Difference between revisions of "Conjugation"

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<ref>http://en.wikipedia.org/wiki/Inner_automorphism</ref> or
 
<ref>http://en.wikipedia.org/wiki/Inner_automorphism</ref> or
 
[[Topological conjugacy]]
  
[[Topological conjugacy]]
−
<ref>http://en.wikipedia.org/wiki/Topological conjugacy</ref>. This conjugation should be specified with some function $P$; it is assumed that the inverse function $Q=P^{-1}$ exists. Then, for some function $f$, its conjugate $g$ is expressed with
 +
<ref>http://en.wikipedia.org/wiki/Topological conjugacy</ref>. This conjugation should be specified with some function \(P\); it is assumed that the inverse function \(Q=P^{-1}\) exists. Then, for some function \(f\), its conjugate \(g\) is expressed with
   
−
$(1) ~ ~ ~
 +
\((1) ~ ~ ~
 
g(z)=P(f(Q(z)))=P\circ f\circ Q (z)
  
g(z)=P(f(Q(z)))=P\circ f\circ Q (z)
  +
\)
−
$
  
   
−
In general, no specific properties of the transform function $P$ (except existing of the inverse funcion) are assumed. Wikipedia suggests the following illustration of the application of function $P$ to the argument of the function and the inverse function $Q$ to the result:"raise umbrella, walk through rain, lower umbrella" has a different result from just "walk through rain". However in some cases, the operations $P$ and $f$ commute, and $P$ can be "drawn through" function $f$, for example,
+
In general, no specific properties of the transform function \(P\) (except existing of the inverse funcion) are assumed. Wikipedia suggests the following illustration of the application of function \(P\) to the argument of the function and the inverse function \(Q\) to the result:"raise umbrella, walk through rain, lower umbrella" has a different result from just "walk through rain". However in some cases, the operations \(P\) and \(f\) commute, and \(P\) can be "drawn through" function \(f\), for example,
 
"take off left glove, take off right glove, put on left glove" may have the same effect as "take off right glove only".
  
"take off left glove, take off right glove, put on left glove" may have the same effect as "take off right glove only".
   
−
Other types of the conjugation specified below are just special cases of the expression (1) for some specific $P$ and $Q$.
 +
Other types of the conjugation specified below are just special cases of the expression (1) for some specific \(P\) and \(Q\).
   
 
==Complex conjugation==
  
==Complex conjugation==
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The result of operation is also [[complex number]].
 
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 $z^*$.
 +
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,
  
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 $a\times (b^*)$
 +
id est, performed before infix operation. For example, \(ab^*\) is interpreted as \(a\times (b^*)\)
−
but not as $(ab)^*$.
 +
but not as \((ab)^*\).
   
−
For complex number $z=\Re(z) + \mathrm i \Im(z)$, its complex conjugation is $z^*=\Re(z) - \mathrm i \Im(z)$
 +
For complex number \(z=\Re(z) + \mathrm i \Im(z)\), its complex conjugation is \(z^*=\Re(z) - \mathrm i \Im(z)\)
   
 
In [[programming language]]s, the special name for the operation of complex conjugation is required, while the
  
In [[programming language]]s, 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.
 +
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 [[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 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.)
  
(In [[Mathematica]], the argument of a function is always written in rectangular parenthesis.)
   
−
With complex conjugation, operation [[square of modulus]], id est, $z\mapsto |z|^2$,
 +
With complex conjugation, operation [[square of modulus]], id est, \(z\mapsto |z|^2\),
−
can be expressed as $z z^*$.
 +
can be expressed as \(z z^*\).
   
−
Similarly, the [[Real part]] and the [[Imaginary part]] of a number $z$ can be expressed with
+
Similarly, the [[Real part]] and the [[Imaginary part]] of a number \(z\) can be expressed with
   
−
$\Re(z)=\frac{1}{2}(z+z^*)$ ,
+
\(\Re(z)=\frac{1}{2}(z+z^*)\) ,
   
−
$\Im(z)=\frac{1}{2}(z-z^*)$
 +
\(\Im(z)=\frac{1}{2}(z-z^*)\)
   
 
Complex conjugation is not a [[holomorphic function]].
  
Complex conjugation is not a [[holomorphic function]].
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Linear conjugation is special kind of conjugation (1),
  
Linear conjugation is special kind of conjugation (1),
−
characterized in that, that $P$ is linear function; $P(z)=a+bz$ for some numbers $a$ and $b$, then
 +
characterized in that, that \(P\) is linear function; \(P(z)=a+bz\) for some numbers \(a\) and \(b\), then
−
$P^{-1}(z)=b^{-1}\times (z-a)=\frac{1}{b}(z-a)=(z-a)/b$
 +
\(P^{-1}(z)=b^{-1}\times (z-a)=\frac{1}{b}(z-a)=(z-a)/b\)
   
−
Often, it is assumed, that $a$ and $b$ are just [[complex number]]s, and $b\ne 0$.
 +
Often, it is assumed, that \(a\) and \(b\) are just [[complex number]]s, and \(b\ne 0\).
−
Then, linear conjugation $f$ of a function $F$ can be written as follows:
 +
Then, linear conjugation \(f\) of a function \(F\) can be written as follows:
   
−
$f(z)=a+b~ F(b^{-1} \times (z-a))$
 +
\(f(z)=a+b~ F(b^{-1} \times (z-a))\)
   
−
Character $\times$ is used above to avoid confusion with the case when $b$ denotes some function and
 +
Character \(\times\) is used above to avoid confusion with the case when \(b\) denotes some function and
−
$b^-1$ is the inverse function <ref>About the notations:
+
\(b^-1\) is the inverse function <ref>About the notations:
−
For [[barbarian]]s, 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 $z-a$.
 +
For [[barbarian]]s, 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 <nomathjax>
 +
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 <nomathjax>
 
\times</nomathjax> and does not require its searching in the menu's Labyrinth.
  
\times</nomathjax> and does not require its searching in the menu's Labyrinth.
 
</ref>.
  
</ref>.
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==Complex conjugation of a function==
  
==Complex conjugation of a function==
   
−
For any function $F$, its [[complex conjugation]] or [[complex conjugate]] is function $G$ such that
+
For any function \(F\), its [[complex conjugation]] or [[complex conjugate]] is function \(G\) such that
   
−
$G(z)=F^*(z)= \Big(F(z^*)\Big)^*$
 +
\(G(z)=F^*(z)= \Big(F(z^*)\Big)^*\)
   
−
For a function $F$, expression
 +
For a function \(F\), expression
   
−
$F^*(z)$
 +
\(F^*(z)\)
   
 
should not be confused with
 
should not be confused with
   
−
$F(z)^*$
 +
\(F(z)^*\)
   
 
nor with
  
nor with
   
−
$F(z^*)$
 +
\(F(z^*)\)
   
 
At the use of the superscript after the name of a function, the special care is required:
  
At the use of the superscript after the name of a function, the special care is required:
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if the character of is just prime, then, perhaps, it denotes derivative,
  
if the character of is just prime, then, perhaps, it denotes derivative,
   
−
$\displaystyle
 +
\(\displaystyle
−
F'(z)=\lim_{t\rightarrow 0} \frac{F(z+t)-F(z)}{t}$
 +
F'(z)=\lim_{t\rightarrow 0} \frac{F(z+t)-F(z)}{t}\)
   
 
If the character is number, it indicates the iteration number;
  
If the character is number, it indicates the iteration number;
−
for example, $F^2(z)=F(F(z))$.
 +
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.
  
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.
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==Self-inverse==
  
==Self-inverse==
−
Complex conjugation is self-inverse, id est, $f^{**}=f$.
 +
Complex conjugation is self-inverse, id est, \(f^{**}=f\).
   
 
This applies both to functions and to numbers.
  
This applies both to functions and to numbers.
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==Hermitian conjugation==
  
==Hermitian conjugation==
   
−
$|\psi\rangle^\dagger = \langle \psi |$
 +
\(|\psi\rangle^\dagger = \langle \psi |\)
   
 
==References==
  
==References==
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[[Complex number]] ,
  
[[Complex number]] ,
 
[[Operator]] ,
  
[[Operator]] ,
−
[[Mathematical notations]]
 +
[[Mathematical notations]],
  +
[[Quantum mechanics]]
   
 
[[Category:Mathematical notation]]
  
[[Category:Mathematical notation]]
 
[[Category:Conjugation]]
  
[[Category:Conjugation]]
 
[[Category:Holomorphic function]]
  
[[Category:Holomorphic function]]
  +
[[Category:Quantum mechanics]]

Latest revision as of 18:46, 30 July 2019

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

In addition,there are more meanings of verb conjugation [1], that are not yet used in TORI.

Inner automorphism

By default, in TORI, Conjugation refers to the inner automorphism [2] or Topological conjugacy [3]. This conjugation should be specified with some function \(P\); it is assumed that the inverse function \(Q=P^{-1}\) exists. Then, for some function \(f\), its conjugate \(g\) is expressed with

\((1) ~ ~ ~ g(z)=P(f(Q(z)))=P\circ f\circ Q (z) \)

In general, no specific properties of the transform function \(P\) (except existing of the inverse funcion) are assumed. Wikipedia suggests the following illustration of the application of function \(P\) to the argument of the function and the inverse function \(Q\) to the result:"raise umbrella, walk through rain, lower umbrella" has a different result from just "walk through rain". However in some cases, the operations \(P\) and \(f\) commute, and \(P\) can be "drawn through" function \(f\), for example, "take off left glove, take off right glove, put on left glove" may have the same effect as "take off right glove only".

Other types of the conjugation specified below are just special cases of the expression (1) for some specific \(P\) and \(Q\).

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 part 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.

Linear conjugation

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

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:

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

Character \(\times\) is used above to avoid confusion with the case when \(b\) denotes some function and \(b^-1\) is the inverse function [4].

Complex conjugation of a function

For any function \(F\), its complex conjugation or complex conjugate is function \(G\) such that

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

For a function \(F\), expression

\(F^*(z)\)

should not be confused with

\(F(z)^*\)

nor with

\(F(z^*)\)

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,

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

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.

Hermitian conjugation

\(|\psi\rangle^\dagger = \langle \psi |\)

References

  1. ↑ http://en.wikipedia.org/wiki/Conjugation
  2. ↑ http://en.wikipedia.org/wiki/Inner_automorphism
  3. ↑ http://en.wikipedia.org/wiki/Topological conjugacy
  4. ↑ About the notations: 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.

Keywords

Holomorphic function , Complex number , Operator , Mathematical notations, Quantum mechanics

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