SciTech-Math-Complex Analysis复分析: Complex复数 + De Moivre's Formula:帝魔服公式 + Euler's Formula:欧拉公式

https://www.desmos.com/calculator/v1nugr08y5

https://mathvault.ca/euler-formula/

https://www.britannica.com/science/Eulers-formula

复数域的:

• 一切代数恒等式 仍像 实数域的 成立;
• 不等式 是不能定义像实数域比较大小的不等式.
  
  事实上, 如果复数域可以比较大小,那么必然推导出\(-1 > 0\):
  
  \(\begin{array}{ccl}
  hypothesis\ i > 0 &\ \Rightarrow\ & (i)^2 > 0 &\Rightarrow \ (+i)^2=-1 > 0 &\ \Rightarrow \ contradiction \\
  hypothesis\ i < 0 &\ \Rightarrow\ & -i > 0 &\Rightarrow \ (-i)^2=-1 > 0 &\ \Rightarrow \ contradiction \\
  \end{array}\)
  
  so\(\ ONLY\) real number field \(R\) has it's \(inequalities\).

De Moivre's Formula

\(Abraham\ de\ Moivre,\ French\ mathematician\)

• \(\large (r_{1}\ cis\ \theta_{1})^n =r^{n}\ cis\ n*\theta\)
  
  OR \(\large [\ r(\cos{\theta}+i*{\sin{\theta}})\ ]^{n} = r^{n}[\ \cos{(n*\theta)} + i*\sin{(n*\theta)}\ ]\)
• It lets us multiply a complex number by itself (as many times as we want) in one go!
  
  Let's learn about it, and also discover a much neater way to write it.
• Thanks to Abraham de Moivre so we have this useful formula.

Euler's Formula

https://mathvault.ca/euler-formula/, \(Leonhard\ Euler,\ Swiss\ mathematician\)

We can also create de Moivre's Formula with some help from Leonhard Euler!

• Euler's Formula for complex numbers says:
  
  \(\large f(\theta) = e^{i\theta} = 1\ cis\ \theta = \cos{\theta} + i \cdot \sin{\theta}\)
  
  \(\large f'(\theta)=\frac{d(e^{i\theta})}{d(\theta)} = i * e^{i\theta}\)
  
  \(\large f'(\theta)= i * f(\theta)\), compares \(\large f'(\theta)\) to \(\large f(\theta)\) at arbitrary polar form point \(\large (1,\theta)\):

  • only the \(\large angle\) in radians rotated \(\large \frac{\pi}{2}\) counterclockwise from \(\large \theta\), and become \(\large (\theta + \frac{\pi}{2})\)
  • and the \(\large magnitude\) remains.

  \(\large \begin{array}{ccl} & \because & f(\theta) &=& e^{i\theta} &=& \cos{\theta} + i \cdot \sin{\theta} \\
  & & f'(\theta) &=& i * f(\theta) &=& i*e^{i\theta}\end{array}\)

  \(\large \begin{array}{ccl} & \therefore & & & & & & & & & \\
  & & f(\theta) &=& e^{i*\theta} & & & & & && \\
  & & f(1) &=& e^{i*1} &=& e^i \ ,&\ f'(1) &=& i * e^{i} &=& i * e^{i} \\
  & & f(0) &=& e^{i*0} &=& 1 \ ,&\ f'(0) &=& i * 1 &=& i \\
  & & f(\frac{\pi}{2}) &=& e^{i*\frac{\pi}{2}} &=& i \ ,&\ f'(\frac{\pi}{2}) &=& i * i &=& -1 \\
  & & f(\pi) &=& e^{i*\pi} &=& -1 \ ,&\ \ f'({\pi}) &=& i * -1 &=& -i \\
  & & f(\frac{3\pi}{2}) &=& e^{i*\frac{3\pi}{2}} &=& -i \ ,&\ f'(\frac{3\pi}{2}) &=& i * -i &=& 1 \end{array}\)

• Euler’s Identity
  
  Euler’s identity is the most beautiful equation in mathematics. It is written as:
  
  \(\large e^{i\pi} + 1 = 0\)

  where it showcases five of the most important constants in mathematics. These are:

  • The \(0\): additive identity
  • The \(1\): unity
  • The \(\large \pi\): Pi constant (ratio of a circle’s circumference to its diameter)
  • The \(\large e\): base of natural logarithm
  • The \(\large i\): imaginary unit

  Among these:

  • three types of numbers are represented: \(\{integers\}\), \(\{irrational\ numbers\}\) and \(\{imaginary\ numbers\}\).
  • Three of the basic mathematical operations are also represented: \(\large addition\), \(\large multiplication\) and \(\large exponentiation\).

  We obtain Euler’s identity:

  • by starting with Euler’s formula \(\large e^{i\theta} = \cos{\theta} + i\cdot\sin{\theta}\)
  • and by setting \(\large \theta = \pi\) and sending the subsequent \(-1\) to the left-hand side.
  • The intermediate form \(\large e^{i \pi} = -1\) is common in the context of trigonometric unit circle in the complex plane:
    
    it corresponds to the point on the unit circle whose angle with respect to the positive real axis is \(\pi\).

Complex Numbers:

• The "\(unit\ imaginary\ number\) when squared equals −1:
  
  $ i^2 = -1\ \ \Rightarrow\ \ i = \pm \sqrt{-1}$

• Special case:

  • $\large i = \cos{\frac{\pi}{2}} + i*\sin{\frac{\pi}{2}} $
  • $ x = r\ cis\ \theta = r(\cos{\theta}+i*{\sin{\theta}}),\ \forall\ z \in C,\ r \in R,\ \theta \in [0,2\pi)$
  • $ x * i = r\ cis \ (\theta +\frac{\pi}{2}) = i * r(\cos{\theta}+i*{\sin{\theta}}) = r * [ \cos{(\theta + \frac{\pi}{2})} + i \sin{(\theta + \frac{\pi}{2})} ] $
    
    ONLY the \(\large angle\ in\ radians\) become \((\theta + \frac{\pi}{2})\), and the \(\large magnitude\) remains.
  • \(\large \forall\ x \in R, \ f(x) = e^{kx}, we\ have\ f'(x)=\frac{d(e^{kx})}{d(x)} = k * e^{kx}, since\ (e^x)' = e^{x}\)
  • \(\large \forall\ \theta \in C, \ f(\theta) = e^{i\theta}, we\ have\ f'(\theta)=\frac{d(e^{i\theta})}{d(\theta)} = i*e^{i\theta}\)

• What is a Complex number?
  
  a Complex Number is a combination of a Real Number and Imaginary Number;

  • $ x = a + b*i,\ Cartesian\ Form$, in Cartesian coordinates.

  • $ x = r(\cos{\theta}+i*{\sin{\theta}}),\ Polar\ Form$, in Polar coordinates.
    
    In fact, a common way to write a complex number in \(Polar form\) is
    
    \(a + b*i = r(\cos\theta + i*\sin\theta)\)
    
    And "\(\cos{\theta} + i*\sin{\theta}\)" is often shortened to "\(cis\ θ\)" OR "\(\angle \theta\)", so:
    
    \(a + b*i = r\ cis\ \theta = r \angle \theta\)
    
    where: \(cis\ \theta\) OR \(\angle \theta\) is just shorthand for \(\cos\theta + i*\sin\theta\)

  • conversions:

    • From \(Cartesian\) to \(Polar\)( use \(3 + 4i\) as a example):
      
      $ r = \sqrt{a^{2} + b^{2}} = \sqrt{3^{2} + 4^{2}} = \sqrt{25} = 5$
      
      $ \theta = \arctan{(b/a)} = \arctan{(4/3)} = 0.9273 (to\ 4\ decimals)$

       import math as m
       a, b = 3, 4
       rho = m.sqrt(m.pow(a, 2) + m.pow(b, 2))
       theta = m.atan(b/a);
      

    • From \(Polar\) to \(Cartesian\):
      
      $ a = r * \cos\theta = 5.0 * \cos{(0.92729..)} = 3.0,\ (at\ perfect\ accuracy)$
      
      $ b = r * \sin\theta = 5.0 * \sin{(0.92729...)} = 4.0,\ (at\ perfect\ accuracy)$

       import math as m
       rho, theta = 5.0, 0.9272952180016122
       a = rho * m.cos(theta)
       b = rho * m.sin(theta)
      

  • In other words the complex number \(3 + 4i\) can also be shown as distance 5 and angle 0.927 radians.

    \(Cartesian\ Form\)	\(Polar\ Form\)

• 两个复数的乘积结果: 模相乘(模等于两者模相乘), 角相加(弧角等于两者弧角相加)
  
  \(极坐标\) 表示：
  
  \(\large \begin{array}{ccl}
  z_{1} &=& r_{1} \angle \theta_{1}\ , \\
  z_{2} &=& r_{2} \angle \theta_{2} \\
  z_{1}*z_{2} &=& r_{1}*r_{2} \angle (\theta_{1}+\theta_{2}) \\
  OR \\
  z_{1} &=& r_{1}(\cos{\theta_{1}}+i*{\sin{\theta_{1}}}) \\
  z_{2} &=& r_2 (\cos{\theta_{2}}+i*\sin{\theta_{2}}) \\
  z_{1}*z_{2} &=& r_{1} * r_2 [\cos{(\theta_{1}+\theta_{2})} + i*\sin{(\theta_{1}+\theta_{2})} ] \end{array}\)

Complex Numbers in Exponential Form

• Cartesian Form: At this point, we already know that a complex number \(z\) can be expressed in Cartesian coordinates as \(x + iy\), where \(x\) and \(y\) are respectively the \(real\ part\) and the \(imaginary\ part\) of \(z\).
• Polar Form: Indeed, the same complex number can also be expressed in Polar coordinates as \(r(\cos \theta + i \sin \theta)\), where \(r\) is the \(magnitude\) of its distance to the origin, and \(\theta\) is its \(angle\ in\ radians\) with respect to the positive real axis.
• Exponential Form: it does not end there: thanks to \(Euler’s formula\), every complex number can now be expressed as a \(complex\ exponential\) as follows:
  
  \(z = r(\cos \theta + i \sin \theta) = r e^{i \theta}\)
  
  where \(r\) and \(\theta\) are the same numbers as before.
  
  To go from \((x, y)\) to \((r, \theta)\), we use the formulas \(\begin{align*} r & = \sqrt{x^2 + y^2} \\[4px] \theta & = \operatorname{atan2}(y, x) \end{align*}\),
  
  where \(\operatorname{atan2}(y, x)\) is the two-argument arctangent function with \(\operatorname{atan2}(y, x) = \arctan (\frac{y}{x})\) whenever \(x>0\).
  
  Conversely, to go from \((r, \theta)\) to \((x, y)\), we use the formulas: \(\begin{align*} x & = r \cos \theta \\[4px] y & = r \sin \theta \end{align*}\)
• The \(exponential\ form\ of\ complex\ numbers\) also makes multiplying complex numbers much easier — much like the same way rectangular coordinates make addition easier.
  
  For example, given two complex numbers \(z_1 = r_1 e^{i \theta_1}\) and \(z_2 = r_2 e^{i \theta_2}\),
  
  we can now multiply them together as follows:
  
  \(\begin{align*} z_1 z_2 & = r_1 e^{i \theta_1} \cdot r_2 e^{i \theta_2} \\ & = r_1 r_2 e^{i(\theta_1 + \theta_2)} \end{align*}\)
  
  In the same spirit, we can also divide the same two numbers as follows:
  
  \(\begin{align*} \frac{z1}{z2} & = \frac{r_1 e^{i \theta_1}}{r_2 e^{i \theta_2}} \\ & = \frac{r_1}{r_2} e^{i (\theta_{1}-\theta_2)} \end{align*}\)

Note

• To be sure, these do presuppose properties of exponent such as \(e^{z_1+z_2}=e^{z_1} e^{z_2}\) and \(e^{-z_1} = \frac{1}{e^{z_1}}\), which for example can be established by expanding the power series of \(e^{z_1}\), \(e^{-z_1}\) and \(e^{z_2}\).
• Had we used the \(rectangular\ notation\) \(x + iy\) instead, the same division would have required multiplying by the complex conjugate in the \(numerator\) and \(denominator\).
  
  With the \(polar\ coordinates\), the situation would have been the same (save perhaps worse).
• If anything, the \(exponential form\) sure makes it easier to see that:
  • multiplying two complex numbers is really the same as:
    
    multiplying magnitudes and adding angles,
  • dividing two complex numbers is really the same as:
    
    dividing magnitudes and subtracting angles.

the most remarkable formula in mathematics

Indeed, whether it’s \(Euler’s\ identity\) or \(complex\ logarithm\),

\(Euler’s\ formula\) seems to leave no stone unturned whenever expressions such \(\sin\), \(i\) and \(e\) are involved.

It’s a powerful tool whose mastery can be tremendously rewarding,

and for that reason is a rightful candidate of “the most remarkable formula in mathematics”.

Description	Statement
Euler’s formula	\(e^{ix} = \cos x + i \sin x\)
Euler’s identity	\(e^{i \pi} + 1 = 0\)
Complex number (exponential form)	\(z = r e^{i \theta}\)
Complex exponential	\(e^{x+iy} = e^x (\cos y + i \sin y)\)
Sine (exponential form)	\(\sin x = \dfrac{e^{ix}-e^{-ix}}{2i}\)
Cosine (exponential form)	\(\cos x = \dfrac{e^{ix} + e^{-ix}}{2}\)
Tangent (exponential form)	\(\tan x = \dfrac{e^{ix}-e^{-ix}}{i(e^{ix} + e^{-ix})}\)
Hyperbolic sine (exponential form)	\(\sinh z = \dfrac{\sin iz}{i}\)
Hyperbolic cosine (exponential form)	\(\cosh z = \cos iz\)
Hyperbolic tangent (exponential form)	\(\tanh z = \dfrac{\tan iz}{i}\)
Complex logarithm	\(\ln z = \ln |z| + i\phi\)
General complex exponential	\(a^z = e^{z \ln a}\)
De Moivre’s theorem	\((\cos x + i \sin x)^n = \cos nx + i \sin nx\)
Additive identity of sine	\(\sin (x+y) = \sin x \cos y + \cos x \sin y\)
Additive identity of cosine	\(\cos (x+y) = \cos x \cos y-\sin x \sin y\)