Python, Theory

Python. Complex numbers. The cmath module. Functions for converting to polar coordinates

Complex numbers. The cmath module. Functions for converting to polar coordinates and vice versa. Power and logarithmic functions

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1. Functions to convert to polar coordinates and vice versa

Python has 3 functions that work with polar coordinates:

  • cmath.phase(x) – returns the phase from the x argument as a float;
  • cmath.polar() – returns the representation of x in polar coordinates;
  • cmath.rect() – returns a complex number from polar coordinates.

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1.1. Function cmath.phase(). Get the phase of a complex number

Function

cmath.phase(x)

returns the phase of a complex number x. The function is equivalent to calling

cmath.atan2(x.imag, x.real)

The result of the function is in the range from –π to π.

Example.

# Include the cmath module
import cmath

# Function cmath.phase(x)
# Create a complex number
x = complex(2, -3) # x = 2 - 3*j

# Call the function
res = cmath.phase(x)

# Display the result
print("res = ", res)

Program result

res = -0.982793723247329

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1.2. Function cmath.polar(). Get a polar representation of a complex number

Function

cmath.polar(x)

returns a polar representation of x. The function returns a pair (r, phi), where r is the modulus of x and phi is the phase of x.
Calling a function is equivalent to calling

(abs(x), phase(x))

Example.

# Include the cmath module
import cmath

# Function cmath.polar(x)

# Create a complex number
x = complex(1, -1) # x = 1 - 1*j

# Call the function
res = cmath.polar(x)

# Display the result
print("res = ", res)

Program result

res = (1.4142135623730951, -0.7853981633974483)

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1.3. Function cmath.rect(). Get complex number based on polar coordinates

Function

cmath.rect(r, phi)

allows you to get a complex number based on the modulus r and the phase phi. Calling a function is equivalent to calling

r * (math.cos(phi) + math.sin(phi)*1j)

Example.

# Include cmath module
import cmath

# Function cmath.rect(x)

# Create a complex number as polar coordinates
r = 4 # modulus of number
phi = 1.2 # phase in radians

# Invoke the function
x = cmath.rect(r, phi)

# Display the result
print("x = ", x)

Program result

x = (1.4494310179066945+3.728156343868905j)

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2. Power and logarithmic functions

The cmath module implements the following power and logarithmic functions that operate on complex numbers:

  • cmath.exp(x) – returns the exponent of e raised to the power of x, where x can be a complex number. The exponent e is the basis of the natural logarithm;
  • cmath.log(x) – returns the natural logarithm of the x argument with a given base;
  • cmath.log10(x) – returns the base 10 logarithm of the x argument;
  • cmath.sqrt(x) – returns the square root of the x argument.

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2.1. Function cmath.exp(). Exponent of a complex number

Function

cmath.exp(x)

returns the number e raised to the power of x. The e value is the basis of the natural logarithm. The x value is a complex number.

Example.

# Include the cmath module
import cmath

# Function cmath.exp(x)

# Create a complex number
re = float(input("re = "))
im = float(input("im = "))

x = complex(re, im)

# Call the function
res = cmath.exp(x)

# Display the result
print("res = ", res)

Test example

re = 1
im = 0
res = (2.718281828459045+0j)

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2.2. Function cmath.log(). Logarithm of a complex number

Function

cmath.log(x [, base])

returns the logarithm of a complex number x with the specified base. If base is not specified, then the natural logarithm of the complex number x is returned.

Example.

# Include the cmath module
import cmath

# Function cmath.log()

# Create a complex number
re = float(input("re = "))
im = float(input("im = "))

x = complex(re, im)

# Call function for natural logarithm
res1 = cmath.log(x)
print("res1 = ", res1)

# Call the function for the base 4 logarithm
res2 = cmath.log(x, 4)
print("res2 = ", res2)

Test example

re = 1
im = -3
res1 = (1.151292546497023-1.2490457723982544j)
res2 = (0.8304820237218407-0.9009960708411433j)

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2.3. Function cmath.log10(). Decimal logarithm

Function

cmath.log10(x)

returns the base 10 logarithm of a complex number x.

Example.

# Include the cmath module
import cmath

# Function cmath.log10()

# Create a complex number
re = float(input("re = "))
im = float(input("im = "))

x = complex(re, im)

# Call the function for the decimal logarithm
res = cmath.log10(x)
print("res = ", res)

Test example

re = 1
im = -5
res = (0.707486673985409-0.5964603745259144j)

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2.4. Function cmath.sqrt(). Square root of the number x

Function

cmath.sqrt(x)

returns the square root of a complex number x.

Example.

# Include the cmath module
import cmath

# Function cmath.sqrt()

# Create a complex number
re = float(input("re = "))
im = float(input("im = "))

x = complex(re, im)

# Call the function for the decimal logarithm
res = cmath.sqrt(x)
print("res = ", res)

Test example

re = 2
im = -3
res = (1.6741492280355401-0.8959774761298381j)

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Related topics

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