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 functions. Typically, and are the real and imaginary parts, respectively, of a complex-valued function
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 Suppose that . The complex-valued function is differentiable at any point in the complex plane.
The real part and the imaginary part are
and their partial derivatives are
We see that indeed the Cauchy–Riemann equations are satisfied, and .
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 is of the form
where and . 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 Let
where and are real-valued functions, be a complex-valued function of a complex variable where and are real variables. so the function can also be regarded as a function of real variables and . Then, the complex-derivative of at a point is defined by
provided this limit exists (that is, the limit exists along every path approaching , and does not depend on the chosen path).
A fundamental result of complex analysis is that is complex differentiable at (that is, it has a complex-derivative), if and only if the bivariate real functions and are differentiable at and satisfy the Cauchy–Riemann equations at this point., p. 110-112 (Translated from Russian)
In fact, if the complex derivative exists at , then it may be computed by taking the limit at along the real axis and the imaginary axis, and the two limits must be equal. Along the real axis, the limit is
and along the imaginary axis, the limit is
So, the equality of the derivatives implies
which is the complex form of Cauchy–Riemann equations () at .
(Note that if is complex differentiable at , it is also real differentiable and the Jacobian of at is the complex scalar , regarded as a real-linear map of , since the limit as .)
Conversely, if is differentiable at (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 (real differentiable). This is equivalent to the existence of the following linear approximation where , , , and as .
Since and , the above can be re-written as
Now, if is real, , while if it is imaginary, then . Therefore, the second term is independent of the path of the limit when (and only when) it vanishes identically: , which is precisely the Cauchy–Riemann equations in the complex form. This proof also shows that, in that case,
Note that the hypothesis of real differentiability at the point is essential and cannot be dispensed with. For example,, 2.14 the function , regarded as a complex function with imaginary part identically zero, has both partial derivatives at , 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 as, in addition to the Cauchy–Riemann equations, the partial derivatives of and 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 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 , denoted , is defined by
for real variables and . Defining the two Wirtinger derivatives as the Cauchy–Riemann equations can then be written as a single equation
and the complex derivative of in that case is In this form, the Cauchy–Riemann equations can be interpreted as the statement that a complex function of a complex variable is independent of the 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, defined by
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.
In vector form, the Cauchy–Riemann equations read
where R is the rotation through 90 degrees counterclockwise:
Thus the Cauchy–Riemann equations imply that and have the same magnitude, and their dot product is zero:
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 curves; so these are the streamlines of the flow. The curves are the equipotential curves of the flow.
A holomorphic function can therefore be visualized by plotting the two families of level curves and . 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 , the stationary points of the flow, the equipotential curves of 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 () asserts that is irrotational (its curl is 0):
The first Cauchy–Riemann equation () asserts that the vector field is solenoidal (or divergence-free):
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 in the plane, given by
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 is the matrix of partial derivatives
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, then so do
for any coordinate system such that the pair is orthonormal and positively oriented. As a consequence, in particular, in the system of coordinates given by the polar representation , the equations then take the form
Combining these into one equation for gives
The inhomogeneous Cauchy–Riemann equations consist of the two equations for a pair of unknown functions and of two real variables
for some given functions and defined in an open subset of R2. These equations are usually combined into a single equation
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 as a function . 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.