Manual review of builtin functions: complex-memoryview

Author: heikkitoivonen

docs/builtins/complex_func.md

@@ -9,7 +9,7 @@ The `complex()` function creates complex numbers from numeric values or strings.
 | From two numbers | O(1) | O(1) | real + imaginary |
 | From complex | O(1) | O(1) | Copy or identity |
 | From string | O(n) | O(1) | n = string length |
-| From int/float | O(1) | O(1) | Direct conversion |
+| From int/float | O(1) | O(1) | Direct conversion for floats and small ints; O(n) for arbitrary precision ints |
 
 ## Basic Usage
 
@@ -36,7 +36,7 @@ c = complex(0)      # 0j
 ```python
 # O(1)
 original = complex(3, 4)
-c = complex(original)  # (3+4j) - creates copy
+c = complex(original)  # (3+4j) - returns the same object
 ```
 
 ### From String
@@ -104,58 +104,6 @@ c = 3 + 4j  # Direct syntax (no function call)
 c = complex("3+4j")  # O(5)
 ```
 
-### Engineering Applications
-
-```python
-# O(1) - electrical impedance
-resistance = 100  # Ohms
-reactance = 50    # Ohms
-
-impedance = complex(resistance, reactance)  # (100+50j)
-
-# Magnitude - O(1)
-z = abs(impedance)  # sqrt(100^2 + 50^2) ≈ 111.8
-
-# Phase angle - O(1)
-import math
-phase = math.atan2(impedance.imag, impedance.real)
-```
-
-### Complex Arithmetic
-
-```python
-# O(1) - all operations
-c1 = complex(1, 2)
-c2 = complex(3, 4)
-
-# Basic operations
-sum_val = c1 + c2        # (4+6j)
-diff = c1 - c2           # (-2-2j)
-prod = c1 * c2           # (-5+10j)
-quot = c1 / c2           # (0.44+0.08j)
-
-# Power
-power = c1 ** 2          # (-3+4j)
-```
-
-### Polar to Rectangular Conversion
-
-```python
-# O(1) - convert between forms
-import cmath
-
-# Magnitude and phase
-magnitude = 5
-phase = 0.927  # radians
-
-# Rectangular form
-c = magnitude * cmath.exp(1j * phase)  # (3+4j)
-
-# Back to polar
-mag = abs(c)  # 5
-phi = cmath.phase(c)  # 0.927
-```
-
 ## Performance Patterns
 
 ### vs Tuple Representation
@@ -196,70 +144,6 @@ c = complex("3+4j")  # O(5)
 # For user input, use complex()
 ```
 
-## Practical Examples
-
-### Signal Processing
-
-```python
-# O(n) - process complex signals
-import cmath
-
-def frequency_response(frequencies):
-    responses = []
-    for freq in frequencies:
-        # Complex impedance at frequency
-        impedance = complex(100, 2 * 3.14159 * freq * 0.001)
-        response = abs(impedance)  # O(1)
-        responses.append(response)
-    return responses
-
-freqs = [100, 1000, 10000]
-responses = frequency_response(freqs)  # O(n)
-```
-
-### Quantum Mechanics
-
-```python
-# O(1) - quantum amplitude
-amplitude = complex(0.707, -0.707)  # 1/sqrt(2) * (1 - i)
-
-# Magnitude (probability)
-probability = abs(amplitude) ** 2  # 0.5
-
-# Phase
-import cmath
-phase = cmath.phase(amplitude)  # -π/4
-```
-
-### Electrical Impedance
-
-```python
-# O(1) - impedance calculation
-impedance = complex(50, 75)  # 50 + 75j Ohms
-
-# Admittance (reciprocal)
-admittance = 1 / impedance  # O(1)
-
-# Power calculation
-voltage = complex(230, 0)
-current = voltage / impedance  # O(1)
-power = voltage * current.conjugate()  # O(1)
-```
-
-### Fourier Analysis
-
-```python
-# O(n) - create complex exponentials
-import cmath
-
-def fourier_basis(n, k):
-    # e^(-2πijk/n)
-    factor = -2j * 3.14159 * k / n
-    return cmath.exp(factor)  # O(1)
-
-basis = fourier_basis(8, 2)  # Complex basis function
-```
-
 ## Edge Cases
 
 ### Zero Complex
@@ -295,7 +179,7 @@ c = complex(3, 4)     # (3+4j)
 conj = c.conjugate()  # (3-4j)
 
 # Useful in calculations
-magnitude_sq = c * c.conjugate()  # 9 + 16 = 25
+magnitude_sq = (c * c.conjugate()).real  # 9 + 16 = 25.0
 ```
 
 ### From String Errors