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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)
Constraints: if (is(R : double));

pure nothrow @nogc @safe auto complex(R, I)(const R re, const I im)
Constraints: 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)
Constraints: 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 std.math;
writeln(abs(complex(1.0))); // 1.0
writeln(abs(complex(0.0, 1.0))); // 1.0
writeln(abs(complex(1.0L, -2.0L))); // std.math.sqrt(5.0L)

pure nothrow @nogc @safe T sqAbs(T)(Complex!T z);

pure nothrow @nogc @safe T sqAbs(T)(const T x)
Constraints: 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;
writeln(sqAbs(complex(0.0))); // 0.0
writeln(sqAbs(complex(1.0))); // 1.0
writeln(sqAbs(complex(0.0, 1.0))); // 1.0
assert(approxEqual(sqAbs(complex(1.0L, -2.0L)), 5.0L));
assert(approxEqual(sqAbs(complex(-3.0L, 1.0L)), 10.0L));
assert(approxEqual(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;
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 : isClose, 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 std.math;
auto z = fromPolar(std.math.sqrt(2.0), PI_4);
assert(approxEqual(z.re, 1.0L, real.epsilon));
assert(approxEqual(z.im, 1.0L, real.epsilon));

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 std.math;
writeln(sin(complex(0.0))); // 0.0
writeln(sin(complex(2.0, 0))); // std.math.sin(2.0)

Examples:

static import std.math;
writeln(cos(complex(0.0))); // 1.0
writeln(cos(complex(1.3, 0.0))); // std.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 @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 std.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 : 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 std.math;
writeln(sqrt(complex(0.0))); // 0.0
writeln(sqrt(complex(1.0L, 0))); // std.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 e^x.

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 : isClose, 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 std.math : sqrt, PI, 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 std.math : LN10, PI, isClose, sqrt;

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)
Constraints: 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 : 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 : 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 : isClose, exp, 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 : 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));