Imaginary_t< T > Class Template Reference

Imaginary type, used to represent purely imaginary numbers. More...

#include <cmplx.h>

Inheritance diagram for Imaginary_t< T >:

Collaboration diagram for Imaginary_t< T >:

Public Member Functions
T	abs () const
	Compute absolute value of Complex number.
T	arg () const
	Compute argument of Complex number.
Complex< T >	conj () const
	Compute conjugate of Complex number.
template<typename A >
Complex< T >	conj_mul (const Complex< A > &b) const
	Conjugate multiply method.
Complex< T >	dagger () const
	Compute dagger of Complex number.
Complex< T > &	gaussian_random (double width=1.0)
	Produces complex gaussian random values.
T &	imag ()
	Imaginary part of Complex number.
T	imag () const
	Imaginary part of Complex number.
template<typename T >
Complex< T >	mul_add (const Complex< T > &a, const Complex< T > &b, const Complex< T > &c)
	Multiply add with Complex numbers.
template<typename A >
Complex< T >	mul_conj (const Complex< A > &b) const
	Multiply conjugate method.
template<typename A , typename B , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
bool	operator!= (const A a, const Complex< B > &b)
	Compare non-equality of Scalar and Complex number.
template<typename A , typename B , std::enable_if_t< hila::is_arithmetic< B >::value, int > = 0>
bool	operator!= (const Complex< A > &a, const B b)
	Compare non-equality of Complex number and Scalar.
template<typename A , typename B >
bool	operator!= (const Complex< A > &a, const Complex< B > &b)
	Compare non-equality of two complex numbers.
template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
auto	operator* (const A &a, const Complex< T > &c)
	Multiplication operator Scalar * Complex.
template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
auto	operator* (const Complex< T > &c, const A &a)
	Multiplication operator Complex * Scalar.
template<typename T1 , typename T2 , typename Tr = hila::type_mul<T1, T2>>
Complex< Tr >	operator* (const Complex< T1 > &a, const Complex< T2 > &b)
	Multiplication operator Complex * Complex.
template<typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
Complex< T > &	operator*= (const A a)
	Real multiply assign operator.
template<typename A >
Complex< T > &	operator*= (const Complex< A > &lhs)
	Complex multiply assignment operator.
template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
auto	operator+ (const A &a, const Complex< T > &c)
	Addition operator Scalar + Complex.
template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
auto	operator+ (const Complex< T > &c, const A &a)
	Addition operator Complex + Scalar.
template<typename T1 , typename T2 , typename Tr = hila::type_plus<T1, T2>>
Complex< Tr >	operator+ (const Complex< T1 > &a, const Complex< T2 > &b)
	Addition operator Complex + Complex.
Complex< T > &	operator++ ()
	Single increment operator.
Complex< T >	operator++ (int)
	Multiple increment operator.
template<typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
Complex< T > &	operator+= (const A &a)
	Real addition assignment operator.
template<typename A >
Complex< T > &	operator+= (const Complex< A > &lhs)
	Complex addition assignment operator.
template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
auto	operator- (const A &a, const Complex< T > &c)
	Subtraction operator Scalar - Complex.
template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
auto	operator- (const Complex< T > &c, const A &a)
	Subtraction operator Complex - Scalar.
template<typename T1 , typename T2 , typename Tr = hila::type_plus<T1, T2>>
Complex< Tr >	operator- (const Complex< T1 > &a, const Complex< T2 > &b)
	Subtraction operator Complex - Complex.
Complex< T > &	operator-- ()
	Single decrement operator.
Complex< T >	operator-- (int)
	Multiple decrement operator.
template<typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
Complex< T > &	operator-= (const A &a)
	Real subtraction assignment operator Subtract assign only to real part of Complex number.
template<typename A >
Complex< T > &	operator-= (const Complex< A > &lhs)
	Complex subtraction assignment operator.
template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
auto	operator/ (const A &a, const Complex< T > &c)
	Division operator Scalar / Complex.
template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
auto	operator/ (const Complex< T > &c, const A &a)
	Division operator Complex / Scalar.
template<typename T1 , typename T2 , typename Tr = hila::type_mul<T1, T2>>
Complex< Tr >	operator/ (const Complex< T1 > &a, const Complex< T2 > &b)
	Division operator Complex / Complex.
template<typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
Complex< T > &	operator/= (const A &a)
	Real divide assignment operator.
template<typename A >
Complex< T > &	operator/= (const Complex< A > &lhs)
	Complex divide assignment operator.
template<typename A , typename B , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
bool	operator== (const A a, const Complex< B > &b)
	Compare equality of Scalar and Complex.
template<typename A , typename B , std::enable_if_t< hila::is_arithmetic< B >::value, int > = 0>
bool	operator== (const Complex< A > &a, const B b)
	Compare equality of Complex and scalar.
template<typename A , typename B >
bool	operator== (const Complex< A > &a, const Complex< B > &b)
	Compare equality of two complex numbersTwo numbers are equal, if the real and imaginary components are respectively equal.
Complex< T >	polar (const T r, const T theta)
	Stores and returns Complex number given in polar coordinates.
Complex< T > &	random ()
	Assign random values to Complex real and imaginary part.
T &	real ()
	Real part of Complex number.
T	real () const
	Real part of Complex number.
T	squarenorm () const
	Compute square norm of Complex number.

Related Symbols
(Note that these are not member symbols.)
constexpr Imaginary_t< double >	operator""_i (long double a)
Mathematical functions
Returns Complex T
template<typename T >
Complex< T >	exp (const Complex< T > z)
	\(\exp(z)\)
template<typename T >
Complex< T >	exp (const Imaginary_t< T > im)
	\(\exp(\Im(z))\)
template<typename T , std::enable_if_t< hila::is_arithmetic< T >::value, int > = 0>
Complex< T >	expi (T a)
	\(\exp(i\cdot x)\)
template<typename T >
Complex< T >	log (Complex< T > z)
	\(\log{z}\)
template<typename T >
Complex< T >	sqrt (Complex< T > z)
	\(\sqrt{z}\)
template<typename T >
Complex< T >	cbrt (Complex< T > z)
	\(\sqrt[3]{z}\)
template<typename A , typename B >
auto	pow (Complex< A > z, Complex< B > p)
	pow(z.p) = \(z^p\) = \(exp(p*log(z))\)
template<typename T , typename S , std::enable_if_t< hila::is_arithmetic< S >::value, int > = 0>
Complex< T >	pow (Complex< T > z, S p)
	pow(z.p) with scalar power
template<typename T , typename S , std::enable_if_t< hila::is_arithmetic< S >::value, int > = 0>
Complex< T >	pow (S z, Complex< T > p)
	pow(z.p) with scalar base
template<typename T >
Complex< T >	sin (Complex< T > z)
template<typename T >
Complex< T >	cos (Complex< T > z)
template<typename T >
Complex< T >	tan (Complex< T > z)
	\(\tan(z) = \frac{\sin(z)}{\cos(z)}\) - rely on optimizer to simplify
template<typename T >
Complex< T >	sinh (Complex< T > z)
template<typename T >
Complex< T >	cosh (Complex< T > z)
template<typename T >
Complex< T >	tanh (Complex< T > z)
	\(\tanh(z)\)
template<typename T >
Complex< T >	atan (Complex< T > z)
	\(\arctan(z)\)
template<typename T >
Complex< T >	asin (Complex< T > z)
	\(\arcsin(z)\)
template<typename T >
Complex< T >	acos (Complex< T > z)
	\(\arccos(z)\)
template<typename T >
Complex< T >	atanh (Complex< T > z)
	\(\text{artanh}(z)\)
template<typename T >
Complex< T >	asinh (Complex< T > z)
	\(\text{arsinh}(z)\)
template<typename T >
Complex< T >	acosh (Complex< T > z)
	\(\text{arcosh}(z)\)

Detailed Description

template<typename T>
class Imaginary_t< T >

Imaginary type, used to represent purely imaginary numbers.

Useful for reducing multiply operations in im * complex or im * real -ops Derived from Complex class, so generic complex ops should remain valid Defines only operators * and /, others go via Complex class

Note: Imaginary_t should be used in Field variables

Template Parameters

T	type of imaginary (float/double)

Definition at line 1442 of file cmplx.h.

Member Function Documentation

abs()

template<typename T >

T Complex< T >::abs ( ) const

inlineinherited

Compute absolute value of Complex number.

Essentially sqrt(squarenorm(z)):

\begin{align} |z| = \sqrt{\Re(z)^2 + \Im(z)^2}\end{align}

Returns

T

Definition at line 292 of file cmplx.h.

arg()

template<typename T >

T Complex< T >::arg ( ) const

inlineinherited

Compute argument of Complex number.

\begin{align} \arg(z) = \arctan2(\Im(z)),\Re(z)\end{align}

Returns

T

Definition at line 303 of file cmplx.h.

conj()

template<typename T >

Complex< T > Complex< T >::conj ( ) const

inlineinherited

Compute conjugate of Complex number.

\begin{align} z &= x + i\cdot y \\ \Rightarrow z^* &= x - i\cdot y\end{align}

Returns

Complex<T>

Definition at line 315 of file cmplx.h.

conj_mul()

template<typename T >

template<typename A >

Complex< T > Complex< T >::conj_mul ( const Complex< A > &  b ) const

inlineinherited

Conjugate multiply method.

Conjugate a (*this) and multiply with give argument b: a.conj_mul(b) \( \equiv a^* \cdot b\)

Template Parameters

A	Type for Complex number b

Parameters

b	number to multiply with

Returns

Complex<T>

Definition at line 698 of file cmplx.h.

dagger()

template<typename T >

Complex< T > Complex< T >::dagger ( ) const

inlineinherited

Compute dagger of Complex number.

Alias to Complex::conj

\begin{align} z^* = z^\dagger \end{align}

Returns

Complex<T>

Definition at line 326 of file cmplx.h.

gaussian_random()

template<typename T >

Complex< T > & Complex< T >::gaussian_random ( double  width = 1.0 )

inlineinherited

Produces complex gaussian random values.

Uses hila::gaussrand2 for both real and imaignary part Assigns same random value for both real and imaginary part

Parameters

width

gaussian_random

Returns

Complex<T>&

Definition at line 371 of file cmplx.h.

imag() [1/2]

template<typename T >

T & Complex< T >::imag ( )

inlineinherited

Imaginary part of Complex number.

Pass by reference

Returns

T & Imaginary part

Definition at line 194 of file cmplx.h.

imag() [2/2]

template<typename T >

T Complex< T >::imag ( ) const

inlineinherited

Imaginary part of Complex number.

Pass by value if Complex number is defined as const

Returns

T Imaginary part

Definition at line 175 of file cmplx.h.

mul_add()

template<typename T >

Complex< T > mul_add ( const Complex< T > &  a, const Complex< T > &  b, const Complex< T > &  c  )

inlineinherited

Multiply add with Complex numbers.

Defined as

mul_add(a,b,c) = a*b + c;

Template Parameters

T	Arithmetic type of a,b and c

Parameters

a	Complex number to multiply
b	Complex number to multiply
c	Complex number to add to result of multiplication

Returns

Complex<T>

Definition at line 1184 of file cmplx.h.

mul_conj()

template<typename T >

template<typename A >

Complex< T > Complex< T >::mul_conj ( const Complex< A > &  b ) const

inlineinherited

Multiply conjugate method.

Multiply a (*this) with conjugate of given argument b: a.mul_conj(b) \( \equiv a \cdot b^*\)

Template Parameters

A	Type for Complex number b

Parameters

b	number to conjugate and multiply withv

Returns

Complex<T>

Definition at line 714 of file cmplx.h.

operator!=() [1/3]

template<typename A , typename B , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

bool operator!= ( const A  a, const Complex< B > &  b  )

inlineinherited

Compare non-equality of Scalar and Complex number.

Negation of operator==()

Template Parameters

A	Arithmetic type of a
B	Arithmetic type of b

Parameters

a	Scalar to compare
b	Complex number to compare

Returns

true if values are not arithmetically equal

Definition at line 1293 of file cmplx.h.

operator!=() [2/3]

template<typename A , typename B , std::enable_if_t< hila::is_arithmetic< B >::value, int > = 0>

bool operator!= ( const Complex< A > &  a, const B  b  )

inlineinherited

Compare non-equality of Complex number and Scalar.

Negation of operator==()

Template Parameters

A	Arithmetic type of a
B	Arithmetic type of b

Parameters

a	Complex number to compare
b	Scalar to compare

Returns

true if values are not arithmetically equal

Definition at line 1277 of file cmplx.h.

operator!=() [3/3]

template<typename A , typename B >

bool operator!= ( const Complex< A > &  a, const Complex< B > &  b  )

inlineinherited

Compare non-equality of two complex numbers.

Negation of operator==()

Template Parameters

A	Arithmetic type of a
B	Arithmetic type of b

Parameters

a	Complex number to compare
b	Complex number to compare

Returns

true if values are not arithmetically equal

Definition at line 1261 of file cmplx.h.

operator*() [1/3]

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

auto operator* ( const A &  a, const Complex< T > &  c  )

inlineinherited

Multiplication operator Scalar * Complex.

Multiplication between Complex and scalar is defined in the usual way

\begin{align}z,z_1 &= x + iy \in \mathbb{C}, a \in \mathbb{R} \\ z * a &= (x\cdot a + iy\cdot a)\end{align}

z = a * z_1;

Template Parameters

T	Arithmetic type of c
A	Arithmetic type of a

Parameters

c	Complex number to multiply
a	Scalar to multiply

Returns

auto

Definition at line 1094 of file cmplx.h.

operator*() [2/3]

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

auto operator* ( const Complex< T > &  c, const A &  a  )

inlineinherited

Multiplication operator Complex * Scalar.

Multiplication between Complex and scalar is defined in the usual way

\begin{align}z,z_1 &= x + iy \in \mathbb{C}, a \in \mathbb{R} \\ z_1 * a &= (x\cdot a + iy\cdot a)\end{align}

z = z_1 * a;

Template Parameters

T	Arithmetic type of c
A	Arithmetic type of a

Parameters

c	Complex number to multiply
a	Scalar to multiply

Returns

auto

Definition at line 1074 of file cmplx.h.

operator*() [3/3]

template<typename T1 , typename T2 , typename Tr = hila::type_mul<T1, T2>>

Complex< Tr > operator* ( const Complex< T1 > &  a, const Complex< T2 > &  b  )

inlineinherited

Multiplication operator Complex * Complex.

Defined in the usual way

\begin{align}z, z_1 &= x + iy, z_2 = x' + iy' \in \mathbb{C} \\ z_1 z_2 &= (x + iy)(x' + iy') = (xx'-yy') + i(xy' + yx')\end{align}

z = z_1 * z_2;

Template Parameters

T1	Arithmetic type of a
T2	Arithmetic type of b
Tr	Resulting type after multiplication

Parameters

a
b

Returns

Complex<Tr>

Definition at line 1054 of file cmplx.h.

operator*=() [1/2]

template<typename T >

template<typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

Complex< T > & Complex< T >::operator*= ( const A  a )

inlineinherited

Real multiply assign operator.

Multiply assign by real number to both components of Complex number

Complex<double> z(1,1);
z *= 2; // z.re = 2, z.im = 2

Template Parameters

A	Type for real value a

Parameters

a	Real value to multiply

Returns

Complex<T>&

Definition at line 541 of file cmplx.h.

operator*=() [2/2]

template<typename T >

template<typename A >

Complex< T > & Complex< T >::operator*= ( const Complex< A > &  lhs )

inlineinherited

Complex multiply assignment operator.

Standard Complex number multiplication

\begin{align}z &= x + iy, w = x' + iy' \\ z w &= (x + iy)(x' + iy') = (xx'-yy') + i(xy' + yx')\end{align}

Complex<double> z,w;
//
// z,w get values
//
z*=w; // z = zw as defined above

Template Parameters

A	Type for Complex number multiply

Parameters

lhs	Complex number to multiply

Returns

Complex<T>&

Definition at line 521 of file cmplx.h.

operator+() [1/3]

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

auto operator+ ( const A &  a, const Complex< T > &  c  )

inlineinherited

Addition operator Scalar + Complex.

Defined in the usual way. \(z,z_1 \in \mathbb{C}, a \in \mathbb{R}\)

z = a + z_1;

Template Parameters

T	Arithmetic type of c
A	Arithmetic type of a

Parameters

a	Scalar to sum
c	Complex number to sum

Returns

auto

Definition at line 973 of file cmplx.h.

operator+() [2/3]

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

auto operator+ ( const Complex< T > &  c, const A &  a  )

inlineinherited

Addition operator Complex + Scalar.

Defined in the usual way. \(z,z_1 \in \mathbb{C}, a \in \mathbb{R}\)

z = z_1 + a;

Template Parameters

T	Arithmetic type of c
A	Arithmetic type of a

Parameters

c	Complex number to sum
a	Scalar to sum

Returns

auto

Definition at line 955 of file cmplx.h.

operator+() [3/3]

template<typename T1 , typename T2 , typename Tr = hila::type_plus<T1, T2>>

Complex< Tr > operator+ ( const Complex< T1 > &  a, const Complex< T2 > &  b  )

inlineinherited

Addition operator Complex + Complex.

Addition between Complex numbers is defined in the usual way:

\begin{align} z_1, z_2 &\in \mathbf{C} \\ z_1 + z_2 &= \Re(z_1) + \Re(z_2) + [\Im(z_1) + \Im(z_2)]i \end{align}

z = z_1 + z_2;

Template Parameters

T1	Type for first complex number
T2	Type for second complex number
Tr	Conversion type between T1 and T2

Parameters

a	First complex number
b	Second complex number

Returns

Complex<Tr>

Definition at line 936 of file cmplx.h.

operator++() [1/2]

template<typename T >

Complex< T > & Complex< T >::operator++ ( )

inlineinherited

Single increment operator.

Increments real part of Complex number

Complex<double> z(1,1);
z++; // z.re = 2, z.im = 1

Returns

Complex<T>&

Definition at line 619 of file cmplx.h.

operator++() [2/2]

template<typename T >

Complex< T > Complex< T >::operator++ ( int  )

inlineinherited

Multiple increment operator.

Increments real part of Complex number

Complex<double> z(2,1);
z++2; // z.re = 4, z.im = 1

Returns

Complex<T>&

Definition at line 653 of file cmplx.h.

operator+=() [1/2]

template<typename T >

template<typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

Complex< T > & Complex< T >::operator+= ( const A &  a )

inlineinherited

Real addition assignment operator.

Add assign only to real part of Complex number

Complex<double> z(0,0);
z += 1; // z.re = 1, z.im = 0

Template Parameters

A	Type for real value a

Parameters

a	Real value to add

Returns

Complex<T>&

Definition at line 447 of file cmplx.h.

operator+=() [2/2]

template<typename T >

template<typename A >

Complex< T > & Complex< T >::operator+= ( const Complex< A > &  lhs )

inlineinherited

Complex addition assignment operator.

Complex<double> z(0,0);
Complex<double> w(1,1);
z += w; // z.re = 1, z.im = 1

Template Parameters

A	Typpe of Complex number to add

Parameters

lhs	Complex number to add

Returns

Complex<T>&

Definition at line 426 of file cmplx.h.

operator-() [1/3]

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

auto operator- ( const A &  a, const Complex< T > &  c  )

inlineinherited

Subtraction operator Scalar - Complex.

Defined in the usual way. \(z,z_1 \in \mathbb{C}, a \in \mathbb{R}\)

z = a - z_1;

Template Parameters

T	Arithmetic type of c
A	Arithmetic type of a

Parameters

a	Scalar
c	Complex number to subtract

Returns

auto

Definition at line 1032 of file cmplx.h.

operator-() [2/3]

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

auto operator- ( const Complex< T > &  c, const A &  a  )

inlineinherited

Subtraction operator Complex - Scalar.

Defined in the usual way. \(z,z_1 \in \mathbb{C}, a \in \mathbb{R}\)

z = z_1 - a;

Template Parameters

T	Arithmetic type of c
A	Arithmetic type of a

Parameters

c	Complex number
a	Scalar to subtract

Returns

auto

Definition at line 1014 of file cmplx.h.

operator-() [3/3]

template<typename T1 , typename T2 , typename Tr = hila::type_plus<T1, T2>>

Complex< Tr > operator- ( const Complex< T1 > &  a, const Complex< T2 > &  b  )

inlineinherited

Subtraction operator Complex - Complex.

Subtraction between Complex numbers is defined in the usual way

\begin{align} z_1, z_2 &\in \mathbf{C} \\ z_1 - z_2 &= \Re(z_1) - \Re(z_2) + [\Im(z_1) - \Im(z_2)]i\end{align}

z = z_1 - z_2;

Template Parameters

T1	Arithmetic type of a
T2	Arithmetic type of b
Tr	Resulting type after subtraction

Parameters

a	First complex number
b	Second complex number to subtract

Returns

Complex<Tr>

Definition at line 996 of file cmplx.h.

operator--() [1/2]

template<typename T >

Complex< T > & Complex< T >::operator-- ( )

inlineinherited

Single decrement operator.

Decrement real part of Complex number

Complex<double> z(1,1);
z--; // z.re = 0, z.im = 1

Returns

Complex<T>&

Definition at line 636 of file cmplx.h.

operator--() [2/2]

template<typename T >

Complex< T > Complex< T >::operator-- ( int  )

inlineinherited

Multiple decrement operator.

Decrement real part of Complex number

Complex<double> z(2,1);
z--2; // z.re = 0, z.im = 1

Returns

Complex<T>&

Definition at line 672 of file cmplx.h.

operator-=() [1/2]

template<typename T >

template<typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

Complex< T > & Complex< T >::operator-= ( const A &  a )

inlineinherited

Real subtraction assignment operator Subtract assign only to real part of Complex number.

Complex<double> z(0,0);
z -= 1; // z.re = -1, z.im = 0

Template Parameters

A	Type for real value a

Parameters

a	Real value to subtract

Returns

Complex<T>&

Definition at line 487 of file cmplx.h.

operator-=() [2/2]

template<typename T >

template<typename A >

Complex< T > & Complex< T >::operator-= ( const Complex< A > &  lhs )

inlineinherited

Complex subtraction assignment operator.

Complex<double> z(0,0);
Complex<double> w(1,1);
z -= w; // z.re = -1, z.im = -1

Template Parameters

A	Type for complex number to subtract

Parameters

lhs	Complex number to subtract

Returns

Complex<T>&

Definition at line 467 of file cmplx.h.

operator/() [1/3]

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

auto operator/ ( const A &  a, const Complex< T > &  c  )

inlineinherited

Division operator Scalar / Complex.

Defined in the usual way

\begin{align} z,z_1=x + iy &\in \mathbb{C}, a \in \mathbb{R} \\ \frac{a}{z_1} &= \frac{az^*}{|z|^2} \end{align}

z = a / z_1;

Template Parameters

T	Arithmetic type of c
A	Arithmetic type of a

Parameters

c	Complex number to divide
a	Scalar to divide with

Returns

auto

Definition at line 1164 of file cmplx.h.

operator/() [2/3]

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

auto operator/ ( const Complex< T > &  c, const A &  a  )

inlineinherited

Division operator Complex / Scalar.

Defined in the usual way

\begin{align}z, z_1=x + iy &\in \mathbb{C}, a \in \mathbb{R} \\ \frac{z_1}{a} &= \frac{x}{a} + i\cdot \frac{y}{a} \end{align}

z = z_1 / a;

Template Parameters

T	Arithmetic type of c
A	Arithmetic type of a

Parameters

c	Complex number to divide
a	Scalar to divide with

Returns

auto

Definition at line 1143 of file cmplx.h.

operator/() [3/3]

template<typename T1 , typename T2 , typename Tr = hila::type_mul<T1, T2>>

Complex< Tr > operator/ ( const Complex< T1 > &  a, const Complex< T2 > &  b  )

inlineinherited

Division operator Complex / Complex.

Defined in the usual way

\begin{align} z,z_1,z_2 &\in \mathbb{C} \\ \frac{z_1}{z_2} &= \frac{z_1\cdot z_2^{*}}{|z_2|^2} \end{align}

z =z_1 / z_2;

Template Parameters

T1	Arithmetic type of a
T2	Arithmetic type of b
Tr	Resulting type after division

Parameters

a	Complex number to divide
b	Complex number to divide with

Returns

Complex<Tr>

Definition at line 1122 of file cmplx.h.

operator/=() [1/2]

template<typename T >

template<typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

Complex< T > & Complex< T >::operator/= ( const A &  a )

inlineinherited

Real divide assignment operator.

Divide assign by real number to both components of Complex number

Complex<double> z(2,2);
z /= 2; // z.re = 1, z.im = 1

Template Parameters

A	Type for real value to divide

Parameters

a	Real value to divide

Returns

Complex<T>&

Definition at line 598 of file cmplx.h.

operator/=() [2/2]

template<typename T >

template<typename A >

Complex< T > & Complex< T >::operator/= ( const Complex< A > &  lhs )

inlineinherited

Complex divide assignment operator.

Standard Complex number division

\begin{align}z &= x + iy, w = x' + iy' \\ \frac{z}{w} &= \frac{x + iy}{x' + iy'} = \frac{(xx'+ yy') + i( yx' - xy')}{|w|^2}\end{align}

Complex<double> z,w;
//
// z,w get values
//
z/=w; // z = z/w as defined above

Template Parameters

A	Type for Complex number divide

Parameters

lhs	Complex number to divide

Returns

Complex<T>&

Definition at line 577 of file cmplx.h.

operator==() [1/3]

template<typename A , typename B , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

bool operator== ( const A  a, const Complex< B > &  b  )

inlineinherited

Compare equality of Scalar and Complex.

Two numbers are equal, if the arithmetic values are equal: thus, complex and real comparison (a + i b) == a is true if b == 0.

Template Parameters

A	Arithmetic type of a
B	Arithmetic type of b

Parameters

b	Complex number to compare
a	Scalar to compare

Returns

true if values compare to equal

Definition at line 1245 of file cmplx.h.

operator==() [2/3]

template<typename A , typename B , std::enable_if_t< hila::is_arithmetic< B >::value, int > = 0>

bool operator== ( const Complex< A > &  a, const B  b  )

inlineinherited

Compare equality of Complex and scalar.

Two numbers are equal, if the arithmetic values are equal: thus, complex and real comparison (a + i b) == a is true if b == 0.

Template Parameters

A	Arithmetic type of a
B	Arithmetic type of b

Parameters

a	Complex number to compare
b	Scalar to compare

Returns

true if values compare to equal

Definition at line 1226 of file cmplx.h.

operator==() [3/3]

template<typename A , typename B >

bool operator== ( const Complex< A > &  a, const Complex< B > &  b  )

inlineinherited

Compare equality of two complex numbersTwo numbers are equal, if the real and imaginary components are respectively equal.

Template Parameters

A	Arithmetic type of a
B	Arithmetic type of b

Parameters

a	Complex number to compare
b	Complex number to compare

Returns

true if values compare to equal

Definition at line 1207 of file cmplx.h.

polar()

template<typename T >

Complex< T > Complex< T >::polar ( const T  r, const T  theta  )

inlineinherited

Stores and returns Complex number given in polar coordinates.

\begin{align} z = r\cdot e^{i\theta} \end{align}

Complex<double> c;
double r = 1;
double theta = 3.14159265358979/2; // pi/2
c.polar(r,theta); // c.re = 0, c.im = 1

Parameters

r	Radius of Complex number
theta	Angle of complex number in radians

Returns

Complex<T> Complex number

Definition at line 346 of file cmplx.h.

random()

template<typename T >

Complex< T > & Complex< T >::random ( )

inlineinherited

Assign random values to Complex real and imaginary part.

Uses hila::random for both real and imaginary part

Returns

Complex<T>&

Definition at line 358 of file cmplx.h.

real() [1/2]

template<typename T >

T & Complex< T >::real ( )

inlineinherited

Real part of Complex number.

Pass by reference

Returns

T & Real part

Definition at line 185 of file cmplx.h.

real() [2/2]

template<typename T >

T Complex< T >::real ( ) const

inlineinherited

Real part of Complex number.

Pass by value if Complex number is defined as const

Returns

T Real part

Definition at line 167 of file cmplx.h.

squarenorm()

template<typename T >

T Complex< T >::squarenorm ( ) const

inlineinherited

Compute square norm of Complex number.

\begin{align} |z|^2 = \Re(z)^2 + \Im(z)^2\end{align}

Returns

T Squared norm

Definition at line 281 of file cmplx.h.

Friends And Related Symbol Documentation

cos()

template<typename T >

Complex< T > cos ( Complex< T >  z )

related

\begin{align}\cos(z) &= \cos(\Re{z})\cos(i \Im(z)) - \sin(\Re{z})\sin(i \Im{z}) \\ &= \cos(\Re{z})\cosh(\Im(z)) - i \sin(\Re{z})\sinh(\Im(z))\end{align}

Definition at line 1621 of file cmplx.h.

cosh()

template<typename T >

Complex< T > cosh ( Complex< T >  z )

related

cosh(z) = cosh(re)cosh(i im) - sinh(re)sinh(i im) = cosh(re)cos(im) - i sinh(re)sin(im)

Definition at line 1641 of file cmplx.h.

operator""_i()

template<typename T >

constexpr Imaginary_t< double > operator""_i ( long double  a )

related

Operators to implement imaginary unit 1_i, enablig expressions 3 + 2_i etc. This is defined as an user-defined literal, which requires an underscore.

Definition at line 1695 of file cmplx.h.

sin()

template<typename T >

Complex< T > sin ( Complex< T >  z )

related

\begin{align}sin(z) &= \sin(\Re(z) + i \Im(z)) \\ &= \sin(\Re(z))\cos(i \Im(z)) + \cos(\Re(z))\sin(i \Im(z)) \\ &= \sin(\Re(z)) \cosh(\Im(z)) + i \cos(\Re(z)) \sinh(\Im(z)) \end{align}

Definition at line 1614 of file cmplx.h.

sinh()

template<typename T >

Complex< T > sinh ( Complex< T >  z )

related

sinh(z) = sinh(re)cosh(i im) + cosh(re)sinh(i im) = sinh(re)cos(im) + i cosh(re)sin(im)

Definition at line 1634 of file cmplx.h.

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The documentation for this class was generated from the following file:

• cmplx.h