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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:
pure nothrow @nogc @safe auto `complex`(R)(R `re`)
if (is(R : double));

pure nothrow @nogc @safe auto `complex`(R, I)(R `re`, 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`, 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'.
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)(Rx `x`, Ry `y`);

this(R : T)(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)(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;
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 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)
```
pure nothrow @nogc @safe Complex!(CommonType!(T, U)) `fromPolar`(T, U)(T `modulus`, 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`);
Trigonometric functions on complex numbers.
Parameters:
 Complex!T `z` A complex number.
Returns:
The sine and cosine of `z`, respectively.
Examples:
```static import std.math;
writeln(sin(complex(0.0))); // 0.0
writeln(sin(complex(2.0L, 0))); // std.math.sin(2.0L)
```
Examples:
```import std.complex;
import std.math;
writeln(cos(complex(0.0))); // 1.0
writeln(cos(complex(1.3L))); // std.math.cos(1.3L)
writeln(cos(complex(0, 5.2L))); // cosh(5.2L)
```
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 that uses std.math.expi. Unlike std.math.expi, which uses the x87 fsincos instruction when possible, this function is no faster than calculating cos(`y`) and sin(`y`) separately.

Examples:
```static import std.math;

writeln(expi(1.3e5L)); // complex(std.math.cos(1.3e5L), std.math.sin(1.3e5L))
writeln(expi(0.0L)); // 1.0L
auto z1 = expi(1.234);
auto z2 = std.math.expi(1.234);
assert(z1.re == z2.re && z1.im == z2.im);
```
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)
```