Cauchy–Riemann equations

This equivalence between differentiability and analyticity is the starting point of all complex analysis.

The Cauchy–Riemann equations first appeared in the work of Jean le Rond d'Alembert.

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.

Geometrically, such a matrix is always the composition of a rotation with a scaling, and in particular preserves angles.

Consequently, a function satisfying the Cauchy–Riemann equations, with a nonzero derivative, preserves the angle between curves in the plane.

(in the real sense) and satisfies the Cauchy-Riemann equations there, then it is complex-differentiable at this point.

, regarded as a complex function with imaginary part identically zero, has both partial derivatives at

Some sources[9][10] state a sufficient condition for the complex differentiability at a point

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.

A standard physical interpretation of the Cauchy–Riemann equations going back to Riemann's work on function theory[11] 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

A holomorphic function can therefore be visualized by plotting the two families of level curves

The streamlines also intersect at the same point, bisecting the angles formed by the equipotential curves.

[12] Suppose that u and v satisfy the Cauchy–Riemann equations in an open subset of R2, and consider the vector field

Owing respectively to Green's theorem and the divergence theorem, such a field is necessarily a 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.)

This is a complex structure in the sense that the square of J is the negative of the 2×2 identity matrix:

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

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

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,

Similarly, some additional assumption is needed besides the Cauchy–Riemann equations (such as continuity), as the following example illustrates[17]

More precisely:[18] This is in fact a special case of a more general result on the regularity of solutions of hypoelliptic partial differential equations.

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.

[19] 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.

The theory of Lie Pseudogroups addresses these kinds of questions.

A visual depiction of a vector X in a domain being multiplied by a complex number z , then mapped by f , versus being mapped by f 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.
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.