In mathematics, the Cauchy–Riemann equations are two partial differential equations that characterize differentiability of complex functions. The equations are and where and are real bivariate differentiable functionss. Typically, and are the real and imaginary parts of a complex variable .

If is complex-differentiable at a complex point , then the partial derivatives of and exist and satisfy the Cauchy–Riemann equations at that point. Conversely, if the functions and are (real) differentiable at and satisfy the Cauchy-Riemann equations there, then is complex-differentiable at .

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

is differentiable at any point

.

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

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

In this form, the equations correspond structurally to the condition that the Jacobian matrix. 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 takes infinitesimal line segments at the intersection of two curves in and rotates them to the corresponding segments in . 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

are real-valued functions, be a complex-valued function. Then, the complex-derivative, and does not depend on the chosen path).

A fundamental result of complex analysis is complex differentiable (that is, it has a complex-derivative), if and only if the bivariate real functions are differentiable and satisfy the Cauchy–Riemann equations at this point.

which is the complex form of Cauchy–Riemann equations (.

, it is also real differentiable and the Jacobian.)

Conversely, if (in the real sense) and satisfies the Cauchy-Riemann equations there, then it is complex-differentiable at this point. Assume that as a function of two real variables and is differentiable at , as .

, the above can be re-written as

is essential and cannot be dispensed with. For example,, 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 be continuous at the point because this continuity condition ensures the existence of the aforementioned linear approximation 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 for real variables . Defining the two Wirtinger derivatives and the complex derivative of In this form, the Cauchy–Riemann equations can be interpreted as the statement that a complex function of a complex variable . As such, we can view analytic functions as true functions of one complex variable () instead of complex functions of two real variables ( and ).

Physical interpretation

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

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 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.

where R is the rotation through 90 degrees counterclockwise:

have the same magnitude, and their dot product 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 curves; so these are the streamlines curves are the equipotential curves of the flow.

A holomorphic function can therefore be visualized by plotting the two families of level curves. Near points where the gradient of u (or, equivalently, v) is not zero, these families form an orthogonal 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 regarded as a (real) two-component vector. Then the second Cauchy–Riemann equation ( is irrotational (its curl

The first Cauchy–Riemann equation () asserts that the vector field is solenoidal (or divergence

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 is both closed and coclosed (a harmonic differential form).

Preservation of complex structure

Another formulation of the Cauchy–Riemann equations involves the complex structure This is a complex structure in the sense that the square of J is the negative of the 2×2 identity matrix. As above, if u(x,y) and v(x,y) are two functions in the plane, put

The Jacobian matrix of f

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 () and () hold for a differentiable pair of functions u and v

for any coordinate system is orthonormal and positively oriented

Combining these into one equation for

The inhomogeneous Cauchy–Riemann equations consist of the two equations for a pair of unknown functions and

for some given functions and defined in an open subset of R2 where f = u + iv and 𝜑 = (α + iβ)/2.

If 𝜑 is Ck, 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 for all ζ ∈ D.

Generalizations

Goursat's theorem and its generalizations

Suppose that is a complex-valued function which is differentiable. 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 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., . Similarly, some additional assumption is needed besides the Cauchy–Riemann equations (such as continuity), as the following example illustrates

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:

1. If and satisfies the Cauchy–Riemann equations weakly, then agrees almost everywhere with an analytic function in .

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

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). The Dirac operator, which can be calculated in the following way:

:

Hence, in traditional notation:

Conformal mappings in higher dimensions

to be a conformal mapping

where Df, and I denotes the identity matrix. For , this system is equivalent to the standard Cauchy–Riemann equations of complex variables, and the solutions are holomorphic functions. In dimension , 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

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

References

Sources

Further reading

External links

• Cauchy–Riemann Equations Module by John H. Mathews