Cauchy–Riemann equations

"Cauchy–Riemann" redirects here. For Cauchy–Riemann manifolds, see CR manifold.

File:Cauchy-Riemann.svg

A visual depiction of a vector X in a domain being multiplied by a complex number z, then mapped by df, versus being mapped by df then being multiplied by z afterwards. If both of these result in the point ending up in the same place for all X and z, then f satisfies the Cauchy–Riemann condition at the basepoint.

Template:Complex analysis sidebar In mathematics, the Cauchy–Riemann equations are two partial differential equations that characterize differentiability of complex functions. The equations are

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} }

(1a)

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x},}

(1b)

where u(x, y) and v(x, y) are real bivariate differentiable functions. Typically, u and v are the real and imaginary parts, respectively, of a complex-valued function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z) = f(x + i y) = u(x, y) + i v(x, y)} of a complex variable z = x + iy.

If f is complex-differentiable at a complex point z = x + iy, then the partial derivatives of u and v exist and satisfy the Cauchy–Riemann equations at that point. Conversely, if the functions u and v are (real) differentiable at z and satisfy the Cauchy-Riemann equations there, then f is complex-differentiable at z.

In this way, the Cauchy-Riemann equations are closely related to differentiability, which is in turn closely related to analyticity. These close relationships are the starting point of complex analysis.

History

The Cauchy–Riemann equations first appeared in the work of Jean le Rond d'Alembert. Later, Leonhard Euler connected this system to the analytic functions. Augustin-Louis Cauchy then used these equations to construct his theory of functions. Bernhard Riemann's dissertation on the theory of functions appeared in 1851.

Simple example

Suppose that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = x + iy} . The complex-valued function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z) = z^2} is differentiable at any point z in the complex plane. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z) = (x + iy)^2 = x^2 - y^2 + 2ixy} The real part Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(x,y)} and the imaginary part Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(x, y)} are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} u(x, y) &= x^2 - y^2 \\ v(x, y) &= 2xy \end{align}} and their partial derivatives are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_x = 2x;\quad u_y = -2y;\quad v_x = 2y;\quad v_y = 2x}

We see that indeed the Cauchy–Riemann equations are satisfied, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_x = v_y} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_y = -v_x} .

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 information: Conformal map

First, the Cauchy–Riemann equations may be written in complex form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle { i \frac{ \partial f }{ \partial x } } = \frac{ \partial f }{ \partial y } . }

(2)

In this form, the equations correspond structurally to the condition that the Jacobian matrix is of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix} a & -b \\ b & a \end{pmatrix},} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = \partial u/\partial x = \partial v/\partial y} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = \partial v/\partial x = -\partial u/\partial y} . A matrix of this form is the matrix representation of a complex number. Geometrically, such a matrix is always the composition of a rotation with a scaling, and in particular preserves angles. The Jacobian of a function f(z) takes infinitesimal line segments at the intersection of two curves in z and rotates them to the corresponding segments in 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.

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z) = u(z) + i \cdot v(z) } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle u} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} are real-valued functions, be a complex-valued function of a complex variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle z = x + i y} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle y} are real variables. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f(z) = f(x + iy) = f(x,y)} so the function can also be regarded as a function of real variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle y} . Then, the complex-derivative of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f } at a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle z_0=x_0+iy_0 } is defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(z_0) =\lim_{\underset{h\in\Complex}{h\to 0}} \frac{f(z_0+h)-f(z_0)}{h} } provided this limit exists (that is, the limit exists along every path approaching Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle z_{0} } , and does not depend on the chosen path).

A fundamental result of complex analysis is that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is complex differentiable at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_0} (that is, it has a complex-derivative), if and only if the bivariate real functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(x+iy)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(x+iy)} are differentiable at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_0,y_0),} and satisfy the Cauchy–Riemann equations at this point.

In fact, if the complex derivative exists at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle z_0} , then it may be computed by taking the limit at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle z_0} along the real axis and the imaginary axis, and the two limits must be equal. Along the real axis, the limit is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}} and along the imaginary axis, the limit is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}.}

So, the equality of the derivatives implies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \left. \frac{\partial f}{\partial x} \right \vert _{z_0} = \left. \frac{\partial f}{\partial y} \right \vert _{z_0}} which is the complex form of Cauchy–Riemann equations (2) at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle z_0} .

(Note that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is complex differentiable at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_0} , it is also real differentiable and the Jacobian of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_0} is the complex scalar Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(z_0)} , regarded as a real-linear map of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb C} , since the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |f(z)-f(z_0)-f'(z_0)(z-z_0)|/|z-z_0|\to 0} as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z\to z_0} .)

Conversely, if f is differentiable at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle z_{0} } (in the real sense) and satisfies the Cauchy-Riemann equations there, then it is complex-differentiable at this point. Assume that f as a function of two real variables x and y is differentiable at z₀ (real differentiable). This is equivalent to the existence of the following linear approximation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta f(z_0) = f(z_0 + \Delta z) - f(z_0) = f_x \,\Delta x + f_y \,\Delta y + \eta(\Delta z)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f_x = \left. \frac{\partial f}{\partial x}\right \vert _{z_0} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f_y = \left. \frac{\partial f}{\partial y} \right \vert _{z_0} } , z = x + iy, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \eta(\Delta z) / |\Delta z| \to 0} as Δz → 0.

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Delta z + \Delta \bar{z}= 2 \, \Delta x } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Delta z - \Delta \bar{z}=2i \, \Delta y } , the above can be re-written as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta f(z_0) = \frac{f_x - if_y}{2} \, \Delta z + \frac{f_x + if_y}{2} \, \Delta \bar{z} + \eta(\Delta z)\, } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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). }

Now, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Delta z} is real, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Delta\bar z/\Delta z = 1} , while if it is imaginary, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Delta\bar z/\Delta z=-1} . Therefore, the second term is independent of the path of the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Delta z\to 0} when (and only when) it vanishes identically: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f_x + i f_y=0} , which is precisely the Cauchy–Riemann equations in the complex form. This proof also shows that, in that case, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.\frac{df}{dz}\right|_{z_0} = \lim_{\Delta z\to 0}\frac{\Delta f}{\Delta z} = \frac{f_x - i f_y}{2}.}

Note that the hypothesis of real differentiability at the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_0} is essential and cannot be dispensed with. For example, the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textstyle f(x,y) = \sqrt{|xy|}} , regarded as a complex function with imaginary part identically zero, has both partial derivatives at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_0,y_0)=(0,0)} , 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 state a sufficient condition for the complex differentiability at a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_0} as, in addition to the Cauchy–Riemann equations, the partial derivatives of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z) = z^2e^{i/|z|}} 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 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} , denoted Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \bar{z}} , is defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{x + iy} := x - iy} for real variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} . Defining the two Wirtinger derivatives asFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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), } the Cauchy–Riemann equations can then be written as a single equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial f}{\partial\bar{z}} = 0,} and the complex derivative of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f} in that case is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \frac{df}{dz}=\frac{\partial f}{\partial z}.} In this form, the Cauchy–Riemann equations can be interpreted as the statement that a complex function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f} of a complex variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle z} is independent of the variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \bar{z}} . As such, we can view analytic functions as true functions of one complex variable (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle z} ) instead of complex functions of two real variables (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle y} ).

Physical interpretation

File:Contours of holomorphic function.png

Contour plot of a pair u and v satisfying the Cauchy–Riemann equations. Streamlines (v = const, red) are perpendicular to equipotentials (u = const, blue). The point (0,0) is a 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 is that u represents a velocity potential of an incompressible steady fluid flow in the plane, and v is its stream function. Suppose that the pair of (twice 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla u = \frac{\partial u}{\partial x}\mathbf i + \frac{\partial u}{\partial y}\mathbf j.}

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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0.} 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.

In vector form, the Cauchy–Riemann equations read Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \nabla v = R \nabla u where R is the rotation through 90 degrees counterclockwise:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R = R(90^\circ) = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}. }

Thus the Cauchy–Riemann equations imply that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla u} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla v} have the same magnitude, and their dot product is zero: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla u\cdot\nabla v = 0.} Geometrically, 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle v = \text{const}} curves; so these are the streamlines of the flow. The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle u = \text{const}} curves are the equipotential curves of the flow.

A holomorphic function can therefore be visualized by plotting the two families of level curves Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle u=\text{const}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle v=\text{const}} . 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \nabla u=0} , the stationary points of the flow, the equipotential curves of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle u=\text{const}} 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 Pólya & Szegő. Suppose that u and v satisfy the Cauchy–Riemann equations in an open subset of R², and consider the vector field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{f} = \begin{bmatrix} u\\ -v \end{bmatrix}} regarded as a (real) two-component vector. Then the second Cauchy–Riemann equation (1b) asserts that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{f}} is irrotational (its curl is 0): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial (-v)}{\partial x} - \frac{\partial u}{\partial y} = 0.}

The first Cauchy–Riemann equation (1a) asserts that the vector field is solenoidal (or divergence-free): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial u}{\partial x} + \frac{\partial (-v)}{\partial y}=0.}

Owing respectively to Green's theorem and the divergence theorem, such a field is necessarily a conservative one on any simply-connected domain, 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. In magnetostatics, such vector fields model static magnetic fields 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 forms. The pair u and v satisfy the Cauchy–Riemann equations if and only if the one-form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v\,dx + u\, dy} is both closed and coclosed (a harmonic differential form).

Preservation of complex structure

Another formulation of the Cauchy–Riemann equations involves the complex structure in the plane, given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}.} This is a complex structure in the sense that the square of J is the negative of the 2×2 identity matrix: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J^2 = -I} . As above, if u(x,y) and v(x,y) are two functions in the plane, put

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x,y) = \begin{bmatrix}u(x,y)\\v(x,y)\end{bmatrix}.}

The Jacobian matrix of f is the matrix of partial derivatives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}}

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

This interpretation is useful in symplectic geometry, where it is the starting point for the study of pseudoholomorphic curves.

Other representations

Other representations of the Cauchy–Riemann equations occasionally arise in other coordinate systems. If (1a) and (1b) hold for a differentiable pair of functions u and v, then so do Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial u}{\partial n} = \frac{\partial v}{\partial s},\quad \frac{\partial v}{\partial n} = -\frac{\partial u}{\partial s} }

for any coordinate system (n(x, y), s(x, y)) such that the pair Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle (\nabla n,\nabla s)} is orthonormal and positively oriented. As a consequence, in particular, in the system of coordinates given by the polar representation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = r e^{i\theta}} , the equations then take the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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}. }

Combining these into one equation for f gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\partial f \over \partial r} = {1 \over ir}{\partial f \over \partial\theta}.}

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}}

for some given functions α(x, y) and β(x, y) defined in an open subset of R². These equations are usually combined into a single equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial f}{\partial\bar{z}} = \varphi(z,\bar{z})} where f = u + iv and 𝜑 = (α + iβ)/2.

If 𝜑 is C^k, then the inhomogeneous equation is explicitly solvable in any bounded domain D, provided 𝜑 is continuous on the closure of D. Indeed, by the Cauchy integral formula, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}} for all ζ ∈ D.

Generalizations

Goursat's theorem and its generalizations

See also: Cauchy–Goursat theorem

Suppose that f = u + iv is a complex-valued function which is differentiable as a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : \mathbb{R}^2 \rarr \mathbb{R}^2} . 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. In particular, continuous differentiability of f need not be assumed.

The hypotheses of Goursat's theorem can be weakened significantly. If f = u + iv is continuous in an open set Ω and the partial derivatives of f with respect to x and y exist in Ω, and satisfy the Cauchy–Riemann equations throughout Ω, then f is holomorphic (and thus analytic). This result is the Looman–Menchoff theorem.

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) = z⁵/|z|⁴). Similarly, some additional assumption is needed besides the Cauchy–Riemann equations (such as continuity), as the following example illustrates

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z) = \begin{cases} \exp\left(-z^{-4}\right) & \text{if }z \not= 0\\ 0 & \text{if }z = 0 \end{cases}}

which satisfies the Cauchy–Riemann equations everywhere, but fails to be continuous at z = 0.

Nevertheless, if a function satisfies the Cauchy–Riemann equations in an open set in a weak sense, then the function is analytic. More precisely:

If f(z) is locally integrable in an open domain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega \isin \mathbb{C},} and satisfies the Cauchy–Riemann equations weakly, then f agrees almost everywhere with an analytic function in Ω.

This is in fact a special case of a more general result on the regularity of solutions of hypoelliptic partial differential equations.

Several variables

There are Cauchy–Riemann equations, appropriately generalized, in the theory of several complex variables. They form a significant overdetermined system of PDEs. This is done using a straightforward generalization of the Wirtinger derivative, where the function in question is required to have the (partial) Wirtinger derivative with respect to each complex variable vanish.

Complex differential forms

As often formulated, the d-bar operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{\partial}} annihilates holomorphic functions. This generalizes most directly the formulation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\partial f \over \partial \bar z} = 0,} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\partial f \over \partial \bar z} = {1 \over 2}\left({\partial f \over \partial x} + i{\partial f \over \partial y}\right).}

Bäcklund transform

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 solitons and integrable systems.

Definition in Clifford algebra

In the Clifford algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C\ell(2)} , the complex number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = x+iy } is represented as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z \equiv x + J y} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J \equiv \sigma_1 \sigma_2} , (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_1^2=\sigma_2^2=1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_1 \sigma_2 + \sigma_2 \sigma_1 = 0} , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J^2=-1} ). The Dirac operator in this Clifford algebra is defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \equiv \sigma_1 \partial_x + \sigma_2\partial_y} . The function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=u + J v} is considered analytic if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla f = 0} , which can be calculated in the following way:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} 0 & =\nabla f= ( \sigma_1 \partial_x + \sigma_2 \partial_y )(u + \sigma_1 \sigma_2 v) \\[4pt] & =\sigma_1 \partial_x u + \underbrace{\sigma_1 \sigma_1 \sigma_2}_{=\sigma_2} \partial_x v + \sigma_2 \partial_y u + \underbrace{\sigma_2 \sigma_1 \sigma_2}_{=-\sigma_1} \partial_y v =0 \end{align} }

Grouping by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_2} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla f = \sigma_1 ( \partial_x u - \partial_y v) + \sigma_2 ( \partial_x v + \partial_y u) = 0 \Leftrightarrow \begin{cases} \partial_x u - \partial_y v = 0\\[4pt] \partial_x v + \partial_y u = 0 \end{cases}}

Hence, in traditional notation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} \dfrac{ \partial u }{ \partial x } = \dfrac{ \partial v }{ \partial y }\\[12pt] \dfrac{ \partial u }{ \partial y } = -\dfrac{ \partial v }{ \partial x } \end{cases}}

Conformal mappings in higher dimensions

Let Ω be an open set in the Euclidean space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n} . The equation for an orientation-preserving mapping Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:\Omega\to\mathbb{R}^n} to be a conformal mapping (that is, angle-preserving) is that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Df^\mathsf{T} Df = (\det(Df))^{2/n}I}

where Df is the Jacobian matrix, with transpose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Df^\mathsf{T}} , and I denotes the identity matrix. For n = 2, this system is equivalent to the standard Cauchy–Riemann equations of complex variables, and the solutions are holomorphic functions. In dimension n > 2, this is still sometimes called the Cauchy–Riemann system, and Liouville's theorem implies, under suitable smoothness assumptions, that any such mapping is a Möbius transformation.

Lie pseudogroups

See also: Pseudogroup

One might seek to generalize the Cauchy-Riemann equations instead by asking more generally when are the solutions of a system of PDEs closed under composition. The theory of Lie Pseudogroups addresses these kinds of questions.

See also

• List of complex analysis topics
• Cauchy integral theorem
• Morera's theorem
• Wirtinger derivatives

Sources

• Gray, J. D.; Morris, S. A. (April 1978). "When is a Function that Satisfies the Cauchy–Riemann Equations Analytic?". The American Mathematical Monthly. 85 (4): 246–256. doi:10.2307/2321164. JSTOR 2321164.
• Looman, H. (1923). "Über die Cauchy–Riemannschen Differentialgleichungen". Göttinger Nachrichten (in German): 97–108.
• Marsden, A; Hoffman, M (1973). Basic complex analysis. W. H. Freeman.
• Rudin, Walter (1966). Real and complex analysis (3rd ed.). McGraw Hill (published 1987). ISBN 0-07-054234-1.

Further reading

• Ahlfors, Lars (1953). Complex analysis (3rd ed.). McGraw Hill (published 1979). ISBN 0-07-000657-1.
• Template:Springer
• Stewart, Ian; Tall, David (1983). Complex Analysis (1st ed.). CUP (published 1984). ISBN 0-521-28763-4.

External links

• Weisstein, Eric W. "Cauchy–Riemann Equations". MathWorld.
• Cauchy–Riemann Equations Module by John H. Mathews