Cauchy-Riemann equations: Difference between revisions
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In [[complex analysis]], the '''Cauchy-Riemann equations''' are one of the of the basic objects of the theory: they are a system of <var>2n</var> [[partial differential equation]]s, where <var>n</var> is the [[Dimension (vector space)|dimension]] of the [[Complex space|complex ambient space]] ℂ''<sup>n</sup>'' considered. Precisely, their [[Homogeneous equation|homogeneous form]] express a necessary and sufficient condition between the [[Real part|real]] and [[imaginary part]] of a given [[Complex number|complex valued]] function of <var>2n</var> [[real number|real]] [[variable]]s to be a [[Holomorphic function|holomorphic one]]. They are named after [[Augustin-Louis Cauchy]] and [[Bernhard Riemann]] who were the first ones to study and use such equations as a mathematical object "per se", creating a new theory. These equations are sometimes referred as '''Cauchy-Riemann conditions''' or '''Cauchy-Riemann system''': the [[partial differential operator]] appearing on the left side of these equations is usually called the '''Cauchy-Riemann operator'''. | |||
In [[complex analysis]], the '''Cauchy-Riemann equations''' are one of the of the basic objects of the theory: they are a system of <var>2n</var> [[partial differential equation]]s, where <var>n</var> is the [[Dimension (vector space)|dimension]] of the [[Complex space|complex ambient space]] ℂ''<sup>n</sup>'' considered. Precisely, their [[Homogeneous equation|homogeneous form]] express a necessary and sufficient condition between the [[Real part|real]] and [[imaginary part]] of a [[Complex number|complex valued]] function of <var>2n</var> [[real number|real]] [[variable]]s | |||
== Historical note == | == Historical note == | ||
The first introduction and use of the Cauchy-Riemann equations for <var>n</var>=1 is due to [[Jean Le-Rond D'Alembert]] in his 1752 work on [[Fluid dynamics|hydrodynamics]]: this connection between [[complex analysis]] and hydrodynamics is made explicit in classical [[treatise]]s of the latter subject, such as [[Horace Lamb]]'s monumental work. | The first introduction and use of the Cauchy-Riemann equations for <var>n</var>=1 is due to [[Jean Le-Rond D'Alembert]] in his 1752 work on [[Fluid dynamics|hydrodynamics]]<ref>See {{harvnb|D'Alembert|1752}}.</ref>: this connection between [[complex analysis]] and hydrodynamics is made explicit in classical [[treatise]]s of the latter subject, such as [[Horace Lamb]]'s monumental work<ref>See {{harvnb|Lamb|1932}}.</ref>. | ||
== Formal definition == | == Formal definition == | ||
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Let <var>f</var>(<var>x</var>, <var>y</var>) = <var>u</var>(<var>x</var>, <var>y</var>) + <var>i</var><var>v</var>(<var>x</var>, <var>y</var>) a [[Complex number|complex valued]] [[differentiable function]]. Then <var>f</var> satisfies the homogeneous Cauchy-Riemann equations if and only if | Let <var>f</var>(<var>x</var>, <var>y</var>) = <var>u</var>(<var>x</var>, <var>y</var>) + <var>i</var><var>v</var>(<var>x</var>, <var>y</var>) a [[Complex number|complex valued]] [[differentiable function]]. Then <var>f</var> satisfies the homogeneous Cauchy-Riemann equations if and only if | ||
:<math>\left\{ | :<math>\left\{ | ||
\begin{ | \begin{align} | ||
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \\ | \frac{\partial u}{\partial x} &= \frac{\partial v}{\partial y} \\ | ||
\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\\ | \frac{\partial u}{\partial y} &= -\frac{\partial v}{\partial x} \\ | ||
\end{ | \end{align}\right. | ||
\right. | |||
</math> | </math> | ||
Using [[Wirtinger derivatives]] these equation can be written in the following more compact form: | Using [[Wirtinger derivatives]] these equation can be written in the following more compact form: | ||
:<math>\frac{\partial f}{\partial\bar{z}}=0</math> | ::<math>\frac{\partial f}{\partial\bar{z}}=0</math> | ||
===The Cauchy-Riemann equations in ℂ''<sup>n</sup>'' (<var>n</var>>1)=== | ===The Cauchy-Riemann equations in ℂ''<sup>n</sup>'' (<var>n</var>>1)=== | ||
Let <var>f</var>(<var>x<sub>1</sub></var>, <var>y<sub>1</sub></var>,...,<var>x<sub>n</sub></var>, <var>y<sub>n</sub></var>) = <var>u</var>(<var>x<sub>1</sub></var>, <var>y<sub>1</sub></var>,...,<var>x<sub>n</sub></var>, <var>y<sub>n</sub></var>) + <var>i</var><var>v</var>(<var>x<sub>1</sub></var>, <var>y<sub>1</sub></var>,...,<var>x<sub>n</sub></var>, <var>y<sub>n</sub></var>) a [[Complex number|complex valued]] [[differentiable function]]. Then <var>f</var> satisfies the homogeneous Cauchy-Riemann equations if and only if | Let <var>f</var>(<var>x<sub>1</sub></var>, <var>y<sub>1</sub></var>,...,<var>x<sub>n</sub></var>, <var>y<sub>n</sub></var>) = <var>u</var>(<var>x<sub>1</sub></var>, <var>y<sub>1</sub></var>,...,<var>x<sub>n</sub></var>, <var>y<sub>n</sub></var>) + <var>i</var><var>v</var>(<var>x<sub>1</sub></var>, <var>y<sub>1</sub></var>,...,<var>x<sub>n</sub></var>, <var>y<sub>n</sub></var>) a [[Complex number|complex valued]] [[differentiable function]]. Then <var>f</var> satisfies the homogeneous Cauchy-Riemann equations if and only if | ||
:<math>\left\{ | :<math>\left\{ | ||
\begin{ | \begin{align} | ||
\frac{\partial u}{\partial x_1} = \frac{\partial v}{\partial y_1} \\ | \frac{\partial u}{\partial x_1} &= \frac{\partial v}{\partial y_1} \\ | ||
\frac{\partial u}{\partial y_1} = -\frac{\partial v}{\partial x_1}\\ | \frac{\partial u}{\partial y_1} &= -\frac{\partial v}{\partial x_1}\\ | ||
&\vdots\\ | |||
\frac{\partial u}{\partial x_n} = \frac{\partial v}{\partial y_n} \\ | \frac{\partial u}{\partial x_n} &= \frac{\partial v}{\partial y_n} \\ | ||
\frac{\partial u}{\partial y_n} = -\frac{\partial v}{\partial x_n} | \frac{\partial u}{\partial y_n} &= -\frac{\partial v}{\partial x_n} | ||
\end{ | \end{align} | ||
\right. | \right. | ||
</math> | </math> | ||
Again, using [[Wirtinger derivatives]] this system of equation can be written in the following more compact form: | Again, using [[Wirtinger derivatives]] this system of equation can be written in the following more compact form: | ||
:<math>\left\{ | :<math>\left\{ | ||
\begin{ | \begin{align} | ||
\frac{\partial f}{\partial\bar{z_1}} = 0 \\ | \frac{\partial f}{\partial\bar{z_1}} &= 0 \\ | ||
&\vdots\\ | |||
\frac{\partial f}{\partial\bar{z_n}} = 0 | \frac{\partial f}{\partial\bar{z_n}} &= 0 | ||
\end{ | \end{align} | ||
\right. | \right. | ||
</math> | </math> | ||
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===Notations for the case <var>n</var>>1 === | ===Notations for the case <var>n</var>>1 === | ||
In the [[France|French]], [[Italy|Italian]] and [[Russia|Russian]] literature on the subject, the [[Dimension (mathematics)|multi-dimensional]] Cauchy-Riemann system is often identified with the following notation: | In the [[France|French]], [[Italy|Italian]] and [[Russia|Russian]] literature on the subject, the [[Dimension (mathematics)|multi-dimensional]] Cauchy-Riemann system is often identified with the following notation: | ||
:<math>\bar{\partial}f</math> | ::<math>\bar{\partial}f</math> | ||
The Anglo-Saxon literature ([[England|English]] and [[United States of America|North American]]) uses the same symbol for the complex [[differential form]] related to the same operator. | The Anglo-Saxon literature ([[England|English]] and [[United States of America|North American]]) uses the same symbol for the complex [[differential form]] related to the same operator. | ||
== Notes == | |||
{{reflist|2}} | |||
== References == | == References == | ||
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*{{Citation | *{{Citation | ||
| last = Lamb | | last = Lamb | ||
| first = Horace | | first = Sir Horace | ||
| author-link = Horace Lamb | | author-link = Horace Lamb | ||
| year = 1932 | | year = 1932 | ||
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| oclc = | | oclc = | ||
| url = | | url = | ||
}}. | }}.[[Category:Suggestion Bot Tag]] | ||
[[Category: | |||
Latest revision as of 17:00, 25 July 2024
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In complex analysis, the Cauchy-Riemann equations are one of the of the basic objects of the theory: they are a system of 2n partial differential equations, where n is the dimension of the complex ambient space ℂn considered. Precisely, their homogeneous form express a necessary and sufficient condition between the real and imaginary part of a given complex valued function of 2n real variables to be a holomorphic one. They are named after Augustin-Louis Cauchy and Bernhard Riemann who were the first ones to study and use such equations as a mathematical object "per se", creating a new theory. These equations are sometimes referred as Cauchy-Riemann conditions or Cauchy-Riemann system: the partial differential operator appearing on the left side of these equations is usually called the Cauchy-Riemann operator.
Historical note
The first introduction and use of the Cauchy-Riemann equations for n=1 is due to Jean Le-Rond D'Alembert in his 1752 work on hydrodynamics[1]: this connection between complex analysis and hydrodynamics is made explicit in classical treatises of the latter subject, such as Horace Lamb's monumental work[2].
Formal definition
In the following text, it is assumed that ℂn≡ℝ2n, identifying the points of the euclidean spaces on the complex and real fields as follows
The subscripts are omitted when n=1.
The Cauchy-Riemann equations in ℂ (n=1)
Let f(x, y) = u(x, y) + iv(x, y) a complex valued differentiable function. Then f satisfies the homogeneous Cauchy-Riemann equations if and only if
Using Wirtinger derivatives these equation can be written in the following more compact form:
The Cauchy-Riemann equations in ℂn (n>1)
Let f(x1, y1,...,xn, yn) = u(x1, y1,...,xn, yn) + iv(x1, y1,...,xn, yn) a complex valued differentiable function. Then f satisfies the homogeneous Cauchy-Riemann equations if and only if
Again, using Wirtinger derivatives this system of equation can be written in the following more compact form:
Notations for the case n>1
In the French, Italian and Russian literature on the subject, the multi-dimensional Cauchy-Riemann system is often identified with the following notation:
The Anglo-Saxon literature (English and North American) uses the same symbol for the complex differential form related to the same operator.
Notes
- ↑ See D'Alembert 1752.
- ↑ See Lamb 1932.
References
- Burckel, Robert B. (1979), An Introduction to Classical Complex Analysis. Vol. 1, Lehrbucher und Monographien aus dem Gebiete der exakten Wissenschaften. Mathematische Reihe, vol. 64, Basel–Stuttgart–New York–Tokyo: Birkhäuser Verlag, ISBN 3-7643-0989-X [e].
- D'Alembert, Jean Le-Rond (1752), Essai d'une nouvelle théorie de la résistance des fluides, Paris: David [e] (in French).
- Hörmander, Lars (1990), An Introduction to Complex Analysis in Several Variables, North–Holland Mathematical Library, vol. 7 (3rd (Revised) ed.), Amsterdam–London–New York–Tokyo: North-Holland, Zbl 0685.32001, ISBN 0-444-88446-7 [e].
- Lamb, Sir Horace (1932), Hydrodynamics, Cambridge Mathematical Library (1995 paperback reprint of the 6th ed.), Cambridge: Cambridge University Press, Zbl 0828.01012, ISBN 0-521-45868-4 [e].