Complex Numbers¶

The complex types, functions and arithmetic operations are defined in the header file cml/complex.h.

Representation of complex numbers¶

Complex numbers are represented using the type cml_complex_t. The internal representation of this type may vary across platforms and should not be accessed directly. The functions and macros described below allow complex numbers to be manipulated in a portable way.

For reference, the default form of the cml_complex_t type is given by the following struct:

typedef struct
{
        union
        {
                double p[2];
                double parts[2];
                struct
                {
                        double re;
                        double im;
                };
                struct
                {
                        double real;
                        double imaginary;
                };
        };
} cml_complex_t;

The real and imaginary part are stored in contiguous elements of a two element array. This eliminates any padding between the real and imaginary parts, parts[0] and parts[1], allowing the struct to be mapped correctly onto packed complex arrays.

cml_complex_t complex(double x, double y)¶: This function uses the rectangular Cartesian components $(x,y)$ to return the complex number $z = x + y i$ . An inline version of this function is used when HAVE_INLINE is defined.

cml_complex_t cml_complex_polar(double r, double theta)¶: This function returns the complex number $z = r \exp(i \theta) = r (\cos(\theta) + i \sin(\theta))$ from the polar representation (r, theta).

creal(z)¶
cimag(z)¶: These macros return the real and imaginary parts of the complex number z.

Properties of complex numbers¶

double cml_complex_arg(cml_complex_t z)¶: This function returns the argument of the complex number z, $\arg(z)$ , where $-\pi < \arg(z) <= \pi$ .

double cml_complex_abs(cml_complex_t z)¶: This function returns the magnitude of the complex number z, $|z|$ .

double cml_complex_abs2(cml_complex_t z)¶: This function returns the squared magnitude of the complex number z, $|z|^2$ .

double cml_complex_logabs(cml_complex_t z)¶: This function returns the natural logarithm of the magnitude of the complex number z, $\log|z|$ . It allows an accurate evaluation of $\log|z|$ when $|z|$ is close to one. The direct evaluation of log(cml_complex_abs(z)) would lead to a loss of precision in this case.

Complex arithmetic operators¶

cml_complex_t cml_complex_add(cml_complex_t a, cml_complex_t b)¶: This function returns the sum of the complex numbers a and b, $z=a+b$ .

cml_complex_t cml_complex_sub(cml_complex_t a, cml_complex_t b)¶: This function returns the difference of the complex numbers a and b, $z=a-b$ .

cml_complex_t cml_complex_mul(cml_complex_t a, cml_complex_t b)¶: This function returns the product of the complex numbers a and b, $z=ab$ .

cml_complex_t cml_complex_div(cml_complex_t a, cml_complex_t b)¶: This function returns the quotient of the complex numbers a and b, $z=a/b$ .

cml_complex_t cml_complex_add_real(cml_complex_t a, double x)¶: This function returns the sum of the complex number a and the real number x, $z=a+x$ .

cml_complex_t cml_complex_sub_real(cml_complex_t a, double x)¶: This function returns the difference of the complex number a and the real number x, $z=a-x$ .

cml_complex_t cml_complex_mul_real(cml_complex_t a, double x)¶: This function returns the product of the complex number a and the real number x, $z=ax$ .

cml_complex_t cml_complex_div_real(cml_complex_t a, double x)¶: This function returns the quotient of the complex number a and the real number x, $z=a/x$ .

cml_complex_t cml_complex_add_imag(cml_complex_t a, double y)¶: This function returns the sum of the complex number a and the imaginary number $iy$ , $z=a+iy$ .

cml_complex_t cml_complex_sub_imag(cml_complex_t a, double y)¶: This function returns the difference of the complex number a and the imaginary number $iy$ , $z=a-iy$ .

cml_complex_t cml_complex_mul_imag(cml_complex_t a, double y)¶: This function returns the product of the complex number a and the imaginary number $iy$ , $z=a*(iy)$ .

cml_complex_t cml_complex_div_imag(cml_complex_t a, double y)¶: This function returns the quotient of the complex number a and the imaginary number $iy$ , $z=a/(iy)$ .

cml_complex_t cml_complex_conj(cml_complex_t z)¶: This function returns the complex conjugate of the complex number z, $z^* = x - y i$ .

cml_complex_t cml_complex_inverse(cml_complex_t z)¶: This function returns the inverse, or reciprocal, of the complex number z, $1/z = (x - y i)/(x^2 + y^2)$ .

cml_complex_t cml_complex_negative(cml_complex_t z)¶: This function returns the negative of the complex number z, $-z = (-x) + (-y)i$ .

Elementary Complex Functions¶

cml_complex_t cml_complex_sqrt(cml_complex_t z)¶: This function returns the square root of the complex number z, $\sqrt z$ . The branch cut is the negative real axis. The result always lies in the right half of the complex plane.

cml_complex_t cml_complex_sqrt_real(double x)¶: This function returns the complex square root of the real number x, where x may be negative.

cml_complex_t cml_complex_pow(cml_complex_t z, cml_complex_t a)¶: The function returns the complex number z raised to the complex power a, $z^a$ . This is computed as $\exp(\log(z)*a)$ using complex logarithms and complex exponentials.

cml_complex_t cml_complex_pow_real(cml_complex_t z, double x)¶: This function returns the complex number z raised to the real power x, $z^x$ .

cml_complex_t cml_complex_exp(cml_complex_t z)¶: This function returns the complex exponential of the complex number z, $\exp(z)$ .

cml_complex_t cml_complex_log(cml_complex_t z)¶: This function returns the complex natural logarithm (base $e$ ) of the complex number z, $\log(z)$ . The branch cut is the negative real axis.

cml_complex_t cml_complex_log10(cml_complex_t z)¶: This function returns the complex base-10 logarithm of the complex number z, $\log_{10} (z)$ .

cml_complex_t cml_complex_log_b(cml_complex_t z, cml_complex_t b)¶: This function returns the complex base-b logarithm of the complex number z, $\log_b(z)$ . This quantity is computed as the ratio $\log(z)/\log(b)$ .

Complex Trigonometric Functions¶

cml_complex_t cml_complex_sin(cml_complex_t z)¶: This function returns the complex sine of the complex number z, $\sin(z) = (\exp(iz) - \exp(-iz))/(2i)$ .

cml_complex_t cml_complex_cos(cml_complex_t z)¶: This function returns the complex cosine of the complex number z, $\cos(z) = (\exp(iz) + \exp(-iz))/2$ .

cml_complex_t cml_complex_tan(cml_complex_t z)¶: This function returns the complex tangent of the complex number z, $\tan(z) = \sin(z)/\cos(z)$ .

cml_complex_t cml_complex_sec(cml_complex_t z)¶: This function returns the complex secant of the complex number z, $\sec(z) = 1/\cos(z)$ .

cml_complex_t cml_complex_csc(cml_complex_t z)¶: This function returns the complex cosecant of the complex number z, $\csc(z) = 1/\sin(z)$ .

cml_complex_t cml_complex_cot(cml_complex_t z)¶: This function returns the complex cotangent of the complex number z, $\cot(z) = 1/\tan(z)$ .

Inverse Complex Trigonometric Functions¶

cml_complex_t cml_complex_asin(cml_complex_t z)¶: This function returns the complex arcsine of the complex number z, $\arcsin(z)$ . The branch cuts are on the real axis, less than $-1$ and greater than $1$ .

cml_complex_t cml_complex_asin_real(double z)¶: This function returns the complex arcsine of the real number z, $\arcsin(z)$ . For $z$ between $-1$ and $1$ , the function returns a real value in the range $[-\pi/2,\pi/2]$ . For $z$ less than $-1$ the result has a real part of $-\pi/2$ and a positive imaginary part. For $z$ greater than $1$ the result has a real part of $\pi/2$ and a negative imaginary part.

cml_complex_t cml_complex_acos(cml_complex_t z)¶: This function returns the complex arccosine of the complex number z, $\arccos(z)$ . The branch cuts are on the real axis, less than $-1$ and greater than $1$ .

cml_complex_t cml_complex_acos_real(double z)¶: This function returns the complex arccosine of the real number z, $\arccos(z)$ . For $z$ between $-1$ and $1$ , the function returns a real value in the range $[0,\pi]$ . For $z$ less than $-1$ the result has a real part of $\pi$ and a negative imaginary part. For $z$ greater than $1$ the result is purely imaginary and positive.

cml_complex_t cml_complex_atan(cml_complex_t z)¶: This function returns the complex arctangent of the complex number z, $\arctan(z)$ . The branch cuts are on the imaginary axis, below $-i$ and above $i$ .

cml_complex_t cml_complex_asec(cml_complex_t z)¶: This function returns the complex arcsecant of the complex number z, $\arcsec(z) = \arccos(1/z)$ .

cml_complex_t cml_complex_asec_real(double z)¶: This function returns the complex arcsecant of the real number z, $\arcsec(z) = \arccos(1/z)$ .

cml_complex_t cml_complex_acsc(cml_complex_t z)¶: This function returns the complex arccosecant of the complex number z, $\arccsc(z) = \arcsin(1/z)$ .

cml_complex_t cml_complex_acsc_real(double z)¶: This function returns the complex arccosecant of the real number z, $\arccsc(z) = \arcsin(1/z)$ .

cml_complex_t cml_complex_acot(cml_complex_t z)¶: This function returns the complex arccotangent of the complex number z, $\arccot(z) = \arctan(1/z)$ .

Complex Hyperbolic Functions¶

cml_complex_t cml_complex_sinh(cml_complex_t z)¶: This function returns the complex hyperbolic sine of the complex number z, $\sinh(z) = (\exp(z) - \exp(-z))/2$ .

cml_complex_t cml_complex_cosh(cml_complex_t z)¶: This function returns the complex hyperbolic cosine of the complex number z, $\cosh(z) = (\exp(z) + \exp(-z))/2$ .

cml_complex_t cml_complex_tanh(cml_complex_t z)¶: This function returns the complex hyperbolic tangent of the complex number z, $\tanh(z) = \sinh(z)/\cosh(z)$ .

cml_complex_t cml_complex_sech(cml_complex_t z)¶: This function returns the complex hyperbolic secant of the complex number z, $\sech(z) = 1/\cosh(z)$ .

cml_complex_t cml_complex_csch(cml_complex_t z)¶: This function returns the complex hyperbolic cosecant of the complex number z, $\csch(z) = 1/\sinh(z)$ .

cml_complex_t cml_complex_coth(cml_complex_t z)¶: This function returns the complex hyperbolic cotangent of the complex number z, $\coth(z) = 1/\tanh(z)$ .

Inverse Complex Hyperbolic Functions¶

cml_complex_t cml_complex_asinh(cml_complex_t z)¶: This function returns the complex hyperbolic arcsine of the complex number z, $\arcsinh(z)$ . The branch cuts are on the imaginary axis, below $-i$ and above $i$ .

cml_complex_t cml_complex_acosh(cml_complex_t z)¶: This function returns the complex hyperbolic arccosine of the complex number z, $\arccosh(z)$ . The branch cut is on the real axis, less than $1$ . Note that in this case we use the negative square root in formula 4.6.21 of Abramowitz & Stegun giving $\arccosh(z)=\log(z-\sqrt{z^2-1})$ .

cml_complex_t cml_complex_acosh_real(double z)¶: This function returns the complex hyperbolic arccosine of the real number z, $\arccosh(z)$ .

cml_complex_t cml_complex_atanh(cml_complex_t z)¶: This function returns the complex hyperbolic arctangent of the complex number z, $\arctanh(z)$ . The branch cuts are on the real axis, less than $-1$ and greater than $1$ .

cml_complex_t cml_complex_atanh_real(double z)¶: This function returns the complex hyperbolic arctangent of the real number z, $\arctanh(z)$ .

cml_complex_t cml_complex_asech(cml_complex_t z)¶: This function returns the complex hyperbolic arcsecant of the complex number z, $\arcsech(z) = \arccosh(1/z)$ .

cml_complex_t cml_complex_acsch(cml_complex_t z)¶: This function returns the complex hyperbolic arccosecant of the complex number z, $\arccsch(z) = \arcsinh(1/z)$ .

cml_complex_t cml_complex_acoth(cml_complex_t z)¶: This function returns the complex hyperbolic arccotangent of the complex number z, $\arccoth(z) = \arctanh(1/z)$ .