2 \\ &= (\sqrt{a^2 + b^2})^2 \\ &= |z|^2 \end{aligned} \)

\(\blacksquare\)

Properties of the complex conjugate and the absolute value of a complex number#

Additivity of the complex conjugate#

Let \(w, z \in \mathbb{C}\).
\(\overline{w + z} = \bar{w} + \bar{z}\)

Proof

\( \begin{aligned} w + z &= (a + bi) + (c + di) \\ &= (a + c) + (b + d)i \\ \overline{w + z} &= (a + c) - (b + d)i \\ &= a + c - bi - di \\ &= (a - bi) + (c - di) \\ &= \bar{w} + \bar{z} \\ \end{aligned} \)

\(\blacksquare\)

Multiplicativity of the complex conjugate#

Let \(w, z \in \mathbb{C}\).
\(\overline{w z} = \bar{w} \bar{z}\)

Proof

\( \begin{aligned} w z &= (a + bi) (c + di) \\ &= ac + adi + bci + bdi^2 \\ &= (ac - bd) + (ad + bc)i \\ \overline{w z} &= (ac - bd) - (ad + bc)i \\ &= ac - bd - adi - bci \\ &= ac - adi - bci + bdi^2 \\ &= (a - bi) (c - di) \\ &= \bar{w} \bar{z} \\ \end{aligned} \)

\(\blacksquare\)

Conjugate of the conjugate#

Let \(z \in \mathbb{C}\).
\(\bar{\bar{z}} = z\)

Proof

\( \begin{aligned} \bar{z} &= a - bi \\ &= (a) + (-b)i \\ \bar{\bar{z}} &= (a) - (-b)i \\ &= a + bi \\ &= z \\ \end{aligned} \)

\(\blacksquare\)

Boundedness#

Let \(z \in \mathbb{C}\).
\(|a| \le |z|\) and \(|b| \le |z|\)

Proof

\( \begin{aligned} |a| &= \sqrt{a^2} \\ &\le \sqrt{a^2 + b^2} \\ &= |z| \end{aligned} \)

\(\blacksquare\)

Absolute value of the complex conjugate#

Let \(z \in \mathbb{C}\).
\(|\bar{z}| = |z|\)

Proof

\( \begin{aligned} \bar{z} &= a - bi \\ &= (a) + (-b)i \\ |\bar{z}| &= \sqrt{(a)^2 + (-b)^2} \\ &= \sqrt{a^2 + b^2} \\ &= |z| \\ \end{aligned} \)

\(\blacksquare\)

Multiplicativity of the absolute value#

Let \(w, z \in \mathbb{C}\).
\(|w||z| = |wz|\)

Proof

\( \begin{aligned} wz &= (a + bi) (c + di) \\ &= ac + adi + bci + bdi^2 \\ &= (ac - bd) + (ad + bc)i \\ |wz| &= \sqrt{(ac - bd)^2 + (ad + bc)^2} \\ &= \sqrt{(ac)^2 - 2abcd + (bd)^2 + (ad)^2 + 2abcd + (bc)^2} \\ &= \sqrt{(ac)^2 + (bd)^2 + (ad)^2 + (bc)^2} \\ &= \sqrt{(ac)^2 + (ad)^2 + (bd)^2 + (bc)^2} \\ &= \sqrt{a^2(c^2 + d^2) + b^2(c^2 + d^2)} \\ &= \sqrt{(a^2 + b^2)(c^2 + d^2)} \\ &= \sqrt{a^2 + b^2} \sqrt{c^2 + d^2} \\ &= |w||z| \\ \end{aligned} \)

\(\blacksquare\)

Triangle Inequality#

Let \(w, z \in \mathbb{C}\).
\(|w + z| \le |w| + |z|\)

Proof

\( \begin{aligned} |w + z|^2 &= (w + z)(\bar{w} + \bar{z}) \\ &= w \bar{w} + z \bar{z} + w \bar{z} + z \bar{w} \\ &= |w|^2 + |z|^2 + w \bar{z} + \overline{w \bar{z}} \\ &= |w|^2 + |z|^2 + w \bar{z} + 2 \,\text{Re}\, (w \bar{z}) \\ &\le |w|^2 + |z|^2 + w \bar{z} + 2 |w \bar{z}| \\ &= |w|^2 + |z|^2 + w \bar{z} + 2 |w| |z| \\ &= (|w| + |z|)^2 \\ |w + z| &\le |w| + |z| \\ \end{aligned} \)

\(\blacksquare\)

Euler’s Formula#

\( \boxed{ \begin{aligned} e^{i\theta} &=\cos\theta+i\sin\theta \\ e^{i\pi}+1 &=0 \end{aligned} } \)

\( \begin{aligned} e^x &=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}+\frac{x^7}{7!}+\frac{x^8}{8!}+\frac{x^9}{9!}+... &&\text{Taylor series} \\ x&=i\theta \\ e^{i\theta}&=1+(i\theta)+\frac{(i\theta)^2}{2!}+\frac{(i\theta)^3}{3!}+\frac{(i\theta)^4}{4!}+\frac{(i\theta)^5}{5!}+\frac{(i\theta)^6}{6!}+\frac{(i\theta)^7}{7!}+\frac{(i\theta)^8}{8!}+\frac{(i\theta)^9}{9!}+... \\ &=1+i\theta+\frac{(-1)\theta^2}{2!}+\frac{(-i)\theta^3}{3!}+\frac{(1)\theta^4}{4!}+\frac{(i)\theta^5}{5!}+\frac{(-1)\theta^6}{6!}+\frac{(-i)\theta^7}{7!}+\frac{(1)\theta^8}{8!}+\frac{(i)\theta^9}{9!}+... \\ &=1+i\theta-\frac{\theta^2}{2!}-\frac{i\theta^3}{3!}+\frac{\theta^4}{4!}+\frac{i\theta^5}{5!}-\frac{\theta^6}{6!}-\frac{i\theta^7}{7!}+\frac{\theta^8}{8!}+\frac{i\theta^9}{9!}-... \\ &=\left(1-\frac{\theta^2}{2!}+\frac{\theta^4}{4!}-\frac{\theta^6}{6!}+\frac{\theta^8}{8!}-...\right) +\left(i\theta-\frac{i\theta^3}{3!}+\frac{i\theta^5}{5!}-\frac{i\theta^7}{7!}+\frac{i\theta^9}{9!}-...\right) \\ &=\left(1-\frac{\theta^2}{2!}+\frac{\theta^4}{4!}-\frac{\theta^6}{6!}+\frac{\theta^8}{8!}-...\right) +i\left(\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\frac{\theta^7}{7!}+\frac{\theta^9}{9!}-...\right) \\ &=\cos\theta+i\sin\theta \\ \theta&=\pi \\ e^{i\pi} &=\cos\pi+i\sin\pi \\ &=-1+0 \end{aligned} \)

\(i^i\)#

\( \boxed{ i^i=\,\,e^{-\frac{\pi}{2}} } \)

\( \begin{aligned} e^{i\pi} &=-1 \\ \sqrt{e^{i\pi}} &=\sqrt{-1} \\ e^{i\frac{\pi}{2}} &=i \\ (e^{i\frac{\pi}{2}})^i &=i^i \\ e^{i^2\frac{\pi}{2}}&= \\ e^{-\frac{\pi}{2}}&= \end{aligned} \)

The Unit Circle in the Complex Plane#

Complex exponentials form circles!

Hide code cell source 
fig = plt.figure(dpi=200);
ax  = plt.subplot(111,projection='rectilinear');
ax.set_aspect(1);

ax.add_patch(plt.Circle((0,0),1,edgecolor='b',facecolor='None'));
ax.scatter( 1, 0); ax.text( 1*1.1, 0, '$1$', horizontalalignment='left');
ax.scatter( 0, 1); ax.text( 0, 1*1.1, '$i$', verticalalignment='bottom');
ax.scatter(-1, 0); ax.text(-1*1.1, 0, '$-1$', horizontalalignment='right');
ax.scatter( 0,-1); ax.text( 0,-1*1.1, '$-i$', verticalalignment='top');

ax.set_xlabel('Re');
ax.set_xticks(ticks=np.arange(-2,3,1));
ax.set_xlim(-2,2);
ax.set_ylabel('Im');
ax.set_yticks(ticks=np.arange(-2,3,1),labels=['-2i','-i','0','i','2i']);
ax.set_ylim(-2,2);
ax.set_title('$e^{i\\theta}$');
../../../_images/3b6eb33fdc252dfaa6aaf596620407cdd61e3880981c0f49a42c7eaca5434cf7.png

\( \begin{aligned} w^0,w^1,...,w^{n-1} \,\,\,\text{where}\,\,\, w=e^{\frac{2\pi}{n}i} \,\,\,\text{and}\,\,\, n=N \end{aligned} \)

N=8
unitCircle = {math.e ** (2*math.pi * 1j * n/N) for n in range(N)}

for i,c in enumerate(unitCircle):
  display(Math(
    fr'w^{i} = {c.real} + {c.imag}i'
  ))

pC([
  unitCircle,
],-2,2,dpi=200)
\[\displaystyle w^0 = 0.7071067811865476 + 0.7071067811865475i\]
\[\displaystyle w^1 = 1.0 + 0.0i\]
\[\displaystyle w^2 = -0.7071067811865475 + 0.7071067811865476i\]
\[\displaystyle w^3 = -0.7071067811865477 + -0.7071067811865475i\]
\[\displaystyle w^4 = 0.7071067811865474 + -0.7071067811865477i\]
\[\displaystyle w^5 = -1.0 + 1.2246467991473532e-16i\]
\[\displaystyle w^6 = 6.123233995736766e-17 + 1.0i\]
\[\displaystyle w^7 = -1.8369701987210297e-16 + -1.0i\]
../../../_images/35e887a725a2aa74cf13247d7bff6e141c31cc7b68c747add79093b4bce88127.png

Polar Representation of Complex Numbers#

\( \boxed{ \begin{aligned} z=re^{\theta i} \,\,\,\text{where}\,\,\, r=|z| \end{aligned} } \)

zs={
  0+1j,
  1+2j,
  2+3j,
  3+4j,
  4+5j,
  5+6j,
}

pC([
  zs,
  {uC(z) for z in zs},
],-5,5,v='y')
../../../_images/29a74aab22047c555777459459377ae6da998c1a7313b9b5712591a981b72613.png

Transformations via Complex Operations#

Rotation#

To rotate a complex number \(z\) where \(\text{arg}\,\,\,z=\theta\) by \(\tau\) radians

\( \begin{aligned} re^{(\theta+\tau)i}=re^{\theta i}re^{\tau i}=ze^{\tau i} \end{aligned} \)

S={
  2   +1.25j,
  2   +1.50j,
  2   +1.75j,
  2   +2j,
  2.25+2j,
  2.50+2j,
  2.75+2j,
  3   +2j,
  2   +1j,
  2.25+1j,
  2.5 +1j,
  2.75+1j,
  3   +1j,
  3   +1.25j,
  3   +1.50j,
  3   +1.75j,
}

pC([
  S,
  {rotate(c,np.pi) for c in S},
  {rotate(c,np.pi/2) for c in S},
  {rotate(c,1) for c in S},
],-6,6,v='n')
../../../_images/75e436eaa9fbcba451c353cd9c91a804515f69c0fc1377d6505171233409f9ea.png 
imgRGB = mpimg.imread(fname='gandalf.png')
fig = plt.figure();
ax  = plt.subplot();
ax.imshow(X=imgRGB);
ax.grid(False);
#ax.axis('off');

---------------------------------------------------------------------------
FileNotFoundError                         Traceback (most recent call last)
Cell In[27], line 1
----> 1 imgRGB = mpimg.imread(fname='gandalf.png')
      2 fig = plt.figure();
      3 ax  = plt.subplot();

File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:1525, in imread(fname, format)
   1518 if isinstance(fname, str) and len(parse.urlparse(fname).scheme) > 1:
   1519     # Pillow doesn't handle URLs directly.
   1520     raise ValueError(
   1521         "Please open the URL for reading and pass the "
   1522         "result to Pillow, e.g. with "
   1523         "``np.array(PIL.Image.open(urllib.request.urlopen(url)))``."
   1524         )
-> 1525 with img_open(fname) as image:
   1526     return (_pil_png_to_float_array(image)
   1527             if isinstance(image, PIL.PngImagePlugin.PngImageFile) else
   1528             pil_to_array(image))

File ~/anaconda3/envs/ml/lib/python3.11/site-packages/PIL/ImageFile.py:125, in ImageFile.__init__(self, fp, filename)
    121 self.decodermaxblock = MAXBLOCK
    123 if is_path(fp):
    124     # filename
--> 125     self.fp = open(fp, "rb")
    126     self.filename = fp
    127     self._exclusive_fp = True

FileNotFoundError: [Errno 2] No such file or directory: 'gandalf.png'

#img=np.flip(np.dot(imgRGB,[0.2989,0.5870,0.1140]),0)
img=np.dot(imgRGB,[0.2989,0.5870,0.1140])
img.shape

(1200, 952)

fig = plt.figure();
ax  = plt.subplot();
ax.imshow(X=img,cmap=mpl.pyplot.get_cmap(name='gray'),vmin=0,vmax=1);
ax.grid(False);
#ax.axis('off');
ax.set_xlim(-952,952);
ax.set_ylim(1200,-1200);
../../../_images/db9d7611cbabdfac05fe6f7325489c3874cfb8d20815bad59e0d081f14c9bf2e.png

Resources#

[Y] Sanderson, Grant (3Blue1Brown). (02 Apr 2023). “Why π is in the normal distribution (beyond integral tricks)”. YouTube.

  • Zaczyński, Bartosz. “Simplify Complex Numbers with Python”. Real Python. Page.

Terms#

  • [W] Complex Analysis

  • [W] Complex Logarithm

  • [W] Complex Number

  • [W] Complex Plane

  • [W] Euler’s Formula

  • [W] Imaginary Unit

  • [W] Lacunary Value

  • [W] Phasor

  • [W] Roots of Unity

def solve1 (a,b,c):
  "solves ax + b = c"
  return (c-b)/a

display(
  solve1(10,5,30),
  solve1(10+5j,5,20),
)

2.5

(1.2-0.6j)
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