将给定的复数转换为极坐标形式并执行所有算术运算

给定两个笛卡尔形式的复数Z1和Z2 ，任务是将给定的复数转换为极坐标形式，并对它们执行所有算术运算（加、减、乘、除）。

例子：

Input: Z1 = (2, 3), Z2 = (4, 6)
Output:
Polar form of the first Complex Number: (3.605551275463989, 0.9827937232473292)
Polar form of the Second Complex Number: (7.211102550927978, 0.9827937232473292)
Addition of two Complex Numbers: (10.816653826391967, 0.9827937232473292)
Subtraction of two Complex Numbers: (3.605551275463989, -0.9827937232473292)
Multiplication of two Complex Numbers: (25.999999999999996, 1.9655874464946583)
Division of two Complex Numbers: (0.5, 0.0)

Input: Z1 = (1, 1), Z2 = (2, 2)
Output:
Polar form of the first Complex Number: (1.4142135623730951, 0.7853981633974482)
Polar form of the Second Complex Number: (2.8284271247461903, 0.7853981633974482)
Addition of two Complex Numbers: (4.242640687119286, 0.7853981633974482)
Subtraction of two Complex Numbers: (1.4142135623730951, -0.7853981633974482)
Multiplication of two Complex Numbers: (4.000000000000001, 1.5707963267948963)
Division of two Complex Numbers: (0.5, 0.0)

方法：可以根据复数的以下性质解决给定的问题：

• 笛卡尔形式的复数Z表示为：

$Z = a+ i\times b$,
where a, b € R and b is known as the imaginary part of the complex number and $i = \sqrt(-1)$

• 复数Z的极坐标形式是：

$Z = r(sin(\theta) + i*cos(\theta))$

$r = \sqrt{a^{2}+ b^{2}}$
$\theta = tan^{-1}(b/a)$
$\theta = Sin^{-1} (b/r)$
$\theta = Cos^{-1} (a/r)$

where, r is known as modules of a complex number and
$\theta$ is the angle made with the positive X axis.

• 在复数的极坐标形式的表达中，以r为常用表现$Sin(\theta)+ i*Cos(\theta) = e^{i(\theta)}$表达式变成：
  • $Z = r*e^{i\theta}$ ，这被称为复数的欧拉形式。
  • 欧拉和极坐标形式都表示为： $(r, \theta)$ .
• 可以使用欧拉形式完成两个复数的乘法和除法：

For Multiplication:

      $Z = (r_{1}*e^{\theta_{1}})*(r_{2}*e^{\theta_{2}})$
=> $Z = (r_{1}*r_{2})*e^{ \theta_{1} + \theta_{2}}$

For Division:

      $Z = (r_{1}*e^{\theta_{1}})\div (r_{2}*e^{\theta_{2}})$
=> $Z = (r_{1}\div r_{2})*e^{ \theta_{1} - \theta_{2}}$

请按照以下步骤解决问题：

• 使用上面讨论的公式将复数转换为极坐标并将其打印为$(r, \theta)$ .
• 定义一个函数say Addition(Z1, Z2)来执行加法运算：
  • 通过将两个实部Z1和Z2相加求出复数的实部，并将其存储在变量a 中。
  • 通过将复数Z1和Z2 的两个虚部相加，求出复数的虚部，并将其存储在变量b 中。
  • 将复数的笛卡尔形式转换为极坐标形式并打印出来。
• 定义一个函数say Subtraction(Z1, Z2)来执行减法运算：
  • 通过减去两个实部Z1和Z2 来找到复数的实部，并将其存储在变量a 中。
  • 通过将复数Z1和Z2 的两个虚部相减，求出复数的虚部，并将其存储在变量b 中。
  • 将复数的笛卡尔形式转换为极坐标形式并打印出来。
• 打印两个复数Z1和Z2的乘法为$Z = (r_{1}*r_{2})*e^{ \theta_{1} + \theta_{2}}$
• 将两个复数Z1和Z2的除法打印为$Z = (r_{1}\div r_{2})*e^{ \theta_{1} - \theta_{2}}$

下面是上述方法的实现：

Python3

# Python program for the above approach
import math
  
# Function to find the polar form
# of the given Complex Number
def get_polar_form(z):
    
    # Z is in cartesian form
    re, im = z
  
    # Stores the modulo of complex number
    r = (re * re + im * im) ** 0.5
  
    # If r is greater than 0
    if r:
        theta = math.asin(im / r)
        return (r, theta)
        
    # Otherwise
    else:
        return (0, 0)
  
# Function to add two complex numbers
def Addition(z1, z2):
    
    # Z is in polar form
    r1, theta1 = z1
    r2, theta2 = z2
  
    # Real part of complex number
    a = r1 * math.cos(theta1) + r2 * math.cos(theta2)
      
    # Imaginary part of complex Number
    b = r1 * math.sin(theta1) + r2 * math.sin(theta2)
      
    # Find the polar form
    return get_polar_form((a, b))
  
# Function to subtract two
# given complex numbers
def Subtraction(z1, z2):
    
    # Z is in polar form
    r1, theta1 = z1
    r2, theta2 = z2
  
    # Real part of the complex number
    a = r1 * math.cos(theta1) - r2 * math.cos(theta2)
      
    # Imaginary part of complex number
    b = r1 * math.sin(theta1) - r2 * math.sin(theta2)
  
    # Converts (a, b) to polar
    # form and return
    return get_polar_form((a, b))
  
# Function to multiply two complex numbers
def Multiplication(z1, z2):
    
    # z is in polar form
    r1, theta1 = z1
    r2, theta2 = z2
  
    # Return the multiplication of Z1 and Z2
    return (r1 * r2, theta1 + theta2)
  
  
# Function to divide two complex numbers
def Division(z1, z2):
    
    # Z is in the polar form
    r1, theta1 = z1
    r2, theta2 = z2
  
    # Return the division of Z1 and Z2
    return (r1 / r2, theta1-theta2)
  
  
# Driver Code
if __name__ == "__main__":
    
    z1 = (2, 3)
    z2 = (4, 6)
  
    # Convert into Polar Form
    z1_polar = get_polar_form(z1)
    z2_polar = get_polar_form(z2)
  
    print("Polar form of the first")
    print("Complex Number: ", z1_polar)
    print("Polar form of the Second")
    print("Complex Number: ", z2_polar)
  
    print("Addition of two complex")
    print("Numbers: ", Addition(z1_polar, z2_polar))
      
    print("Subtraction of two ")
    print("complex Numbers: ",
           Subtraction(z1_polar, z2_polar))
      
    print("Multiplication of two ")
    print("Complex Numbers: ",
           Multiplication(z1_polar, z2_polar))
            
    print("Division of two complex ")
    print("Numbers: ", Division(z1_polar, z2_polar))

输出：

Polar form of the first
Complex Number:  (3.605551275463989, 0.9827937232473292)
Polar form of the Second
Complex Number:  (7.211102550927978, 0.9827937232473292)
Addition of two complex
Numbers:  (10.816653826391967, 0.9827937232473292)
Subtraction of two 
complex Numbers:  (3.605551275463989, -0.9827937232473292)
Multiplication of two 
Complex Numbers:  (25.999999999999996, 1.9655874464946583)
Division of two complex 
Numbers:  (0.5, 0.0)

时间复杂度： O(1)
辅助空间： O(1)