Complex math - Python Full Tutorial

Python - Complex math

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Advanced complex arithmetic The module `cmath` includes additional functions to use complex numbers. `import cmath` This module can calculate the phase of a complex number, in radians: `z = 2+3j # A complex number cmath.phase(z) # 0.982793723247329` It allows the conversion between the cartesian (rectangular) and polar representations of complex numbers: `cmath.polar(z) # (3.605551275463989, 0.982793723247329) cmath.rect(2, cmath.pi/2) # (0+2j)` The module contains the complex version of Exponential and logarithmic functions (as usual, `log` is the natural logarithm and `log10` the decimal logarithm): `cmath.exp(z) # (-7.315110094901103+1.0427436562359045j) cmath.log(z) # (1.2824746787307684+0.982793723247329j) cmath.log10(-100) # (2+1.3643763538418412j)` Square roots: `cmath.sqrt(z) # (1.6741492280355401+0.8959774761298381j)` Trigonometric functions and their inverses: `cmath.sin(z) # (9.15449914691143-4.168906959966565j) cmath.cos(z) # (-4.189625690968807-9.109227893755337j) cmath.tan(z) # (-0.003764025641504249+1.00323862735361j) cmath.asin(z) # (0.5706527843210994+1.9833870299165355j) cmath.acos(z) # (1.0001435424737972-1.9833870299165355j) cmath.atan(z) # (1.4099210495965755+0.22907268296853878j) cmath.sin(z)2 + cmath.cos(z)2 # (1+0j)` Hyperbolic functions and their inverses: `cmath.sinh(z) # (-3.59056458998578+0.5309210862485197j) cmath.cosh(z) # (-3.7245455049153224+0.5118225699873846j) cmath.tanh(z) # (0.965385879022133-0.009884375038322495j) cmath.asinh(z) # (0.5706527843210994+1.9833870299165355j) cmath.acosh(z) # (1.9833870299165355+1.0001435424737972j) cmath.atanh(z) # (0.14694666622552977+1.3389725222944935j) cmath.cosh(z)2 - cmath.sin(z)2 # (1+0j) cmath.cosh((0+1j)*z) - cmath.cos(z) # 0j`
Basic complex arithmetic Python has built-in support for complex arithmetic. The imaginary unit is denoted by `j` (opens new window): `z = 2+3j # A complex number w = 1-7j # Another complex number` Complex numbers can be summed, subtracted, multiplied, divided and exponentiated: `z + w # (3-4j) z - w # (1+10j) z * w # (23-11j) z / w # (-0.38+0.34j) z**3 # (-46+9j)` Python can also extract the real and imaginary parts of complex numbers, and calculate their absolute value and conjugate: `z.real # 2.0 z.imag # 3.0 abs(z) # 3.605551275463989 z.conjugate() # (2-3j)` Syntax cmath.rect(AbsoluteValue, Phase)

Advanced complex arithmetic

The module cmath includes additional functions to use complex numbers.

import cmath

This module can calculate the phase of a complex number, in radians:

z = 2+3j # A complex number
cmath.phase(z) # 0.982793723247329

It allows the conversion between the cartesian (rectangular) and polar representations of complex numbers:

cmath.polar(z) # (3.605551275463989, 0.982793723247329)
cmath.rect(2, cmath.pi/2) # (0+2j)

The module contains the complex version of

Exponential and logarithmic functions (as usual, `log` is the natural logarithm and `log10` the decimal logarithm):

  cmath.exp(z) # (-7.315110094901103+1.0427436562359045j)
  cmath.log(z) # (1.2824746787307684+0.982793723247329j)
  cmath.log10(-100) # (2+1.3643763538418412j)

Square roots:

  cmath.sqrt(z) # (1.6741492280355401+0.8959774761298381j)

Trigonometric functions and their inverses:

  cmath.sin(z)  # (9.15449914691143-4.168906959966565j)
  cmath.cos(z)  # (-4.189625690968807-9.109227893755337j)
  cmath.tan(z)  # (-0.003764025641504249+1.00323862735361j)
  cmath.asin(z) # (0.5706527843210994+1.9833870299165355j)
  cmath.acos(z) # (1.0001435424737972-1.9833870299165355j)
  cmath.atan(z) # (1.4099210495965755+0.22907268296853878j)
  cmath.sin(z)**2 + cmath.cos(z)**2 # (1+0j)

Hyperbolic functions and their inverses:

  cmath.sinh(z)  # (-3.59056458998578+0.5309210862485197j)
  cmath.cosh(z)  # (-3.7245455049153224+0.5118225699873846j)
  cmath.tanh(z)  # (0.965385879022133-0.009884375038322495j)
  cmath.asinh(z) # (0.5706527843210994+1.9833870299165355j)
  cmath.acosh(z) # (1.9833870299165355+1.0001435424737972j)
  cmath.atanh(z) # (0.14694666622552977+1.3389725222944935j)
  cmath.cosh(z)**2 - cmath.sin(z)**2  # (1+0j)
  cmath.cosh((0+1j)*z) - cmath.cos(z) # 0j

Basic complex arithmetic

Python has built-in support for complex arithmetic. The imaginary unit is denoted by j (opens new window):

z = 2+3j # A complex number
w = 1-7j # Another complex number

Complex numbers can be summed, subtracted, multiplied, divided and exponentiated:

z + w # (3-4j) 
z - w # (1+10j)
z * w # (23-11j) 
z / w # (-0.38+0.34j)
z**3  # (-46+9j)

Python can also extract the real and imaginary parts of complex numbers, and calculate their absolute value and conjugate:

z.real # 2.0
z.imag # 3.0
abs(z) # 3.605551275463989
z.conjugate() # (2-3j)

Syntax

cmath.rect(AbsoluteValue, Phase)

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