Editing Cauchy–Riemann equations (section)

== Interpretation and reformulation ==
The Cauchy-Riemann equations are one way of looking at the condition for 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.

=== Conformal mappings ===
{{further|Conformal map}}

First, the Cauchy–Riemann equations may be written in complex form
{{NumBlk||<math display="block">{ i \frac{ \partial f }{ \partial x } } = \frac{ \partial f }{ \partial y } . </math>|{{EquationRef|2}}}}

In this form, the equations correspond structurally to the condition that the [[Jacobian matrix]] is of the form
<math display="block">\begin{pmatrix}
  a & -b  \\
  b &  a  
\end{pmatrix},</math>
where <math> a = \partial u/\partial x = \partial v/\partial y</math> and <math> 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 [[Homothetic transformation|scaling]], and in particular preserves [[angle]]s. The Jacobian of a function {{math|''f''(''z'')}} takes infinitesimal line segments at the intersection of two curves in {{math|''z''}} and rotates them to the corresponding segments in {{math|''f''(''z'')}}. 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]].

Moreover, because the composition of a conformal transformation with another conformal transformation is also conformal, the composition of a solution of the Cauchy–Riemann equations with a conformal map must itself solve the Cauchy–Riemann equations. Thus the Cauchy–Riemann equations are conformally invariant.

=== Complex differentiability ===
Let
<math display="block"> f(z) = u(z) + i \cdot v(z) </math>
where <math display="inline">u</math> and <math>v</math> are [[Real-valued function|real-valued functions]], be a [[complex-valued function]] of a complex variable <math display="inline"> z = x + i y</math> where <math display="inline"> x</math> and <math display="inline"> y</math> are real variables. <math display="inline"> f(z) = f(x + iy) = f(x,y)</math> so the function can also be regarded as a function of real variables <math display="inline">x</math> and <math display="inline"> y</math>. Then, the ''complex-derivative'' of <math display="inline"> f </math> at a point <math display="inline"> z_0=x_0+iy_0 </math> is defined by
<math display="block"> f'(z_0) =\lim_{\underset{h\in\Complex}{h\to 0}} \frac{f(z_0+h)-f(z_0)}{h} </math>
provided this limit exists (that is, the limit exists along every path approaching <math display="inline"> z_{0} </math>, and does not depend on the chosen path).

A fundamental result of [[complex analysis]] is that <math>f</math> is [[differentiable function#Differentiability in complex analysis|complex differentiable]] at <math>z_0</math> (that is, it has a complex-derivative), [[if and only if]] the bivariate [[real function]]s <math>u(x+iy)</math> and <math>v(x+iy)</math> are [[differentiable]] at <math>(x_0,y_0),</math> and satisfy the Cauchy–Riemann equations at this point.{{sfn|Rudin|1966}}{{sfn|Marsden|Hoffman|1973}}<ref>{{cite book|first=A.I.|last=Markushevich|title=Theory of functions of a complex variable 1|publisher=Chelsea|year=1977}}, p. 110-112 (Translated from Russian)</ref>

In fact, if the complex derivative exists at <math display="inline"> z_0</math>, then it may be computed by taking the limit at <math display="inline"> z_0</math> along the real axis and the imaginary axis, and the two limits must be equal. Along the real axis, the limit is 
<math display="block">\lim_{\underset{h\in\Reals}{h\to 0}} \frac{f(z_0+h)-f(z_0)}{h} = \left.  \frac{\partial f}{\partial x} \right \vert_{z_0}</math>
and along the imaginary axis, the limit is
<math display="block">\lim_{\underset{h\in \Reals}{h\to 0}} \frac{f(z_0+ih)-f(z_0)}{ih} = \left. \frac{1}{i}\frac{\partial f}{\partial y} \right \vert _{z_0}.</math>

So, the equality of the derivatives implies 
<math display="block">i \left. \frac{\partial f}{\partial x} \right \vert _{z_0} = \left. \frac{\partial f}{\partial y} \right \vert _{z_0}</math>
which is the complex form of Cauchy–Riemann equations ({{EquationNote|2}}) at <math display="inline"> z_0</math>.

(Note that if <math>f</math> is complex differentiable at <math>z_0</math>, it is also real differentiable and the [[Jacobian]] of <math>f</math> at <math>z_0</math> is the complex scalar <math>f'(z_0)</math>, regarded as a real-linear map of <math>\mathbb C</math>, since the limit <math>|f(z)-f(z_0)-f'(z_0)(z-z_0)|/|z-z_0|\to 0</math> as <math>z\to z_0</math>.)

Conversely, if {{mvar|f}} is differentiable at <math display="inline"> z_{0} </math> (in the real sense) and satisfies the Cauchy-Riemann equations there, then it is complex-differentiable at this point. Assume that {{mvar|f}} as a function of two real variables {{mvar|x}} and {{mvar|y}} is differentiable at {{math|''z''<sub>0</sub>}} (real differentiable). This is equivalent to the existence of the following linear approximation <math display="block"> \Delta f(z_0) = f(z_0 + \Delta z) - f(z_0) = f_x \,\Delta x + f_y \,\Delta y + \eta(\Delta z)</math>where <math display="inline"> f_x = \left. \frac{\partial f}{\partial x}\right \vert _{z_0} </math>, <math display="inline"> f_y = \left. \frac{\partial f}{\partial y} \right \vert _{z_0} </math>, {{math|1=''z'' = ''x'' + ''iy''}}, and <math display="inline">\eta(\Delta z) / |\Delta z| \to 0</math> as {{math|Δ''z'' → 0}}. 

Since <math display="inline"> \Delta z + \Delta \bar{z}= 2 \, \Delta x </math> and <math display="inline"> \Delta z - \Delta \bar{z}=2i \, \Delta y </math>, the above can be re-written as

<math display="block"> \Delta f(z_0) = \frac{f_x - if_y}{2} \, \Delta z + \frac{f_x + if_y}{2} \, \Delta \bar{z} + \eta(\Delta z)\, </math><math display="block">\frac{\Delta f}{\Delta z} = \frac{f_x -i f_y}{2}+ \frac{f_x + i f_y}{2}\cdot \frac{\Delta\bar{z}}{\Delta z} + \frac{\eta(\Delta z)}{\Delta z}, \;\;\;\;(\Delta z \neq 0). </math>  

Now, if <math display="inline">\Delta z</math> is real, <math display="inline">\Delta\bar z/\Delta z = 1</math>, while if it is imaginary, then <math display="inline">\Delta\bar z/\Delta z=-1</math>. Therefore, the second term is independent of the path of the limit <math display="inline">\Delta z\to 0</math> when (and only when) it vanishes identically: <math display="inline">f_x + i f_y=0</math>, which is precisely the Cauchy–Riemann equations in the complex form. This proof also shows that, in that case,
<math display="block">\left.\frac{df}{dz}\right|_{z_0} = \lim_{\Delta z\to 0}\frac{\Delta f}{\Delta z} = \frac{f_x - i f_y}{2}.</math>

Note that the hypothesis of real differentiability at the point <math>z_0</math> is essential and cannot be dispensed with. For example,<ref>{{cite book|first=E|last=Titchmarsh|title=The theory of functions|year=1939|publisher=Oxford University Press}}, 2.14</ref> the function <math>f(x,y) = \sqrt{|xy|}</math>, regarded as a complex function with imaginary part identically zero, has both partial derivatives at <math>(x_0,y_0)=(0,0)</math>, and it moreover satisfies the Cauchy–Riemann equations at that point, but it is not differentiable in the sense of real functions (of several variables), and so the first condition, that of real differentiability, is not met. Therefore, this function is not complex differentiable.

Some sources<ref>{{Cite book |last1=Arfken |first1=George B. |title=Mathematical Methods for Physicists: A Comprehensive Guide |last2=Weber |first2=Hans J. |last3=Harris |first3=Frank E. |publisher=Academic Press |year=2013 |isbn=978-0-12-384654-9 |edition=7th |pages=471–472 |language=English |chapter=11.2 CAUCHY-RIEMANN CONDITIONS}}
</ref><ref>{{Cite book |last=Hassani |first=Sadri |title=Mathematical Physics: A Modern Introduction to Its Foundations |publisher=Springer |year=2013 |isbn=978-3-319-01195-0 |edition=2nd |pages=300–301 |language=English |chapter=10.2 Analytic Functions}}</ref> state a sufficient condition for the complex differentiability at a point <math>z_0</math> as, in addition to the Cauchy–Riemann equations, the partial derivatives of <math>u</math> and <math>v</math> be continuous at the point because this continuity condition ensures the existence of the aforementioned linear approximation. Note that it is not a necessary condition for the complex differentiability. For example, the function <math>f(z) = z^2e^{i/|z|}</math> is complex differentiable at 0, but its real and imaginary parts have discontinuous partial derivatives there. Since complex differentiability is usually considered in an open set, where it in fact implies continuity of all partial derivatives (see [[#Goursat's theorem and its generalizations|below]]), this distinction is often elided in the literature.

=== Independence of the complex conjugate ===
The above proof suggests another interpretation of the Cauchy–Riemann equations. The [[complex conjugate]] of <math>z</math>, denoted <math display="inline">\bar{z}</math>, is defined by
<math display="block">\overline{x + iy} := x - iy</math>
for real variables ''<math>x</math>'' and <math>y</math>. Defining the two [[Wirtinger derivatives]] as<math display="block"> \frac{\partial}{\partial z}
  = \frac{1}{2} \left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right), \;\;\; \frac{\partial}{\partial\bar{z}}
  = \frac{1}{2} \left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right),
</math>
the Cauchy–Riemann equations can then be written as a single equation
<math display="block">\frac{\partial f}{\partial\bar{z}} = 0,</math>
and the complex derivative of ''<math display="inline">f</math>'' in that case is <math display="inline">\frac{df}{dz}=\frac{\partial f}{\partial z}.</math> In this form, the Cauchy–Riemann equations can be interpreted as the statement that a complex function ''<math display="inline">f</math>'' of a complex variable ''<math display="inline">z</math>'' is independent of the variable <math display="inline">\bar{z}</math>. As such, we can view analytic functions as true functions of ''one'' complex variable (''<math display="inline">z</math>'') instead of complex functions of ''two'' real variables (''<math display="inline">x</math>'' and ''<math display="inline">y</math>'').

=== Physical interpretation ===
[[File:Contours of holomorphic function.png|right|thumb|[[Contour plot]] of a pair ''u'' and ''v'' satisfying the Cauchy–Riemann equations. Streamlines (''v''&nbsp;=&nbsp;const, red) are perpendicular to equipotentials (''u''&nbsp;=&nbsp;const, blue). The point (0,0) is a [[Critical point (mathematics)|stationary point]] of the potential flow, with six streamlines meeting, and six equipotentials also meeting and bisecting the angles formed by the streamlines.]]
A standard physical interpretation of the Cauchy–Riemann equations going back to Riemann's work on function theory{{r|Klein1893_see}} is that ''u'' represents a [[velocity potential]] of an incompressible [[potential flow|steady fluid flow]] in the plane, and ''v'' is its [[stream function]]. Suppose that the pair of (twice [[Differentiable function|continuously differentiable]]) functions ''u'' and ''v'' satisfies the Cauchy–Riemann equations. We will take ''u'' to be a velocity potential, meaning that we imagine a flow of fluid in the plane such that the [[velocity vector]] of the fluid at each point of the plane is equal to the [[gradient]] of ''u'', defined by
<math display="block">\nabla u = \frac{\partial u}{\partial x}\mathbf i + \frac{\partial u}{\partial y}\mathbf j.</math>

By differentiating the Cauchy–Riemann equations for the functions ''u'' and ''v'', with the [[symmetry of second derivatives]], one shows that ''u'' solves [[Laplace's equation]]:
<math display="block">\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0.</math>
That is, ''u'' is a [[harmonic function]]. This means that the [[divergence]] of the gradient is zero, and so the fluid is incompressible.

The function ''v'' also satisfies the Laplace equation, by a similar analysis. Also, the Cauchy–Riemann equations imply that the [[dot product]] <math display="inline">\nabla u\cdot\nabla v = 0</math> (<math display="inline">\nabla u\cdot\nabla v = \frac{\partial u}{\partial x} \cdot \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \cdot \frac{\partial v}{\partial y} = \frac{\partial u}{\partial x} \cdot \frac{\partial v}{\partial x} - \frac{\partial u}{\partial x} \cdot \frac{\partial v}{\partial x}  = 0</math>), i.e., the direction of the maximum slope of ''u'' and that of ''v'' are orthogonal to each other. This implies that the gradient of ''u'' must point along the <math display="inline">v = \text{const}</math> curves; so these are the [[Streamlines, streaklines, and pathlines|streamlines]] of the flow. The <math display="inline">u = \text{const}</math> curves are the [[equipotential curve]]s of the flow.

A holomorphic function can therefore be visualized by plotting the two families of [[level curve]]s <math display="inline">u=\text{const}</math> and <math display="inline">v=\text{const}</math>. Near points where the gradient of ''u'' (or, equivalently, ''v'') is not zero, these families form an [[orthogonal]] family of curves. At the points where <math display="inline">\nabla u=0</math>, the stationary points of the flow, the equipotential curves of <math display="inline">u=\text{const}</math> intersect. The streamlines also intersect at the same point, bisecting the angles formed by the equipotential curves.

=== Harmonic vector field ===
Another interpretation of the Cauchy–Riemann equations can be found in [[Problems and Theorems in Analysis|Pólya & Szegő]].{{r|PólyaSzegő1978}} Suppose that ''u'' and ''v'' satisfy the Cauchy–Riemann equations in an open subset of '''R'''<sup>2</sup>, and consider the [[vector field]]
<math display="block">\bar{f} = \begin{bmatrix} u\\ -v \end{bmatrix}</math>
regarded as a (real) two-component vector. Then the second Cauchy–Riemann equation ({{EquationNote|1b}}) asserts that <math>\bar{f}</math> is [[irrotational vector field|irrotational]] (its [[Curl (mathematics)|curl]] is 0):
<math display="block">\frac{\partial (-v)}{\partial x} - \frac{\partial u}{\partial y} = 0.</math>

The first Cauchy–Riemann equation ({{EquationNote|1a}}) asserts that the vector field is [[solenoidal vector field|solenoidal]] (or [[divergence]]-free):
<math display="block">\frac{\partial u}{\partial x} + \frac{\partial (-v)}{\partial y}=0.</math>

Owing respectively to [[Green's theorem]] and the [[divergence theorem]], such a field is necessarily a [[conservative vector field|conservative]] one, and it is free from sources or sinks, having net flux equal to zero through any open domain without holes. (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]].{{r|Chanson2007}} 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.

This interpretation can equivalently be restated in the language of [[differential form]]s. The pair ''u'' and ''v'' satisfy the Cauchy–Riemann equations if and only if the [[one-form]] <math>v\,dx + u\, dy</math> is both [[Closed and exact differential forms|closed]] and [[codifferential|coclosed]] (a [[Hodge theory|harmonic differential form]]).

=== Preservation of complex structure ===
Another formulation of the Cauchy–Riemann equations involves the [[linear complex structure|complex structure]] in the plane, given by
<math display="block">J = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}.</math>
This is a complex structure in the sense that the square of ''J'' is the negative of the 2×2 identity matrix: <math>J^2 = -I</math>. As above, if ''u''(''x'',''y'') and ''v''(''x'',''y'') are two functions in the plane, put

<math display="block">f(x,y) = \begin{bmatrix}u(x,y)\\v(x,y)\end{bmatrix}.</math>

The [[Jacobian matrix]] of ''f'' is the matrix of partial derivatives
<math display="block">Df(x,y) = \begin{bmatrix}
\dfrac{\partial u}{\partial x} & \dfrac{\partial u}{\partial y} \\[5pt]
\dfrac{\partial v}{\partial x} & \dfrac{\partial v}{\partial y}
\end{bmatrix}</math>

Then the pair of functions ''u'', ''v'' satisfies the Cauchy–Riemann equations if and only if the 2×2 matrix ''Df'' commutes with ''J''.{{r|KobayashiNomizu1969_PropIX22}}

This interpretation is useful in [[symplectic geometry]], where it is the starting point for the study of [[pseudoholomorphic curve]]s.

=== Other representations ===
Other representations of the Cauchy–Riemann equations occasionally arise in other [[coordinate system]]s. If ({{EquationNote|1a}}) and ({{EquationNote|1b}}) hold for a differentiable pair of functions ''u'' and ''v'', then so do
<math display="block">
  \frac{\partial u}{\partial n} =  \frac{\partial v}{\partial s},\quad
  \frac{\partial v}{\partial n} = -\frac{\partial u}{\partial s}
</math>

for any coordinate system {{math|(''n''(''x'', ''y''), ''s''(''x'', ''y''))}} such that the pair <math display="inline">(\nabla n,\nabla s)</math> is [[orthonormal]] and [[orientation (space)|positively oriented]]. As a consequence, in particular, in the system of coordinates given by the polar representation <math>z = r e^{i\theta}</math>, the equations then take the form
<math display="block">
  {\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>

Combining these into one equation for {{math|''f''}} gives
<math display="block">{\partial f \over \partial r} = {1 \over ir}{\partial f \over \partial\theta}.</math>

The inhomogeneous Cauchy–Riemann equations consist of the two equations for a pair of unknown functions {{math|''u''(''x'', ''y'')}} and {{math|''v''(''x'', ''y'')}} of two real variables
<math display="block">\begin{align}
  \frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} &= \alpha(x, y) \\[4pt]
  \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} &= \beta(x, y)
\end{align}</math>

for some given functions {{math|α(''x'', ''y'')}} and {{math|β(''x'', ''y'')}} defined in an open subset of '''R'''<sup>2</sup>. These equations are usually combined into a single equation
<math display="block">\frac{\partial f}{\partial\bar{z}} = \varphi(z,\bar{z})</math>
where ''f'' = ''u'' + i''v'' and ''𝜑'' = (''α'' + i''β'')/2.

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]],
<math display="block">f\left(\zeta, \bar{\zeta}\right) = \frac{1}{2\pi i} \iint_D \varphi\left(z, \bar{z}\right) \, \frac{dz\wedge d\bar{z}}{z - \zeta}</math>
for all ''ζ'' ∈ ''D''.