Various documentation improvements.

Sources/ComplexModule/Complex+AdditiveArithmetic.swift

@@ -2,7 +2,7 @@
 //
 // This source file is part of the Swift Numerics open source project
 //
-// Copyright (c) 2019-2021 Apple Inc. and the Swift Numerics project authors
+// Copyright (c) 2019-2025 Apple Inc. and the Swift Numerics project authors
 // Licensed under Apache License v2.0 with Runtime Library Exception
 //
 // See https://swift.org/LICENSE.txt for license information
@@ -14,7 +14,7 @@ import RealModule
 extension Complex: AdditiveArithmetic {
   /// The additive identity, with real and imaginary parts both zero.
   ///
-  /// See also: `one`, `i`, `infinity`
+  /// See also: ``one``, ``i``, ``infinity``
   @_transparent
   public static var zero: Complex {
     Complex(0, 0)

Sources/ComplexModule/Complex+AlgebraicField.swift

@@ -2,7 +2,7 @@
 //
 // This source file is part of the Swift Numerics open source project
 //
-// Copyright (c) 2019-2024 Apple Inc. and the Swift Numerics project authors
+// Copyright (c) 2019-2025 Apple Inc. and the Swift Numerics project authors
 // Licensed under Apache License v2.0 with Runtime Library Exception
 //
 // See https://swift.org/LICENSE.txt for license information
@@ -13,6 +13,8 @@ import RealModule
 
 extension Complex: AlgebraicField {
   /// The multiplicative identity `1 + 0i`.
+  ///
+  /// See also: ``zero``, ``i``, ``infinity``
   @_transparent
   public static var one: Complex {
     Complex(1, 0)

Sources/ComplexModule/Complex+Codable.swift

@@ -2,7 +2,7 @@
 //
 // This source file is part of the Swift Numerics open source project
 //
-// Copyright (c) 2019 - 2021 Apple Inc. and the Swift Numerics project authors
+// Copyright (c) 2019-2025 Apple Inc. and the Swift Numerics project authors
 // Licensed under Apache License v2.0 with Runtime Library Exception
 //
 // See https://swift.org/LICENSE.txt for license information

Sources/ComplexModule/Complex+ElementaryFunctions.swift

@@ -2,7 +2,7 @@
 //
 // This source file is part of the Swift Numerics open source project
 //
-// Copyright (c) 2019-2020 Apple Inc. and the Swift Numerics project authors
+// Copyright (c) 2019-2025 Apple Inc. and the Swift Numerics project authors
 // Licensed under Apache License v2.0 with Runtime Library Exception
 //
 // See https://swift.org/LICENSE.txt for license information

Sources/ComplexModule/Complex+Hashable.swift

@@ -2,7 +2,7 @@
 //
 // This source file is part of the Swift Numerics open source project
 //
-// Copyright (c) 2019 - 2021 Apple Inc. and the Swift Numerics project authors
+// Copyright (c) 2019-2025 Apple Inc. and the Swift Numerics project authors
 // Licensed under Apache License v2.0 with Runtime Library Exception
 //
 // See https://swift.org/LICENSE.txt for license information

Sources/ComplexModule/Complex+IntegerLiteral.swift

@@ -2,7 +2,7 @@
 //
 // This source file is part of the Swift Numerics open source project
 //
-// Copyright (c) 2019 - 2021 Apple Inc. and the Swift Numerics project authors
+// Copyright (c) 2019-2025 Apple Inc. and the Swift Numerics project authors
 // Licensed under Apache License v2.0 with Runtime Library Exception
 //
 // See https://swift.org/LICENSE.txt for license information

Sources/ComplexModule/Complex+Numeric.swift

@@ -2,7 +2,7 @@
 //
 // This source file is part of the Swift Numerics open source project
 //
-// Copyright (c) 2019 - 2021 Apple Inc. and the Swift Numerics project authors
+// Copyright (c) 2019-2025 Apple Inc. and the Swift Numerics project authors
 // Licensed under Apache License v2.0 with Runtime Library Exception
 //
 // See https://swift.org/LICENSE.txt for license information
@@ -37,17 +37,20 @@ extension Complex: Numeric {
     self.init(real, 0)
   }
 
-  /// The ∞-norm of the value (`max(abs(real), abs(imaginary))`).
+  /// The infinity-norm of the value (a.k.a. "maximum norm" or "Чебышёв
+  /// [Chebyshev] norm").
   ///
-  /// If you need the Euclidean norm (a.k.a. 2-norm) use the `length` or
-  /// `lengthSquared` properties instead.
+  /// Equal to `max(abs(real), abs(imaginary))`.
   ///
-  /// Edge cases:
-  /// - If `z` is not finite, `z.magnitude` is `.infinity`.
-  /// - If `z` is zero, `z.magnitude` is `0`.
-  /// - Otherwise, `z.magnitude` is finite and non-zero.
+  /// If you need to work with the Euclidean norm (a.k.a. 2-norm) instead,
+  /// use the ``length`` or ``lengthSquared`` properties. If you just need
+  /// to know "how big" a number is, use this property.
+  ///
+  /// **Edge cases:**
   ///
-  /// See also `.length` and `.lengthSquared`.
+  /// - If `z` is not finite, `z.magnitude` is infinity.
+  /// - If `z` is zero, `z.magnitude` is zero.
+  /// - Otherwise, `z.magnitude` is finite and non-zero.
   @_transparent
   public var magnitude: RealType {
     guard isFinite else { return .infinity }

Sources/ComplexModule/Complex+StringConvertible.swift

@@ -2,7 +2,7 @@
 //
 // This source file is part of the Swift Numerics open source project
 //
-// Copyright (c) 2019 - 2021 Apple Inc. and the Swift Numerics project authors
+// Copyright (c) 2019-2025 Apple Inc. and the Swift Numerics project authors
 // Licensed under Apache License v2.0 with Runtime Library Exception
 //
 // See https://swift.org/LICENSE.txt for license information

Sources/ComplexModule/Complex.swift

@@ -2,7 +2,7 @@
 //
 // This source file is part of the Swift Numerics open source project
 //
-// Copyright (c) 2019 - 2021 Apple Inc. and the Swift Numerics project authors
+// Copyright (c) 2019-2025 Apple Inc. and the Swift Numerics project authors
 // Licensed under Apache License v2.0 with Runtime Library Exception
 //
 // See https://swift.org/LICENSE.txt for license information
@@ -11,39 +11,8 @@
 
 import RealModule
 
-/// A complex number represented by real and imaginary parts.
-///
-/// TODO: introductory text on complex numbers
-///
-/// Implementation notes:
-///
-/// This type does not provide heterogeneous real/complex arithmetic,
-/// not even the natural vector-space operations like real * complex.
-/// There are two reasons for this choice: first, Swift broadly avoids
-/// mixed-type arithmetic when the operation can be adequately expressed
-/// by a conversion and homogeneous arithmetic. Second, with the current
-/// typechecker rules, it would lead to undesirable ambiguity in common
-/// expressions (see README.md for more details).
-///
-/// Unlike C's `_Complex` and C++'s `std::complex<>` types, we do not
-/// attempt to make meaningful semantic distinctions between different
-/// representations of infinity or NaN. Any Complex value with at least
-/// one non-finite component is simply "non-finite". In as much as
-/// possible, we use the semantics of the point at infinity on the
-/// Riemann sphere for such values. This approach simplifies the number of
-/// edge cases that need to be considered for multiplication, division, and
-/// the elementary functions considerably.
-///
-/// `.magnitude` does not return the Euclidean norm; it uses the "infinity
-/// norm" (`max(|real|,|imaginary|)`) instead. There are two reasons for this
-/// choice: first, it's simply faster to compute on most hardware. Second,
-/// there exist values for which the Euclidean norm cannot be represented
-/// (consider a number with `.real` and `.imaginary` both equal to
-/// `RealType.greatestFiniteMagnitude`; the Euclidean norm would be
-/// `.sqrt(2) * .greatestFiniteMagnitude`, which overflows). Using
-/// the infinity norm avoids this problem entirely without significant
-/// downsides. You can access the Euclidean norm using the `length`
-/// property.
+// A [complex number](https://en.wikipedia.org/wiki/Complex_number).
+// See Documentation.docc/Complex.md for more details.
 @frozen
 public struct Complex<RealType> where RealType: Real {
   //  A note on the `x` and `y` properties
@@ -53,11 +22,11 @@ public struct Complex<RealType> where RealType: Real {
   //  `.real` and `.imaginary` properties, which wrap this storage and
   //  fixup the semantics for non-finite values.
 
-  /// The real component of the value.
+  /// The storage for the real component of the value.
   @usableFromInline @inline(__always)
   internal var x: RealType
 
-  /// The imaginary part of the value.
+  /// The storage for the imaginary part of the value.
   @usableFromInline @inline(__always)
   internal var y: RealType
 
@@ -95,27 +64,40 @@ extension Complex {
     set { y = newValue }
   }
 
+  /// The raw representation of the value.
+  ///
+  /// Use this when you need the underlying RealType values,
+  /// without fixup for NaN or infinity.
+  public var rawStorage: (x: RealType, y: RealType) {
+    @_transparent
+    get { (x, y) }
+    @_transparent
+    set { (x, y) = newValue }
+  }
+
   /// The raw representation of the real part of this value.
+  @available(*, deprecated, message: "Use rawStorage")
   @_transparent
   public var _rawX: RealType { x }
 
   /// The raw representation of the imaginary part of this value.
+  @available(*, deprecated, message: "Use rawStorage")
   @_transparent
   public var _rawY: RealType { y }
 }
 
 extension Complex {
   /// The imaginary unit.
   ///
-  /// See also `.zero`, `.one` and `.infinity`.
+  /// See also ``zero``, ``one`` and ``infinity``.
   @_transparent
   public static var i: Complex {
     Complex(0, 1)
   }
 
   /// The point at infinity.
   ///
-  /// See also `.zero`, `.one` and `.i`.
+  /// See also ``zero``, ``one`` and ``i``.
   @_transparent
   public static var infinity: Complex {
     Complex(.infinity, 0)
@@ -125,7 +107,7 @@ extension Complex {
   ///
   /// A complex value is finite if neither component is an infinity or nan.
   ///
-  /// See also `.isNormal`, `.isSubnormal` and `.isZero`.
+  /// See also ``isNormal``, ``isSubnormal`` and ``isZero``.
   @_transparent
   public var isFinite: Bool {
     x.isFinite && y.isFinite
@@ -138,7 +120,7 @@ extension Complex {
   /// one of the components is normal if its exponent allows a full-precision
   /// representation.
   ///
-  /// See also `.isFinite`, `.isSubnormal` and `.isZero`.
+  /// See also ``isFinite``, ``isSubnormal`` and ``isZero``.
   @_transparent
   public var isNormal: Bool {
     isFinite && (x.isNormal || y.isNormal)
@@ -150,7 +132,7 @@ extension Complex {
   /// When the result of a computation is subnormal, underflow has occurred and
   /// the result generally does not have full precision.
   ///
-  /// See also `.isFinite`, `.isNormal` and `.isZero`.
+  /// See also ``isFinite``, ``isNormal`` and ``isZero``.
   @_transparent
   public var isSubnormal: Bool {
     isFinite && !isNormal && !isZero
@@ -161,7 +143,7 @@ extension Complex {
   /// A complex number is zero if *both* the real and imaginary components
   /// are zero.
   ///
-  /// See also `.isFinite`, `.isNormal` and `isSubnormal`.
+  /// See also ``isFinite``, ``isNormal`` and ``isSubnormal``.
   @_transparent
   public var isZero: Bool {
     x == 0 && y == 0
@@ -200,7 +182,7 @@ extension Complex {
     self.init(real, 0)
   }
 
-  /// The complex number with specified imaginary part and zero real part.
+  /// The complex number with zero real part and specified imaginary part.
   ///
   /// Equivalent to `Complex(0, imaginary)`.
   @inlinable

Sources/ComplexModule/Documentation.docc/Complex.md

@@ -0,0 +1,67 @@
+# ``Complex``
+
+A complex number type represented by its real and imaginary parts, and equipped
+with the usual arithmetic operators and math functions.
+
+## Overview
+
+You can access these Cartesian components using the real and imaginary
+properties.
+
+```swift
+let z = Complex(1,-1) //  1 - i
+let re = z.real       //  1
+let im = z.imaginary  // -1
+```
+
+All `Complex` numbers with a non-finite component are treated as a single
+"point at infinity," with infinite magnitude and indeterminant phase. Thus,
+the real and imaginary parts of an infinity are nan.
+
+```swift
+let w = Complex<Double>.infinity
+w == -w               // true
+let re = w.real       // .nan
+let im = w.imag       // .nan
+```
+
+See <doc:Infinity> for more details.
+
+The ``magnitude`` property of a complex number is the infinity norm of the
+value (a.k.a. “maximum norm” or “Чебышёв [Chebyshev] norm”). To get the two
+norm (a.k.a. "Euclidean norm"), use the ``length`` property. See
+<doc:Magnitude> for more details.
+
+## Topics 
+
+### Real and imaginary parts
+
+- ``real``
+- ``imaginary``
+- ``rawStorage``
+- ``init(_:_:)``
+- ``init(_:)-5aesj``
+- ``init(imaginary:)``
+
+### Phase, length and magnitude
+
+- ``magnitude``
+- ``length``
+- ``lengthSquared``
+- ``normalized``
+- ``phase``
+- ``polar``
+- ``init(length:phase:)``
+
+### Scaling by real numbers
+- ``multiplied(by:)``
+- ``divided(by:)``
+
+### Complex-specific operations
+- ``conjugate``
+
+### Classification
+- ``isZero``
+- ``isSubnormal``
+- ``isNormal``
+- ``isFinite``