Conjugation

Conjugation is certain type of transformation of mathematical objects. There are several kinds of conjugation.

Complex conjugation

Complex conjugation is function defined at the set of complex numbers.

The result of operation is also complex number.

Often, the complex conjugation is denoted with asterisk * as the superscript; conjugation of a complex number $z$ is written as $z^*$.

If the order of operations is not indicated with parenthesis, the complex conjugation has highest priority, id est, performed before infix operation. For example, $ab^*$ is interpreted as $a\times (b^*)$ but not as $(ab)^*$.

For complex number $z=\Re(z) + \mathrm i \Im(z)$, its complex conjugation is $z^*=\Re(z) - \mathrm i \Im(z)$

In programming languages, the special name for the operation of complex conjugation is required, while the symbol asterisk is used to denote multiplication (instead of symbol $\times$, which is not ascii character and, therefore, not recommended at all for the programming code.

In C++, the complex conjugation is denoted with built-in finction conj.

In Mathematica, the complex conjugation is denoted with built-in finction Conjugate; the complex conjugation of number $z$ is written as Conjugate[z] . (In Mathematica, the argument of a function is always written in rectangular parenthesis.)

With complex conjugation, operation square of modulus, id est, $z\mapsto |z|^2$, can be expressed as $z z^*$.

Similarly, the Real part and the imaginary parts of a number $z$ can be expressed with

$\Re(z)=\frac{1}{2}(z+z^*)$ ,

$\Im(z)=\frac{1}{2}(z-z^*)$

Complex conjugation is not a holomorphic function.

Conjugation of function

Term Conjugate also may refer to some transformation of function, that affect both the argument of the function and its value.

Function $G$ is qualified as conjugation of function $F$, if there exist function $p$ such that

$G=p\circ F\circ p^{-1}$

is est, $G(z)=p\Big(F\big(p^{-1}(z)\big)\Big)$ for some wide range of values of $z$.

Linear conjugation of a function

Linear conjugation is special kind of conjugation of the precious section, characterized in that, that $p$ is linear function; $p(z)=a+bz$ for some numbers $a$ and $b$, then $p^{-1}(z)=b^{-1}\times (z-a)$

Often, it is assumed, that $a$ and $b$ are just complex numbers, and $b\ne 0$. Then, linear conjugation $f$ of a function $F$ can be written as follows:

$f(z)=a+b~ F(b^{-1} \times (z-a))$

For barbarians, who use the same identifier as variable and as name of function, it is better to indicate multiplication with symbol $\times$; over-vice, in expression $b^{-1} (z-a)$ one needs to guess, should function $b^{-1}$ be evaluated with argument $z-a$ or the number $b^{-1}$ should be multiplied by expression $z-a$. In the similar way, colleagues, who use asterisk * instead of multiplication symbol $\times$, in scientific manuscripts, are recommended to install some Latex, where character $\times$ can be generated with built-in command \times and does not require its searching in the menu's Labyrinth.

Complex conjugation of a function

For any function $F$, its complex conjugation or complex conjugate is function $G$ such that

$G(z)= \Big(F(z^*)\Big)^*$

However, at the use of the superscript after the name of a function, the special care is required:

if the character of is just prime, then, perhaps, it denotes derivative,

$\fisplaystye F'(z)=\lim_{t\rightarrow 0} \frac{F(z+t)-F(z)}{t}$

If the character is number, it indicates the iteration number; for example, $F^2(z)=F(F(z))$.

Use in superscripts of some object, that may mean prime, and may mean a number, and may mean asterisk, is not recommended, as it easy may lead to confusion. In particular, this refer to a case, when the number of derivative is some number, that is either too long (and it is difficult to write the corresponding amount of primes) or not yet specified. In this case, the parenthesis are used; so, if the whole superscript appear in parenthesis, then, perhaps, this supersctipt indicate derivative and not iteration.

Self-inverse

Complex conjugation is self-inverse, id est, $f^{**}=f$. this applies both to functions and to numbers.

Keywords

Holomorphic function , Complex number , Operator , Mathematical notations