Difference between revisions of "Cauchy-Riemann equations"
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| − | + | {{MSC|30-XX|32-XX}} | |
| + | {{TEX|done}} | ||
| − | + | Also known as Cauchy-Riemann conditions and D'Alembert-Euler conditions, they are the partial differential equations that must be satisfied by the real and imaginary parts of a complex-valued function $f$ of one (or several) complex variable so that $f$ is [[Holomorphic function|holomorphic]]. | |
| + | |||
| + | |||
| + | ===One complex variable=== | ||
| + | More precisely, assume $D\subset \mathbb C$ is some open set and $f: D \to \mathbb C$ a map with real and imaginary parts given by $u$ and $v$ (i.e. $f(z) = u (z) + i v (z)$). If we introduce the real variables $x,y$ so that $z= x+iy$, we can consider $u$ and $v$ as real functions on a domain in $\mathbb R^2$. If $u$ and $v$ are differentiable (in the real-variable sense), then they solve the Cauchy-Riemann equations if | ||
| + | \begin{equation}\label{e:CR} | ||
| + | \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \qquad | ||
| + | \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}\, . | ||
| + | \end{equation} | ||
| + | Such equations are equivalent to the fact that $f$ is complex-differentiable at every point $z_0\in D$, where its complex derivative is defined by | ||
| + | \[ | ||
| + | f' (z_0) = \lim_{w\in \mathbb C, w\to 0} \frac{f(z_0+w)-f(z_0)}{w}\, . | ||
| + | \] | ||
| + | It then turns out that | ||
| + | \begin{equation}\label{e:f'} | ||
| + | f' = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} | ||
| + | \end{equation} | ||
| + | (and evidently we can use \eqref{e:CR} to derive three other similar formulas). | ||
| + | |||
| + | ====Harmonicity and conjugacy==== | ||
| + | Any pair of solutions of the Cauchy-Riemann equations turn out to be infinitely differentiable and in fact [[Analytic function|analytic]]. Moreover they are also [[Conjugate harmonic functions|conjugate]] [[Harmonic function|harmonic functions]]. Viceversa if $u$ (or $v$) is a given harmonic function on a [[Simply-connected domain|simply connected]] open $D$, then there is a conjugate harmonic function to $u$, that is a $v$ satisfying \eqref{e:CR}. Such $v$ is unique up to addition of a constant. | ||
| + | |||
| + | ====Different systems of coordinates==== | ||
| + | The conditions \eqref{e:CR} can be written, equivalently, for any two orthogonal directions $s$ and $n$, with the same mutual orientation as the $x$- and $y$-axes, in the form: | ||
| + | \[ | ||
| + | \frac{\partial u}{\partial s} = \frac{\partial v}{\partial n} \qquad | ||
| + | \frac{\partial u}{\partial n} = - \frac{\partial v}{\partial s}\, . | ||
| + | \] | ||
| + | For example, in polar coordinates $r, \phi$, for $r\neq 0$ the Cauchy-Riemann equations read: | ||
| + | \[ | ||
| + | \frac{\partial u}{\partial r} = \frac{1}{r} \frac{\partial v}{\partial \phi} \qquad | ||
| + | \frac{1}{r} \frac{\partial u}{\partial \phi} = - \frac{\partial v}{\partial r}\, . | ||
| + | \] | ||
| + | |||
| + | ====$\frac{\partial}{\partial \bar{z}}$ and $\frac{\partial}{\partial z}$ operators==== | ||
| + | Defining the complex differential operators by | ||
| + | \[ | ||
| + | \frac{\partial}{\partial \bar{z}}=\frac{1}{2} \left(\frac{\partial}{\partial x} + i \frac{\partial}{\partial y}\right)\qquad\mbox{and}\qquad | ||
| + | \frac{\partial}{\partial z}=\frac{1}{2} \left(\frac{\partial}{\partial x} - i \frac{\partial}{\partial y}\right)\, , | ||
| + | \] | ||
| + | one can rewrite the Cauchy–Riemann equations \eqref{e:CR} as | ||
| + | \[ | ||
| + | \frac{\partial f}{\partial \bar{z}} = 0 | ||
| + | \] | ||
| + | and the identity \eqref{e:f'} as | ||
| + | \[ | ||
| + | f' = \frac{\partial f}{\partial z}\, . | ||
| + | \] | ||
| + | |||
| + | ====Conformality==== | ||
| + | The Cauchy-Riemann equations are equivalent to the fact that the map $(u,v): D \to \mathbb R^2$ is [[Conformal mapping|conformal]], i.e. it preserves the angles and (locally where it is injective) the orientation. Such condition can indeed be expressed at the differential level with the property that at each point $(x_0, y_0)$ the [[Jacobian|Jacobian matrix]] of $(u,v)$ is a multiple of a rotation. In turn this can be easily seen to be equivalent to \eqref{e:CR}. | ||
| + | |||
| + | ===Several complex variables=== | ||
| + | For analytic functions of several complex variables $z = (z_1, \ldots, z_n)$, with $z_k = x_k + iy_k$ , the Cauchy–Riemann equations is given by the following system of partial differential equations (overdetermined when $n>1$) for the functions | ||
| + | \begin{align*} | ||
| + | & u (x_1, \ldots, x_n, y_1, \ldots , y_n) = {\rm Re}\, f (x_1+iy_1, \ldots, x_n + iy_n)\\ | ||
| + | & v (x_1, \ldots, x_n, y_1, \ldots , y_n) = {\rm Im}\, f (x_1+iy_1, \ldots, x_n + iy_n): | ||
| + | \end{align*} | ||
| + | \begin{equation}\label{e:CR-sys} | ||
| + | \frac{\partial u}{\partial x_k} = \frac{\partial v}{\partial y_k} \qquad | ||
| + | \frac{\partial u}{\partial y_k} = - \frac{\partial v}{\partial x_k}\, \qquad k = 1, \ldots, n\, , | ||
| + | \end{equation} | ||
| + | or, in terms of the complex differential operators: | ||
| + | \[ | ||
| + | \frac{\partial f}{\partial \bar{z}_k} = 0\, . | ||
| + | \] | ||
| + | Each of the two functions $u$ and $v$ satisfying the conditions \eqref{e:CR-sys} (which as in the one-variable case, turn out to be infinitely differentiable and analytic) is a [[Pluriharmonic function|pluriharmonic function]] of the variables $x_k$ and $y_k$. When $n>1$ the pluriharmonic functions constitute a proper subclass of the class of harmonic functions. The conditions \eqref{e:CR-sys} are conjugacy conditions for two pluriharmonic functions $u$ and $v$: knowing one of them, one can determine the other by integration (up to addition of a constant in each connected component of the domain of definition). | ||
| + | |||
| + | ===Historical remarks=== | ||
| + | The conditions \eqref{e:CR} apparently occurred for the first time in the works of [[DAlembert|J. d'Alembert]] {{Cite|DA}}. Their first appearance as a criterion for analyticity was in a paper of L. Euler, delivered at the Petersburg Academy of Sciences in 1777 {{Cite|Eu}}. A.L. Cauchy utilized the conditions \eqref{e:CR} to construct the theory of functions, beginning with a memoir presented to the Paris Academy in 1814 (see {{Cite|Ca}}). The celebrated dissertation of B. Riemann on the fundamentals of function theory dates to 1851 (see {{Cite|Ri}}). | ||
| + | |||
| + | ===References=== | ||
| + | {| | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Al}}|| L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1966) {{MR|0188405}} {{ZBL|0154.31904}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ca}}|| A.L. Cauchy, "Mémoire sur les intégrales définies" , ''Oeuvres complètes Ser. 1'' , '''1''' , Paris (1882) pp. 319–506 | ||
| + | |- | ||
| + | |valign="top"|{{Ref|DA}}|| J. d'Alembert, "Essai d'une nouvelle théorie de la résistance des fluides" , Paris (1752) | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Eu}}|| L. Euler, ''Nova Acta Acad. Sci. Petrop.'' , '''10''' (1797) pp. 3–19 | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ma}}|| A.I. Markushevich, "Theory of functions of a complex variable" , '''1–3''' , Chelsea (1977) (Translated from Russian) {{MR|0444912}} {{ZBL|0357.30002}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ri}}|| B. Riemann, "Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen komplexen Grösse" H. Weber (ed.) , ''Riemann's gesammelte math. Werke'' , Dover, reprint (1953) pp. 3–48 | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Sh}}|| B.V. Shabat, "Introduction of complex analysis" , '''1–2''' , Moscow (1976) (In Russian) {{MR|}} {{ZBL|0799.32001}} {{ZBL|0732.32001}} {{ZBL|0732.30001}} {{ZBL|0578.32001}} {{ZBL|0574.30001}} | ||
| + | |- | ||
| + | |} | ||
Latest revision as of 12:51, 22 March 2023
2020 Mathematics Subject Classification: Primary: 30-XX Secondary: 32-XX [MSN][ZBL]
Also known as Cauchy-Riemann conditions and D'Alembert-Euler conditions, they are the partial differential equations that must be satisfied by the real and imaginary parts of a complex-valued function $f$ of one (or several) complex variable so that $f$ is holomorphic.
One complex variable
More precisely, assume $D\subset \mathbb C$ is some open set and $f: D \to \mathbb C$ a map with real and imaginary parts given by $u$ and $v$ (i.e. $f(z) = u (z) + i v (z)$). If we introduce the real variables $x,y$ so that $z= x+iy$, we can consider $u$ and $v$ as real functions on a domain in $\mathbb R^2$. If $u$ and $v$ are differentiable (in the real-variable sense), then they solve the Cauchy-Riemann equations if \begin{equation}\label{e:CR} \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \qquad \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}\, . \end{equation} Such equations are equivalent to the fact that $f$ is complex-differentiable at every point $z_0\in D$, where its complex derivative is defined by \[ f' (z_0) = \lim_{w\in \mathbb C, w\to 0} \frac{f(z_0+w)-f(z_0)}{w}\, . \] It then turns out that \begin{equation}\label{e:f'} f' = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} \end{equation} (and evidently we can use \eqref{e:CR} to derive three other similar formulas).
Harmonicity and conjugacy
Any pair of solutions of the Cauchy-Riemann equations turn out to be infinitely differentiable and in fact analytic. Moreover they are also conjugate harmonic functions. Viceversa if $u$ (or $v$) is a given harmonic function on a simply connected open $D$, then there is a conjugate harmonic function to $u$, that is a $v$ satisfying \eqref{e:CR}. Such $v$ is unique up to addition of a constant.
Different systems of coordinates
The conditions \eqref{e:CR} can be written, equivalently, for any two orthogonal directions $s$ and $n$, with the same mutual orientation as the $x$- and $y$-axes, in the form: \[ \frac{\partial u}{\partial s} = \frac{\partial v}{\partial n} \qquad \frac{\partial u}{\partial n} = - \frac{\partial v}{\partial s}\, . \] For example, in polar coordinates $r, \phi$, for $r\neq 0$ the Cauchy-Riemann equations read: \[ \frac{\partial u}{\partial r} = \frac{1}{r} \frac{\partial v}{\partial \phi} \qquad \frac{1}{r} \frac{\partial u}{\partial \phi} = - \frac{\partial v}{\partial r}\, . \]
$\frac{\partial}{\partial \bar{z}}$ and $\frac{\partial}{\partial z}$ operators
Defining the complex differential operators by \[ \frac{\partial}{\partial \bar{z}}=\frac{1}{2} \left(\frac{\partial}{\partial x} + i \frac{\partial}{\partial y}\right)\qquad\mbox{and}\qquad \frac{\partial}{\partial z}=\frac{1}{2} \left(\frac{\partial}{\partial x} - i \frac{\partial}{\partial y}\right)\, , \] one can rewrite the Cauchy–Riemann equations \eqref{e:CR} as \[ \frac{\partial f}{\partial \bar{z}} = 0 \] and the identity \eqref{e:f'} as \[ f' = \frac{\partial f}{\partial z}\, . \]
Conformality
The Cauchy-Riemann equations are equivalent to the fact that the map $(u,v): D \to \mathbb R^2$ is conformal, i.e. it preserves the angles and (locally where it is injective) the orientation. Such condition can indeed be expressed at the differential level with the property that at each point $(x_0, y_0)$ the Jacobian matrix of $(u,v)$ is a multiple of a rotation. In turn this can be easily seen to be equivalent to \eqref{e:CR}.
Several complex variables
For analytic functions of several complex variables $z = (z_1, \ldots, z_n)$, with $z_k = x_k + iy_k$ , the Cauchy–Riemann equations is given by the following system of partial differential equations (overdetermined when $n>1$) for the functions \begin{align*} & u (x_1, \ldots, x_n, y_1, \ldots , y_n) = {\rm Re}\, f (x_1+iy_1, \ldots, x_n + iy_n)\\ & v (x_1, \ldots, x_n, y_1, \ldots , y_n) = {\rm Im}\, f (x_1+iy_1, \ldots, x_n + iy_n): \end{align*} \begin{equation}\label{e:CR-sys} \frac{\partial u}{\partial x_k} = \frac{\partial v}{\partial y_k} \qquad \frac{\partial u}{\partial y_k} = - \frac{\partial v}{\partial x_k}\, \qquad k = 1, \ldots, n\, , \end{equation} or, in terms of the complex differential operators: \[ \frac{\partial f}{\partial \bar{z}_k} = 0\, . \] Each of the two functions $u$ and $v$ satisfying the conditions \eqref{e:CR-sys} (which as in the one-variable case, turn out to be infinitely differentiable and analytic) is a pluriharmonic function of the variables $x_k$ and $y_k$. When $n>1$ the pluriharmonic functions constitute a proper subclass of the class of harmonic functions. The conditions \eqref{e:CR-sys} are conjugacy conditions for two pluriharmonic functions $u$ and $v$: knowing one of them, one can determine the other by integration (up to addition of a constant in each connected component of the domain of definition).
Historical remarks
The conditions \eqref{e:CR} apparently occurred for the first time in the works of J. d'Alembert [DA]. Their first appearance as a criterion for analyticity was in a paper of L. Euler, delivered at the Petersburg Academy of Sciences in 1777 [Eu]. A.L. Cauchy utilized the conditions \eqref{e:CR} to construct the theory of functions, beginning with a memoir presented to the Paris Academy in 1814 (see [Ca]). The celebrated dissertation of B. Riemann on the fundamentals of function theory dates to 1851 (see [Ri]).
References
| [Al] | L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1966) MR0188405 Zbl 0154.31904 |
| [Ca] | A.L. Cauchy, "Mémoire sur les intégrales définies" , Oeuvres complètes Ser. 1 , 1 , Paris (1882) pp. 319–506 |
| [DA] | J. d'Alembert, "Essai d'une nouvelle théorie de la résistance des fluides" , Paris (1752) |
| [Eu] | L. Euler, Nova Acta Acad. Sci. Petrop. , 10 (1797) pp. 3–19 |
| [Ma] | A.I. Markushevich, "Theory of functions of a complex variable" , 1–3 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002 |
| [Ri] | B. Riemann, "Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen komplexen Grösse" H. Weber (ed.) , Riemann's gesammelte math. Werke , Dover, reprint (1953) pp. 3–48 |
| [Sh] | B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001 |
Cauchy-Riemann equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy-Riemann_equations&oldid=22265