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# std.complex

This module contains the Complex type, which is used to represent complex numbers, along with related mathematical operations and functions.
Complex will eventually replace the built-in types cfloat, cdouble, creal, ifloat, idouble, and ireal.
Authors:
Lars Tandle Kyllingstad, Don Clugston
License:
Boost License 1.0

Source std/complex.d

pure nothrow @nogc @safe auto `complex`(R)(const R `re`)
if (is(R : double));
pure nothrow @nogc @safe auto `complex`(R, I)(const R `re`, const I `im`)
if (is(R : double) && is(I : double));
  Helper function that returns a complex number with the specified real and imaginary parts.
  Parameters:

  R | (template parameter) type of real part of complex number
  I | (template parameter) type of imaginary part of complex number
  R `re` | real part of complex number to be constructed
  I `im` | (optional) imaginary part of complex number, 0 if omitted.

  Returns:
  Complex instance with real and imaginary parts set to the values provided as input. If neither `re` nor `im` are floating-point numbers, the return type will be Complex!double. Otherwise, the return type is deduced using std.traits.CommonType!(R, I).
  Examples:

      auto a = complex(1.0);
      static assert(is(typeof(a) == Complex!double));
      writeln(a.re); // 1.0
      writeln(a.im); // 0.0

      auto b = complex(2.0L);
      static assert(is(typeof(b) == Complex!real));
      writeln(b.re); // 2.0L
      writeln(b.im); // 0.0L

      auto c = complex(1.0, 2.0);
      static assert(is(typeof(c) == Complex!double));
      writeln(c.re); // 1.0
      writeln(c.im); // 2.0

      auto d = complex(3.0, 4.0L);
      static assert(is(typeof(d) == Complex!real));
      writeln(d.re); // 3.0
      writeln(d.im); // 4.0L

      auto e = complex(1);
      static assert(is(typeof(e) == Complex!double));
      writeln(e.re); // 1
      writeln(e.im); // 0

      auto f = complex(1L, 2);
      static assert(is(typeof(f) == Complex!double));
      writeln(f.re); // 1L
      writeln(f.im); // 2

      auto g = complex(3, 4.0L);
      static assert(is(typeof(g) == Complex!real));
      writeln(g.re); // 3
      writeln(g.im); // 4.0L

struct `Complex`(T) if (isFloatingPoint!T);
  A complex number parametrised by a type T, which must be either float, double or real.
  T `re`;
    The real part of the number.
  T `im`;
    The imaginary part of the number.
  const @safe string `toString`();
  const void `toString`(Writer, Char)(scope Writer `w`, ref scope const FormatSpec!Char `formatSpec`)
  if (isOutputRange!(Writer, const(Char)[]));
    Converts the complex number to a string representation.
    The second form of this function is usually not called directly; instead, it is used via std.string.format, as shown in the examples below. Supported format characters are 'e', 'f', 'g', 'a', and 's'.
    See the std.format and std.string.format documentation for more information.
    Examples:

        auto c = complex(1.2, 3.4);

        // Vanilla toString formatting:
        writeln(c.toString()); // "1.2+3.4i"

        // Formatting with std.string.format specs: the precision and width
        // specifiers apply to both the real and imaginary parts of the
        // complex number.
        import std.format : format;
        writeln(format("%.2f", c)); // "1.20+3.40i"
        writeln(format("%4.1f", c)); // " 1.2+ 3.4i"

  this(R : T)(Complex!R `z`);
  this(Rx : T, Ry : T)(const Rx `x`, const Ry `y`);
  this(R : T)(const R `r`);
    Construct a complex number with the specified real and imaginary parts. In the case where a single argument is passed that is not complex, the imaginary part of the result will be zero.
pure nothrow @nogc @safe T `abs`(T)(Complex!T `z`);
  Parameters:

  Complex!T `z` | A complex number.

  Returns:
  The absolute value (or modulus) of `z`.
  Examples:

      static import core.math;
      writeln(abs(complex(1.0))); // 1.0
      writeln(abs(complex(0.0, 1.0))); // 1.0
      writeln(abs(complex(1.0L, -2.0L))); // core.math.sqrt(5.0L)

pure nothrow @nogc @safe T `sqAbs`(T)(Complex!T `z`);
pure nothrow @nogc @safe T `sqAbs`(T)(const T `x`)
if (isFloatingPoint!T);
  Parameters:

  Complex!T `z` | A complex number.
  T `x` | A real number.

  Returns:
  The squared modulus of `z`. For genericity, if called on a real number, returns its square.
  Examples:

      import std.math.operations : isClose;
      writeln(sqAbs(complex(0.0))); // 0.0
      writeln(sqAbs(complex(1.0))); // 1.0
      writeln(sqAbs(complex(0.0, 1.0))); // 1.0
      assert(isClose(sqAbs(complex(1.0L, -2.0L)), 5.0L));
      assert(isClose(sqAbs(complex(-3.0L, 1.0L)), 10.0L));
      assert(isClose(sqAbs(complex(1.0f,-1.0f)), 2.0f));

pure nothrow @nogc @safe T `arg`(T)(Complex!T `z`);
  Parameters:

  Complex!T `z` | A complex number.

  Returns:
  The argument (or phase) of `z`.
  Examples:

      import std.math.constants : PI_2, PI_4;
      writeln(arg(complex(1.0))); // 0.0
      writeln(arg(complex(0.0L, 1.0L))); // PI_2
      writeln(arg(complex(1.0L, 1.0L))); // PI_4

pure nothrow @nogc @safe T `norm`(T)(Complex!T `z`);
  Extracts the norm of a complex number.
  Parameters:

  Complex!T `z` | A complex number

  Returns:
  The squared magnitude of `z`.
  Examples:

      import std.math.operations : isClose;
      import std.math.constants : PI;
      writeln(norm(complex(3.0, 4.0))); // 25.0
      writeln(norm(fromPolar(5.0, 0.0))); // 25.0
      assert(isClose(norm(fromPolar(5.0L, PI / 6)), 25.0L));
      assert(isClose(norm(fromPolar(5.0L, 13 * PI / 6)), 25.0L));

pure nothrow @nogc @safe Complex!T `conj`(T)(Complex!T `z`);
  Parameters:

  Complex!T `z` | A complex number.

  Returns:
  The complex conjugate of `z`.
  Examples:

      writeln(conj(complex(1.0))); // complex(1.0)
      writeln(conj(complex(1.0, 2.0))); // complex(1.0, -2.0)

Complex!T `proj`(T)(Complex!T `z`);
  Returns the projection of `z` onto the Riemann sphere.
  Parameters:

  Complex!T `z` | A complex number

  Returns:
  The projection of `z` onto the Riemann sphere.
  Examples:

      writeln(proj(complex(1.0))); // complex(1.0)
      writeln(proj(complex(double.infinity, 5.0))); // complex(double.infinity, 0.0)
      writeln(proj(complex(5.0, -double.infinity))); // complex(double.infinity, -0.0)

pure nothrow @nogc @safe Complex!(CommonType!(T, U)) `fromPolar`(T, U)(const T `modulus`, const U `argument`);
  Constructs a complex number given its absolute value and argument.
  Parameters:

  T `modulus` | The modulus
  U `argument` | The argument

  Returns:
  The complex number with the given modulus and argument.
  Examples:

      import core.math;
      import std.math.operations : isClose;
      import std.math.algebraic : sqrt;
      import std.math.constants : PI_4;
      auto z = fromPolar(core.math.sqrt(2.0), PI_4);
      assert(isClose(z.re, 1.0L));
      assert(isClose(z.im, 1.0L));

pure nothrow @nogc @safe Complex!T `sin`(T)(Complex!T `z`);
pure nothrow @nogc @safe Complex!T `cos`(T)(Complex!T `z`);
pure nothrow @nogc @safe Complex!T `tan`(T)(Complex!T `z`);
  Trigonometric functions on complex numbers.
  Parameters:

  Complex!T `z` | A complex number.

  Returns:
  The sine, cosine and tangent of `z`, respectively.
  Examples:

      static import core.math;
      writeln(sin(complex(0.0))); // 0.0
      writeln(sin(complex(2.0, 0))); // core.math.sin(2.0)

  Examples:

      static import core.math;
      static import std.math;
      writeln(cos(complex(0.0))); // 1.0
      writeln(cos(complex(1.3, 0.0))); // core.math.cos(1.3)
      writeln(cos(complex(0.0, 5.2))); // std.math.cosh(5.2)

  Examples:

      static import std.math;
      assert(ceqrel(tan(complex(1.0, 0.0)), complex(std.math.tan(1.0), 0.0)) >= double.mant_dig - 2);
      assert(ceqrel(tan(complex(0.0, 1.0)), complex(0.0, std.math.tanh(1.0))) >= double.mant_dig - 2);

pure nothrow @nogc @safe Complex!T `asin`(T)(Complex!T `z`);
pure nothrow @nogc @safe Complex!T `acos`(T)(Complex!T `z`);
pure nothrow @nogc @safe Complex!T `atan`(T)(Complex!T `z`);
  Inverse trigonometric functions on complex numbers.
  Parameters:

  Complex!T `z` | A complex number.

  Returns:
  The arcsine, arccosine and arctangent of `z`, respectively.
  Examples:

      import std.math.operations : isClose;
      import std.math.constants : PI;
      writeln(asin(complex(0.0))); // 0.0
      assert(isClose(asin(complex(0.5L)), PI / 6));

  Examples:

      import std.math.operations : isClose;
      import std.math.constants : PI;
      import std.math.trigonometry : std_math_acos = acos;
      writeln(acos(complex(0.0))); // std_math_acos(0.0)
      assert(isClose(acos(complex(0.5L)), PI / 3));

  Examples:

      import std.math.operations : isClose;
      import std.math.constants : PI;
      writeln(atan(complex(0.0))); // 0.0
      assert(isClose(atan(sqrt(complex(3.0L))), PI / 3));
      assert(isClose(atan(sqrt(complex(3.0f))), float(PI) / 3));

pure nothrow @nogc @safe Complex!T `sinh`(T)(Complex!T `z`);
pure nothrow @nogc @safe Complex!T `cosh`(T)(Complex!T `z`);
pure nothrow @nogc @safe Complex!T `tanh`(T)(Complex!T `z`);
  Hyperbolic trigonometric functions on complex numbers.
  Parameters:

  Complex!T `z` | A complex number.

  Returns:
  The hyperbolic sine, cosine and tangent of `z`, respectively.
  Examples:

      static import std.math;
      writeln(sinh(complex(0.0))); // 0.0
      writeln(sinh(complex(1.0L))); // std.math.sinh(1.0L)
      writeln(sinh(complex(1.0f))); // std.math.sinh(1.0f)

  Examples:

      static import std.math;
      writeln(cosh(complex(0.0))); // 1.0
      writeln(cosh(complex(1.0L))); // std.math.cosh(1.0L)
      writeln(cosh(complex(1.0f))); // std.math.cosh(1.0f)

  Examples:

      import std.math.operations : isClose;
      import std.math.trigonometry : std_math_tanh = tanh;
      writeln(tanh(complex(0.0))); // 0.0
      assert(isClose(tanh(complex(1.0L)), std_math_tanh(1.0L)));
      assert(isClose(tanh(complex(1.0f)), std_math_tanh(1.0f)));

pure nothrow @nogc @safe Complex!T `asinh`(T)(Complex!T `z`);
pure nothrow @nogc @safe Complex!T `acosh`(T)(Complex!T `z`);
pure nothrow @nogc @safe Complex!T `atanh`(T)(Complex!T `z`);
  Inverse hyperbolic trigonometric functions on complex numbers.
  Parameters:

  Complex!T `z` | A complex number.

  Returns:
  The hyperbolic arcsine, arccosine and arctangent of `z`, respectively.
  Examples:

      import std.math.operations : isClose;
      import std.math.trigonometry : std_math_asinh = asinh;
      writeln(asinh(complex(0.0))); // 0.0
      assert(isClose(asinh(complex(1.0L)), std_math_asinh(1.0L)));
      assert(isClose(asinh(complex(1.0f)), std_math_asinh(1.0f)));

  Examples:

      import std.math.operations : isClose;
      import std.math.trigonometry : std_math_acosh = acosh;
      writeln(acosh(complex(1.0))); // 0.0
      assert(isClose(acosh(complex(3.0L)), std_math_acosh(3.0L)));
      assert(isClose(acosh(complex(3.0f)), std_math_acosh(3.0f)));

  Examples:

      import std.math.operations : isClose;
      import std.math.trigonometry : std_math_atanh = atanh;
      writeln(atanh(complex(0.0))); // 0.0
      assert(isClose(atanh(complex(0.5L)), std_math_atanh(0.5L)));
      assert(isClose(atanh(complex(0.5f)), std_math_atanh(0.5f)));

pure nothrow @nogc @trusted Complex!real `expi`(real `y`);
  Parameters:

  real `y` | A real number.

  Returns:
  The value of cos(y) + i sin(y).

  Note `expi` is included here for convenience and for easy migration of code.

  Examples:

      import core.math : cos, sin;
      writeln(expi(0.0L)); // 1.0L
      writeln(expi(1.3e5L)); // complex(cos(1.3e5L), sin(1.3e5L))

pure nothrow @nogc @safe Complex!real `coshisinh`(real `y`);
  Parameters:

  real `y` | A real number.

  Returns:
  The value of cosh(y) + i sinh(y)

  Note `coshisinh` is included here for convenience and for easy migration of code.

  Examples:

      import std.math.trigonometry : cosh, sinh;
      writeln(coshisinh(3.0L)); // complex(cosh(3.0L), sinh(3.0L))

pure nothrow @nogc @safe Complex!T `sqrt`(T)(Complex!T `z`);
  Parameters:

  Complex!T `z` | A complex number.

  Returns:
  The square root of `z`.
  Examples:

      static import core.math;
      writeln(sqrt(complex(0.0))); // 0.0
      writeln(sqrt(complex(1.0L, 0))); // core.math.sqrt(1.0L)
      writeln(sqrt(complex(-1.0L, 0))); // complex(0, 1.0L)
      writeln(sqrt(complex(-8.0, -6.0))); // complex(1.0, -3.0)

pure nothrow @nogc @trusted Complex!T `exp`(T)(Complex!T `x`);
  Calculates ex.
  Parameters:

  Complex!T `x` | A complex number

  Returns:
  The complex base e exponential of `x`

  Special Values
  x | exp(x)
  --- | ---
  (±0, +0) | (1, +0)
  (any, +∞) | (NAN, NAN)
  (any, NAN) | (NAN, NAN)
  (+∞, +0) | (+∞, +0)
  (-∞, any) | (±0, cis(x.im))
  (+∞, any) | (±∞, cis(x.im))
  (-∞, +∞) | (±0, ±0)
  (+∞, +∞) | (±∞, NAN)
  (-∞, NAN) | (±0, ±0)
  (+∞, NAN) | (±∞, NAN)
  (NAN, +0) | (NAN, +0)
  (NAN, any) | (NAN, NAN)
  (NAN, NAN) | (NAN, NAN)

  Examples:

      import std.math.operations : isClose;
      import std.math.constants : PI;

      writeln(exp(complex(0.0, 0.0))); // complex(1.0, 0.0)

      auto a = complex(2.0, 1.0);
      writeln(exp(conj(a))); // conj(exp(a))

      auto b = exp(complex(0.0L, 1.0L) * PI);
      assert(isClose(b, -1.0L, 0.0, 1e-15));

pure nothrow @nogc @safe Complex!T `log`(T)(Complex!T `x`);
  Calculate the natural logarithm of x. The branch cut is along the negative axis.
  Parameters:

  Complex!T `x` | A complex number

  Returns:
  The complex natural logarithm of `x`

  Special Values
  x | log(x)
  --- | ---
  (-0, +0) | (-∞, π)
  (+0, +0) | (-∞, +0)
  (any, +∞) | (+∞, π/2)
  (any, NAN) | (NAN, NAN)
  (-∞, any) | (+∞, π)
  (+∞, any) | (+∞, +0)
  (-∞, +∞) | (+∞, 3π/4)
  (+∞, +∞) | (+∞, π/4)
  (±∞, NAN) | (+∞, NAN)
  (NAN, any) | (NAN, NAN)
  (NAN, +∞) | (+∞, NAN)
  (NAN, NAN) | (NAN, NAN)

  Examples:

      import core.math : sqrt;
      import std.math.constants : PI;
      import std.math.operations : isClose;

      auto a = complex(2.0, 1.0);
      writeln(log(conj(a))); // conj(log(a))

      auto b = 2.0 * log10(complex(0.0, 1.0));
      auto c = 4.0 * log10(complex(sqrt(2.0) / 2, sqrt(2.0) / 2));
      assert(isClose(b, c, 0.0, 1e-15));

      writeln(log(complex(-1.0L, 0.0L))); // complex(0.0L, PI)
      writeln(log(complex(-1.0L, -0.0L))); // complex(0.0L, -PI)

pure nothrow @nogc @safe Complex!T `log10`(T)(Complex!T `x`);
  Calculate the base-10 logarithm of x.
  Parameters:

  Complex!T `x` | A complex number

  Returns:
  The complex base 10 logarithm of `x`
  Examples:

      import core.math : sqrt;
      import std.math.constants : LN10, PI;
      import std.math.operations : isClose;

      auto a = complex(2.0, 1.0);
      writeln(log10(a)); // log(a) / log(complex(10.0))

      auto b = log10(complex(0.0, 1.0)) * 2.0;
      auto c = log10(complex(sqrt(2.0) / 2, sqrt(2.0) / 2)) * 4.0;
      assert(isClose(b, c, 0.0, 1e-15));

      assert(ceqrel(log10(complex(-100.0L, 0.0L)), complex(2.0L, PI / LN10)) >= real.mant_dig - 1);
      assert(ceqrel(log10(complex(-100.0L, -0.0L)), complex(2.0L, -PI / LN10)) >= real.mant_dig - 1);

pure nothrow @nogc @safe Complex!T `pow`(T, Int)(Complex!T `x`, const Int `n`)
if (isIntegral!Int);
pure nothrow @nogc @trusted Complex!T `pow`(T)(Complex!T `x`, const T `n`);
pure nothrow @nogc @trusted Complex!T `pow`(T)(Complex!T `x`, Complex!T `y`);
pure nothrow @nogc @trusted Complex!T `pow`(T)(const T `x`, Complex!T `n`);
  Calculates xn. The branch cut is on the negative axis.
  Parameters:

  Complex!T `x` | base
  Int `n` | exponent

  Returns:
  `x` raised to the power of `n`
  Examples:

      import std.math.operations : isClose;

      auto a = complex(1.0, 2.0);
      writeln(pow(a, 2)); // a * a
      writeln(pow(a, 3)); // a * a * a
      writeln(pow(a, -2)); // 1.0 / (a * a)
      assert(isClose(pow(a, -3), 1.0 / (a * a * a)));

      auto b = complex(2.0);
      assert(ceqrel(pow(b, 3), exp(3 * log(b))) >= double.mant_dig - 1);

  Examples:

      import std.math.operations : isClose;
      writeln(pow(complex(0.0), 2.0)); // complex(0.0)
      writeln(pow(complex(5.0), 2.0)); // complex(25.0)

      auto a = pow(complex(-1.0, 0.0), 0.5);
      assert(isClose(a, complex(0.0, +1.0), 0.0, 1e-16));

      auto b = pow(complex(-1.0, -0.0), 0.5);
      assert(isClose(b, complex(0.0, -1.0), 0.0, 1e-16));

  Examples:

      import std.math.operations : isClose;
      import std.math.exponential : exp;
      import std.math.constants : PI;
      auto a = complex(0.0);
      auto b = complex(2.0);
      writeln(pow(a, b)); // complex(0.0)

      auto c = complex(0.0L, 1.0L);
      assert(isClose(pow(c, c), exp((-PI) / 2)));

  Examples:

      import std.math.operations : isClose;
      writeln(pow(2.0, complex(0.0))); // complex(1.0)
      writeln(pow(2.0, complex(5.0))); // complex(32.0)

      auto a = pow(-2.0, complex(-1.0));
      assert(isClose(a, complex(-0.5), 0.0, 1e-16));

      auto b = pow(-0.5, complex(-1.0));
      assert(isClose(b, complex(-2.0), 0.0, 1e-15));

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