Complex Conjugate Of A Function

Fundamental operation on circuitous numbers

Geometric representation (Argand diagram) of ${\displaystyle z}$ and its conjugate ${\displaystyle {\overline {z}}}$ in the complex plane. The complex cohabit is found by reflecting ${\displaystyle z}$ across the existent axis.

In mathematics, the complex conjugate of a complex number is the number with an equal existent role and an imaginary part equal in magnitude only opposite in sign. That is, (if ${\displaystyle a}$ and ${\displaystyle b}$ are real, then) the complex cohabit of ${\displaystyle a+bi}$ is equal to ${\displaystyle a-bi.}$ The complex conjugate of ${\displaystyle z}$ is often denoted as ${\displaystyle {\overline {z}}}$ or ${\displaystyle z^{*}}$ .

In polar grade, the conjugate of ${\displaystyle re^{i\varphi }}$ is ${\displaystyle re^{-i\varphi }.}$ This tin can exist shown using Euler's formula.

The production of a complex number and its conjugate is a real number: ${\displaystyle a^{2}+b^{2}}$  (or ${\displaystyle r^{2}}$ in polar coordinates).

If a root of a univariate polynomial with real coefficients is complex, and then its complex cohabit is also a root.

Notation [edit]

The complex cohabit of a complex number ${\displaystyle z}$ is written as ${\displaystyle {\overline {z}}}$ or ${\displaystyle z^{*}.}$ The first annotation, a vinculum, avoids confusion with the notation for the cohabit transpose of a matrix, which tin can exist idea of as a generalization of the circuitous conjugate. The second is preferred in physics, where dagger (†) is used for the conjugate transpose, equally well equally electrical engineering and computer applied science, where bar notation can be confused for the logical negation ("Non") Boolean algebra symbol, while the bar note is more common in pure mathematics. If a complex number is represented as a ${\displaystyle 2\times 2}$ matrix, the notations are identical.^{[ clarification needed ]}

Properties [edit]

The following properties utilise for all complex numbers ${\displaystyle z}$ and ${\displaystyle due west,}$ unless stated otherwise, and can be proved past writing ${\displaystyle z}$ and ${\displaystyle west}$ in the form ${\displaystyle a+bi.}$

For any ii complex numbers, conjugation is distributive over improver, subtraction, multiplication and division:^{[ref 1]}

$${\displaystyle {\begin{aligned}{\overline {z+w}}&={\overline {z}}+{\overline {west}},\\{\overline {z-w}}&={\overline {z}}-{\overline {west}},\\{\overline {zw}}&={\overline {z}}\;{\overline {due west}},\quad {\text{and}}\\{\overline {\left({\frac {z}{w}}\right)}}&={\frac {\overline {z}}{\overline {w}}},\quad {\text{if }}w\neq 0.\finish{aligned}}}$$

A complex number is equal to its complex conjugate if its imaginary part is zero, that is, if the number is real. In other words, real numbers are the only stock-still points of conjugation.

Conjugation does not change the modulus of a complex number: ${\displaystyle \left|{\overline {z}}\right|=|z|.}$

Conjugation is an involution, that is, the conjugate of the conjugate of a circuitous number ${\displaystyle z}$ is ${\displaystyle z.}$ In symbols, ${\displaystyle {\overline {\overline {z}}}=z.}$ ^{[ref 1]}

The production of a complex number with its cohabit is equal to the square of the number'southward modulus:

$${\displaystyle z{\overline {z}}={\left|z\correct|}^{2}.}$$

This allows like shooting fish in a barrel computation of the multiplicative changed of a circuitous number given in rectangular coordinates:

$${\displaystyle z^{-1}={\frac {\overline {z}}{{\left|z\right|}^{two}}},\quad {\text{ for all }}z\neq 0.}$$

Conjugation is commutative under limerick with exponentiation to integer powers, with the exponential function, and with the natural logarithm for nonzero arguments:

$${\displaystyle {\overline {z^{n}}}=\left({\overline {z}}\right)^{n},\quad {\text{ for all }}n\in \mathbb {Z} }$$

^{[note 1]}

$${\displaystyle \exp \left({\overline {z}}\correct)={\overline {\exp(z)}}}$$

$${\displaystyle \ln \left({\overline {z}}\correct)={\overline {\ln(z)}}{\text{ if }}z{\text{ is not-zip }}}$$

If ${\displaystyle p}$ is a polynomial with real coefficients and ${\displaystyle p(z)=0,}$ so ${\displaystyle p\left({\overline {z}}\right)=0}$ as well. Thus, non-real roots of real polynomials occur in circuitous cohabit pairs (run into Complex conjugate root theorem).

In general, if ${\displaystyle \varphi }$ is a holomorphic part whose restriction to the real numbers is existent-valued, and ${\displaystyle \varphi (z)}$ and ${\displaystyle \varphi ({\overline {z}})}$ are defined, then

$${\displaystyle \varphi \left({\overline {z}}\right)={\overline {\varphi (z)}}.\,\!}$$

The map ${\displaystyle \sigma (z)={\overline {z}}}$ from ${\displaystyle \mathbb {C} }$ to ${\displaystyle \mathbb {C} }$ is a homeomorphism (where the topology on ${\displaystyle \mathbb {C} }$ is taken to be the standard topology) and antilinear, if ane considers ${\displaystyle \mathbb {C} }$ as a complex vector space over itself. Even though it appears to be a well-behaved function, it is non holomorphic; it reverses orientation whereas holomorphic functions locally preserve orientation. It is bijective and uniform with the arithmetical operations, and hence is a field automorphism. As it keeps the existent numbers stock-still, information technology is an element of the Galois group of the field extension ${\displaystyle \mathbb {C} /\mathbb {R} .}$ This Galois group has only two elements: ${\displaystyle \sigma }$ and the identity on ${\displaystyle \mathbb {C} .}$ Thus the but two field automorphisms of ${\displaystyle \mathbb {C} }$ that leave the real numbers fixed are the identity map and complex conjugation.

Use as a variable [edit]

Once a complex number ${\displaystyle z=x+yi}$ or ${\displaystyle z=re^{i\theta }}$ is given, its conjugate is sufficient to reproduce the parts of the ${\displaystyle z}$ -variable:

Furthermore, ${\displaystyle {\overline {z}}}$ can exist used to specify lines in the plane: the gear up

$${\displaystyle \left\{z:z{\overline {r}}+{\overline {z}}r=0\right\}}$$

is a line through the origin and perpendicular to ${\displaystyle {r},}$ since the existent role of ${\displaystyle z\cdot {\overline {r}}}$ is zip simply when the cosine of the angle between ${\displaystyle z}$ and ${\displaystyle {r}}$ is cypher. Similarly, for a fixed complex unit ${\displaystyle u=east^{ib},}$ the equation

$${\displaystyle {\frac {z-z_{0}}{{\overline {z}}-{\overline {z_{0}}}}}=u^{2}}$$

determines the line through ${\displaystyle z_{0}}$ parallel to the line through 0 and ${\displaystyle u.}$

These uses of the conjugate of ${\displaystyle z}$ equally a variable are illustrated in Frank Morley'south book Inversive Geometry (1933), written with his son Frank Vigor Morley.

Generalizations [edit]

The other planar real unital algebras, dual numbers, and split-complex numbers are also analyzed using complex conjugation.

For matrices of complex numbers, ${\textstyle {\overline {\mathbf {AB} }}=\left({\overline {\mathbf {A} }}\right)\left({\overline {\mathbf {B} }}\right),}$ where ${\textstyle {\overline {\mathbf {A} }}}$ represents the element-by-element conjugation of ${\displaystyle \mathbf {A} .}$ ^{[ref 2]} Contrast this to the property ${\textstyle \left(\mathbf {AB} \right)^{*}=\mathbf {B} ^{*}\mathbf {A} ^{*},}$ where ${\textstyle \mathbf {A} ^{*}}$ represents the conjugate transpose of ${\textstyle \mathbf {A} .}$

Taking the conjugate transpose (or adjoint) of circuitous matrices generalizes complex conjugation. Even more general is the concept of adjoint operator for operators on (peradventure infinite-dimensional) complex Hilbert spaces. All this is subsumed past the *-operations of C*-algebras.

I may also define a conjugation for quaternions and carve up-quaternions: the cohabit of ${\textstyle a+bi+cj+dk}$ is ${\textstyle a-bi-cj-dk.}$

All these generalizations are multiplicative only if the factors are reversed:

$${\displaystyle {\left(zw\right)}^{*}=w^{*}z^{*}.}$$

Since the multiplication of planar existent algebras is commutative, this reversal is not needed there.

There is besides an abstract notion of conjugation for vector spaces ${\textstyle Five}$ over the complex numbers. In this context, whatever antilinear map ${\textstyle \varphi :5\to V}$ that satisfies

1. ${\displaystyle \varphi ^{2}=\operatorname {id} _{V}\,,}$ where ${\displaystyle \varphi ^{2}=\varphi \circ \varphi }$ and ${\displaystyle \operatorname {id} _{V}}$ is the identity map on ${\displaystyle V,}$
2. ${\displaystyle \varphi (zv)={\overline {z}}\varphi (five)}$ for all ${\displaystyle v\in V,z\in \mathbb {C} ,}$ and
3. ${\displaystyle \varphi \left(v_{1}+v_{2}\right)=\varphi \left(v_{ane}\right)+\varphi \left(v_{2}\correct)\,}$ for all ${\displaystyle v_{1}v_{two},\in V,}$

is chosen a complex conjugation, or a real construction. As the involution ${\displaystyle \varphi }$ is antilinear, it cannot be the identity map on ${\displaystyle Five.}$

Of course, ${\textstyle \varphi }$ is a ${\textstyle \mathbb {R} }$ -linear transformation of ${\textstyle V,}$ if 1 notes that every complex space ${\displaystyle V}$ has a real form obtained past taking the same vectors as in the original space and restricting the scalars to be real. The above properties actually define a existent structure on the complex vector infinite ${\displaystyle V.}$ ^[1]

One example of this notion is the cohabit transpose operation of complex matrices defined higher up. However, on generic complex vector spaces, in that location is no canonical notion of complex conjugation.

See also [edit]

• Absolute square
• Complex conjugate line – Operation in complex geometry
• Complex cohabit representation
• Complex cohabit vector space
• Composition algebra – Type of algebras, possibly non associative
• Conjugate (square roots)
• Hermitian part – Type of complex function
• Wirtinger derivatives – Concept in complex analysis

References [edit]

1. ^ ^{a b} Friedberg, Stephen; Insel, Arnold; Spence, Lawrence (2018), Linear Algebra (five ed.), ISBN978-0134860244 , Appendix D
2. ^ Arfken, Mathematical Methods for Physicists, 1985, pg. 201

note [edit]

1. ^ Encounter Exponentiation#Non-integer powers of complex numbers.

Bibliography [edit]

• Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988. ISBN 0-387-19078-iii. (antilinear maps are discussed in section 3.three).

1. ^ Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988, p. 29

Complex Conjugate Of A Function,

Source: https://en.wikipedia.org/wiki/Complex_conjugate

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