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| In the field of [[complex analysis]] in [[mathematics]], the '''Cauchy–Riemann equations''', named after [[Augustin Louis Cauchy|Augustin Cauchy]] and [[Bernhard Riemann]], consist of a system of two [[partial differential equation]]s which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a [[complex function]] to be [[complex differentiable]], that is [[holomorphic function|holomorphic]]. This system of equations first appeared in the work of [[Jean le Rond d'Alembert]] {{harv|d'Alembert|1752}}. Later, [[Leonhard Euler]] connected this system to the [[analytic functions]] {{harv|Euler|1797}}. {{harvtxt|Cauchy|1814}} then used these equations to construct his theory of functions. Riemann's dissertation {{harv|Riemann|1851}} on the theory of functions appeared in 1851.
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| The Cauchy–Riemann equations on a pair of real-valued functions of two real variables '''''u'''''('''''x''''','''''y''''') and '''''v'''''('''''x''''','''''y''''') are the two equations:
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| :(1a){{quad}} <math>\dfrac{ \partial u }{ \partial x } = \dfrac{ \partial v }{ \partial y }</math>
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| :(1b){{quad}}<math>\dfrac{ \partial u }{ \partial y } = -\dfrac{ \partial v }{ \partial x }</math>
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| Typically '''''u''''' and '''''v''''' are taken to be the [[real part|real]] and [[imaginary part]]s respectively of a [[complex number|complex]]-valued function of a single complex variable '''''z''''' = '''''x''''' + '''''iy''''', '''''f'''''('''''x''''' + i'''''y''''') = '''''u'''''('''''x''''','''''y''''') + i'''''v'''''('''''x''''','''''y'''''). Suppose that '''''u''''' and '''''v''''' are real-[[differentiable]] at a point in an [[open subset]] of '''C '''( '''C''' is the set of complex numbers), which can be considered as functions from '''R'''<sup>2</sup> to '''R'''. This implies that the partial derivatives of '''''u''''' and '''''v''''' exist (although they need not be continuous) and we can approximate small variations of '''''f''''' linearly. Then '''''f''''' = '''''u''''' + i'''''v''''' is complex-[[differentiable]] at that point if and only if the partial derivatives of '''''u''''' and '''''v''''' satisfy the Cauchy–Riemann equations (1a) and (1b) at that point. The sole existence of partial derivatives satisfying the Cauchy–Riemann equations is not enough to ensure complex differentiability at that point. It is necessary that u and v be real differentiable, which is a stronger condition than the existence of the partial derivatives, but it is not necessary that these partial derivatives be continuous.
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| [[holomorphic|Holomorphy]] is the property of a complex function of being differentiable at every point of an open and connected subset of '''C''' (this is called a [[Domain (mathematical analysis)|domain]] in '''C'''). Consequently, we can assert that a complex function '''''f''''', whose real and imaginary parts '''''u''''' and '''''v''''' are real-differentiable functions, is [[holomorphic]] if and only if, equations (1a) and (1b) are satisfied throughout the [[Domain (mathematical analysis)|domain]] we are dealing with.
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| The reason why [[Euler]] and some other authors relate the Cauchy–Riemann equations with [[analytic function|analyticity]] is that a major theorem in [[complex analysis]] says that [[holomorphic functions are analytic]] and vice versa. This means that, in complex analysis, a function that is complex-differentiable in a whole domain (holomorphic) is the same as an analytic function. This is not true for real differentiable functions.
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| == Interpretation and reformulation ==
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| The equations are one way of looking at the condition on a function to be differentiable in the sense of [[complex analysis]]: in other words they encapsulate the notion of [[function of a complex variable]] by means of conventional [[differential calculus]]. In the theory there are several other major ways of looking at this notion, and the translation of the condition into other language is often needed. | |
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| === Conformal mappings ===
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| First, the Cauchy–Riemann equations may be written in complex form
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| :(2){{quad}}<math>{ i \dfrac{ \partial f }{ \partial x } } = \dfrac{ \partial f }{ \partial y } . </math>
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| In this form, the equations correspond structurally to the condition that the [[Jacobian matrix]] is of the form
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| :<math>
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| \begin{pmatrix}
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| a & -b \\
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| b & \;\; a
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| \end{pmatrix},
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| </math> | |
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| where <math>\scriptstyle a=\partial u/\partial x=\partial v/\partial y</math> and <math>\scriptstyle b=\partial v/\partial x=-\partial u/\partial y</math>. A matrix of this form is the [[Complex number#Matrix representation of complex numbers|matrix representation of a complex number]]. Geometrically, such a matrix is always the [[function composition|composition]] of a [[rotation]] with a [[homothety|scaling]], and in particular preserves [[angle]]s. Consequently, a function satisfying the Cauchy–Riemann equations, with a nonzero derivative, preserves the angle between curves in the plane. That is, the Cauchy–Riemann equations are the conditions for a function to be [[conformal map|conformal]].
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| === Complex differentiability ===
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| Suppose that
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| : <math> f(z) = u(z) + i \cdot v(z) </math>
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| is a function of a complex number ''z''. Then the complex derivative of ''f'' at a point ''z''<sub>0</sub> is defined by | |
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| : <math>\lim_{\underset{h\in\mathbf{C}}{h\to 0}} \frac{f(z_0+h)-f(z_0)}{h} = f'(z_0)</math>
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| provided this limit exists.
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| If this limit exists, then it may be computed by taking the limit as ''h'' → 0 along the real axis or imaginary axis; in either case it should give the same result. Approaching along the real axis, one finds
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| :<math>\lim_{\underset{h\in\mathbf{R}}{h\to 0}} \frac{f(z_0+h)-f(z_0)}{h} = \frac{\partial f}{\partial x}(z_0).</math>
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| On the other hand, approaching along the imaginary axis,
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| :<math>\lim_{\underset{h\in \mathbf{R}}{h\to 0}} \frac{f(z_0+ih)-f(z_0)}{ih} =\frac{1}{i}\frac{\partial f}{\partial y}(z_0).</math>
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| The equality of the derivative of ''f'' taken along the two axes is
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| :<math>i\frac{\partial f}{\partial x}(z_0)=\frac{\partial f}{\partial y}(z_0),</math>
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| which are the Cauchy–Riemann equations (2) at the point ''z''<sub>0</sub>. | |
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| Conversely, if ''f'' : '''C''' → '''C''' is a function which is differentiable when regarded as a function on '''R'''<sup>2</sup>, then ''f'' is complex differentiable if and only if the Cauchy–Riemann equations hold. In other words, if u and v are real-differentiable functions of two real variables, obviously ''u'' + ''iv'' is a (complex-valued) real-differentiable function, but ''u'' + ''iv'' is complex-differentiable if and only if the Cauchy–Riemann equations hold.
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| Indeed, following {{harvtxt|Rudin|1966}}, suppose ''f'' is a complex function defined in an open set Ω ⊂ '''C'''. Then, writing {{nowrap|''z'' {{=}} ''x'' + i''y''}} for every ''z'' ∈ Ω, one can also regard Ω as an open subset of '''R'''<sup>2</sup>, and ''f'' as a function of two real variables ''x'' and ''y'', which maps Ω ⊂ '''R'''<sup>2</sup> to '''C'''. We consider the Cauchy–Riemann equations at ''z'' = 0 assuming ''f''(''z'') = 0, just for notational simplicity – the proof is identical in general case. So assume ''f'' is differentiable at 0, as a function of two real variables from Ω to '''C'''. This is equivalent to the existence of two complex numbers α and β (which are the partial derivatives of ''f'') such that we have the linear approximation
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| :<math> f(z) = \alpha x + \beta y + \eta(z)z \,</math>
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| where ''z'' = ''x'' + ''iy'' and η(''z'') → 0 as ''z'' → ''z''<sub>0</sub> = 0. Since <math> z+\bar{z}=2x </math> and <math> z-\bar{z}=2iy </math>, the above can be re-written as
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| :<math> f(z) = \frac{\alpha - i\beta}{2}z + \frac{\alpha + i\beta}{2}\bar{z} + \eta(z) z\, </math> | |
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| Defining the two [[Wirtinger derivatives]] as
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| :<math> \frac{\partial}{\partial z} = \frac{1}{2} \Bigl( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \Bigr), \;\;\; \frac{\partial}{\partial\bar{z}}= \frac{1}{2} \Bigl( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \Bigr),</math>
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| the above equality can be written as
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| :<math> \frac{f(z)}{z} =\left(\frac{\partial f}{\partial z} \right)(0) + \left(\frac{\partial f}{\partial\bar{z}}\right)(0) \cdot \frac{\bar{z}}{z} + \eta(z), \;\;\;\;(z \neq 0). </math>
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| For real values of ''z'', we have <math> \bar{z}/z = 1</math> and for purely imaginary ''z'' we have <math> \bar{z}/z = -1 </math> hence ''f''(''z'')/''z'' has a limit at 0 (''i.e.'', ''f'' is complex differentiable at 0) if and only if <math> \scriptstyle (\partial f/\partial\bar{z})(0) = 0 </math>. But this is exactly the Cauchy–Riemann equations, thus ''f'' is differentiable at 0 if and only if the Cauchy–Riemann equations hold at 0.
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| === Independence of the complex conjugate ===
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| The above proof suggests another interpretation of the Cauchy–Riemann equations. The [[complex conjugate]] of ''z'', denoted <math>\bar{z}</math>, is defined by
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| :<math>\overline{x + iy} := x - iy</math>
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| for real ''x'' and ''y''. The Cauchy–Riemann equations can then be written as a single equation | |
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| :(3){{quad}}<math>\dfrac{\partial f}{\partial\bar{z}} = 0</math>
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| by using the [[Wirtinger derivatives|Wirtinger derivative with respect to the conjugate variable]]. In this form, the Cauchy–Riemann equations can be interpreted as the statement that ''f'' is independent of the variable <math>\bar{z}</math>. As such, we can view analytic functions as true functions of ''one'' complex variable as opposed to complex functions of ''two'' real variables.
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| === Physical interpretation ===
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| One interpretation of the Cauchy–Riemann equations {{harv|Pólya|Szegö|1978}} does not involve complex variables directly. Suppose that ''u'' and ''v'' satisfy the Cauchy–Riemann equations in an open subset of '''R'''<sup>2</sup>, and consider the [[vector field]]
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| :<math>\bar{f} = \begin{bmatrix}u\\ -v\end{bmatrix}</math>
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| regarded as a (real) two-component vector. Then the second Cauchy–Riemann equation (1b) asserts that <math>\bar{f}</math> is [[irrotational vector field|irrotational]] (its [[Curl (mathematics)|curl]] is 0):
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| :<math>\frac{\partial (-v)}{\partial x} - \frac{\partial u}{\partial y} = 0.</math>
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| The first Cauchy–Riemann equation (1a) asserts that the vector field is [[solenoidal vector field|solenoidal]] (or [[divergence]]-free): | |
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| :<math>\frac{\partial u}{\partial x} + \frac{\partial (-v)}{\partial y}=0.</math>
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| Owing respectively to [[Green's theorem]] and the [[divergence theorem]], such a field is necessarily [[conservative vector field|conserved]] and free from sources or sinks, having net flux equal to zero through any open domain. (These two observations combine as real and imaginary parts in [[Cauchy's integral theorem]].) In [[fluid dynamics]], such a vector field is a [[potential flow]] {{harv|Chanson|2007}}. In [[magnetostatics]], such vector fields model static [[magnetic field]]s on a region of the plane containing no current. In [[electrostatics]], they model static electric fields in a region of the plane containing no electric charge.
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| === Other representations ===
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| Other representations of the Cauchy–Riemann equations occasionally arise in other [[coordinate system]]s. If (1a) and (1b) hold for a differentiable pair of functions ''u'' and ''v'', then so do
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| :<math>\frac{\partial u}{\partial n} = \frac{\partial v}{\partial s},\quad \frac{\partial v}{\partial n} = -\frac{\partial u}{\partial s}</math>
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| for any coordinate system {{nowrap|(''n''(''x'', ''y''), ''s''(''x'', ''y''))}} such that the pair (∇''n'', ∇''s'') is [[orthonormal]] and [[orientation (mathematics)|positively oriented]]. As a consequence, in particular, in the system of coordinates given by the polar representation {{nowrap|''z'' {{=}} ''r'' ''e''<sup>iθ</sup>}}, the equations then take the form
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| :<math>{ \partial u \over \partial r } = {1 \over r}{ \partial v \over \partial \theta},\quad{ \partial v \over \partial r } = -{1 \over r}{ \partial u \over \partial \theta}.</math>
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| Combining these into one equation for ''f'' gives
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| :<math>{\partial f \over \partial r} = {1 \over i r}{\partial f \over \partial \theta}.</math>
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| The inhomogeneous Cauchy–Riemann equations consist of the two equations for a pair of unknown functions ''u''(''x'',''y'') and ''v''(''x'',''y'') of two real variables
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| :<math>\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y} = \alpha(x,y)</math>
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| :<math>\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x} = \beta(x,y)</math>
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| for some given functions α(''x'',''y'') and β(''x'',''y'') defined in an open subset of '''R'''<sup>2</sup>. These equations are usually combined into a single equation
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| :<math>\frac{\partial f}{\partial\bar{z}} = \varphi(z,\bar{z})</math>
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| where ''f'' = ''u'' + i''v'' and φ = (α + iβ)/2.
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| If φ is [[continuously differentiable|''C''<sup>''k''</sup>]], then the inhomogeneous equation is explicitly solvable in any bounded domain ''D'', provided φ is continuous on the [[closure (topology)|closure]] of ''D''. Indeed, by the [[Cauchy integral formula]],
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| :<math>f(\zeta,\bar{\zeta}) = \frac{1}{2\pi i}\iint_D \varphi(z,\bar{z})\frac{dz\wedge d\bar{z}}{z-\zeta}</math>
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| for all ζ ∈ ''D''.
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| ==Generalizations==
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| === Goursat's theorem and its generalizations ===
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| {{see also|Cauchy–Goursat theorem}}
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| Suppose that {{nowrap|''f'' {{=}} ''u'' + i''v''}} is a complex-valued function which is [[Fréchet derivative|differentiable]] as a function {{nowrap|''f'' : '''R'''<sup>2</sup> → '''R'''<sup>2</sup>}}. Then '''Goursat's theorem''' asserts that ''f'' is analytic in an open complex domain Ω if and only if it satisfies the Cauchy–Riemann equation in the domain {{harv|Rudin|1966|loc=Theorem 11.2}}. In particular, continuous differentiability of ''f'' need not be assumed {{harv|Dieudonné|1969|loc=§9.10, Ex. 1}}.
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| The hypotheses of Goursat's theorem can be weakened significantly. If {{nowrap|''f'' {{=}} ''u'' + i''v''}} is continuous in an open set Ω and the [[partial derivative]]s of ''f'' with respect to ''x'' and ''y'' exist in Ω, and satisfies the Cauchy–Riemann equations throughout Ω, then ''f'' is holomorphic (and thus analytic). This result is the [[Looman–Menchoff theorem]].
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| The hypothesis that ''f'' obey the Cauchy–Riemann equations throughout the domain Ω is essential. It is possible to construct a continuous function satisfying the Cauchy–Riemann equations at a point, but which is not analytic at the point (e.g., ''f''(''z'') = {{nowrap|''z''<sup>5</sup> / {{!}}z{{!}}<sup>4</sup>)}}. Similarly, some additional assumption is needed besides the Cauchy–Riemann equations (such as continuity), as the following example illustrates {{harv|Looman|1923|p=107}}
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| :<math>f(z) = \begin{cases}
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| \exp(-z^{-4})&\mathrm{if\ }z\not=0\\
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| 0&\mathrm{if\ }z=0
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| \end{cases}</math>
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| which satisfies the Cauchy–Riemann equations everywhere, but fails to be continuous at ''z'' = 0.
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| Nevertheless, if a function satisfies the Cauchy–Riemann equations in an open set in a [[weak derivative|weak sense]], then the function is analytic. More precisely {{harv|Gray|Morris|1978|loc=Theorem 9}}:
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| * If ''f''(''z'') is locally integrable in an open domain Ω ⊂ '''C''', and satisfies the Cauchy–Riemann equations weakly, then ''f'' agrees [[almost everywhere]] with an analytic function in Ω.
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| This is in fact a special case of a more general result on the regularity of solutions of [[hypoelliptic operator|hypoelliptic]] partial differential equations.
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| ===Several variables===
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| There are Cauchy–Riemann equations, appropriately generalized, in the theory of [[several complex variables]]. They form a significant [[overdetermined system]] of PDEs. As often formulated, the ''[[d-bar operator]]''
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| :<math>\bar{\partial}</math>
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| annihilates holomorphic functions. This generalizes most directly the formulation
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| :<math>{\partial f \over \partial \bar z} = 0,</math>
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| where
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| :<math>{\partial f \over \partial \bar z} = {1 \over 2}\left({\partial f \over \partial x} + i{\partial f \over \partial y}\right).</math>
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| === Bäcklund transform ===
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| Viewed as [[conjugate harmonic functions]], the Cauchy–Riemann equations are a simple example of a [[Bäcklund transform]]. More complicated, generally non-linear Bäcklund transforms, such as in the [[sine-Gordon equation]], are of great interest in the theory of [[soliton]]s and [[integrable system]]s.
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| ==See also==
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| *[[List of complex analysis topics]]
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| *[[Morera's theorem]]
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| *[[Wirtinger derivatives]]
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| == References ==
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| *{{citation|first=Lars|last=Ahlfors|authorlink=Lars Ahlfors|title=Complex analysis|publisher=McGraw Hill|year=1953|publication-date=1979|edition=3rd|isbn=0-07-000657-1}}.
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| *{{citation|first=J.|last=d'Alembert|authorlink=Jean le Rond d'Alembert|title=Essai d'une nouvelle théorie de la résistance des fluides|url=http://gallica2.bnf.fr/ark:/12148/bpt6k206036b.modeAffichageimage.f1.langFR.vignettesnaviguer|publication-place=Paris|year=1752}}.
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| *{{citation|first=A.L.|last=Cauchy|authorlink=Augustin Cauchy|title=Mémoire sur les intégrales définies, |series=Oeuvres complètes Ser. 1|volume=1|publication-place=Paris|year=1814|publication-date=1882|pages=319–506}}
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| * {{citation|url=http://espace.library.uq.edu.au/view/UQ:119883|last=Chanson|first=H.|year=2007|title=Le Potentiel de Vitesse pour les Ecoulements de Fluides Réels: la Contribution de Joseph-Louis Lagrange." ('Velocity Potential in Real Fluid Flows: Joseph-Louis Lagrange's Contribution.')|journal=Journal La Houille Blanche|volume=5|pages=127–131|doi=10.1051/lhb:2007072|issn=0018-6368}}.
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| * {{citation|last=Dieudonné|first=Jean Alexander|authorlink=Jean Dieudonné|title=Foundations of modern analysis|publisher=Academic Press|year=1969}}.
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| *{{citation|first=L.|last=Euler|authorlink=Leonhard Euler|journal=Nova Acta Acad. Sci. Petrop.|volume=10|year=1797|pages=3–19}}
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| *{{citation|title=When is a Function that Satisfies the Cauchy–Riemann Equations Analytic?|first1=J. D.|last1=Gray|first2=S. A.|last2=Morris|journal=The American Mathematical Monthly|volume=85|number=4|year=1978|publication-date=April 1978|pages=246–256|jstor=2321164}}.
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| *{{citation|first=H.|last=Looman|title=Über die Cauchy–Riemannschen Differeitalgleichungen|journal=Göttinger Nachrichten|year=1923|pages=97–108}}.
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| * {{citation|last1=Pólya|first1=George|authorlink1=George Pólya|last2=Szegö|first2=Gabor|title=Problems and theorems in analysis I|publisher=Springer|year=1978|isbn=3-540-63640-4}}
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| *{{citation|last=Riemann|first=B.|authorlink=Bernhard Riemann|contribution=Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen komplexen Grösse|editor=H. Weber|title=Riemann's gesammelte math. Werke|publisher=Dover|publication-date=1953|pages=3–48|year=1851}}
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| * {{citation|last=Rudin|first=Walter|authorlink=Walter Rudin|title=Real and complex analysis|year=1966|publisher=McGraw Hill|publication-date=1987|edition=3rd|isbn=0-07-054234-1}}.
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| * {{springer|last=Solomentsev|first=E.D.|title=Cauchy–Riemann conditions|id=c/c020970|year=2001}}
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| * {{citation|last1=Stewart|first1=Ian|last2=Tall|first2=David|authorlink=Ian Stewart (mathematician)|title=Complex Analysis|year=1983|publisher=CUP|publication-date=1984|edition=1st|isbn=0-521-28763-4}}.
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| == External links ==
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| * {{MathWorld | urlname= Cauchy-RiemannEquations | title= Cauchy–Riemann Equations }}
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| * [http://math.fullerton.edu/mathews/c2003/CauchyRiemannMod.html Cauchy–Riemann Equations Module by John H. Mathews]
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| * [http://www.encyclopediaofmath.org/index.php/Cauchy-Riemann_equations Cauchy-Riemann equations] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| {{DEFAULTSORT:Cauchy-Riemann equations}}
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| [[Category:Partial differential equations]]
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| [[Category:Complex analysis]]
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| [[Category:Harmonic functions]]
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| [[Category:Equations]]
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