# Operations on complex numbers

SYNOPSIS

#include "zmatrix.h"
complex zmake(double real, double imag)
complex zconj(complex z)
complex zneg(complex z)
double  zabs(complex z)
complex zsub(complex z1, complex z2)
complex zmlt(complex z1, complex z2)
complex zinv(complex z)
complex zdiv(complex z1, complex z2)
complex zsqrt(complex z)
complex zexp(complex z)
complex zlog(complex z)


DESCRIPTION

These routines provide the basic operations on complex numbers. Complex numbers are represented by the complex data structure which is defined as

typedef struct  {  Real re, im;   } complex;

and the real part of complex z; is z.re and its imaginary part is z.im. Let $z = x+iy$. The routine zmake(real,imag) returns the complex number with real part real and imaginary part imag. The routine zconj(z) returns $\bar z = x - iy$ The routine zneg(z) returns $-z$. The routine zabs(z) returns $|z|=\sqrt{x^2+y^2}$. Note that it is done safely to avoid overflow if $|x|$ or $|y|$ is close to floating point limits. The routine zadd(z1,z2) returns $z_1+z_2$. The routine zsub(z1,z2) returns $z_1-z_2$. The routine zmlt(z1,z2) returns $z_1 z_2$. The routine zinv(z) returns $1/z$. An E_SING error is raised if $z=0$. The routine zdiv(z1,z2) returns $z_1/z_2$. An E_SING error is raised if $z_2=0$. The routine zsqrt(z) returns $\sqrt{z}$. The principle branch is used for a branch cut along the negative real axis, so the real part of $\sqrt{z}$ as computed is not negative. The routine zexp(z) returns $\exp(z)=e^{z}=e^x(\cos y+i\sin y)$. The routine zlog(z) returns $\log(z)$. The principle branch is used for a branch cut along the negative real axis, so the imaginary part of $\log(z)$ lies between or on $\pm\pi$.

EXAMPLE

To compute $\log(z+e^w)/\sqrt{1+z^2}$:

complex w, z, result;
......

where ONE is $1+0i$; ONE = zmake(1.0,0.0);.