where u(x, y) and v(x, y) are real
differentiable bivariate functions.
Typically, u and v are respectively the
real and
imaginary parts of a
complex-valued function f(x + iy) = f(x, y) = u(x, y) + iv(x, y) of a single complex variable z = x + iy where x and y are real variables; u and v are real
differentiable functions of the real variables. Then f is
complex differentiable at a complex point if and only if the partial derivatives of u and v satisfy the Cauchy–Riemann equations at that point.
A
holomorphic function is a complex function that is differentiable at every point of some open subset of the complex plane C. It has been proved that
holomorphic functions are analytic and
analytic complex functions are complex-differentiable. In particular, holomorphic functions are infinitely complex-differentiable.
This equivalence between differentiability and analyticity is the starting point of all
complex analysis.
History
The Cauchy–Riemann equations first appeared in the work of
Jean le Rond d'Alembert.[1] Later,
Leonhard Euler connected this system to the
analytic functions.[2] Cauchy[3] then used these equations to construct his theory of functions. Riemann's dissertation on the theory of functions appeared in 1851.[4]
Simple example
Suppose that . The complex-valued function is differentiable at any point z 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.
First, the Cauchy–Riemann equations may be written in complex form
(2)
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 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
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).
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 f is differentiable at (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
where , , z = x + iy, and as Δz → 0.
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,[8] 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[9][10] 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
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[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
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 (), 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 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.[12] 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 (1b) asserts that is
irrotational (its
curl is 0):
The first Cauchy–Riemann equation (1a) 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, 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.[13] 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.
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.[14]
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
for any coordinate system (n(x, y), s(x, y)) 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 f gives
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
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
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,
Suppose that f = u + iv is a complex-valued function which is
differentiable as a function f : R2 → R2. 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.[15] In particular, continuous differentiability of f need not be assumed.[16]
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[17]
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:[18]
If f(z) is locally integrable in an open domain Ω ⊂ 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.
In the
Clifford algebra, the complex number is represented as where , (, so ). The
Dirac operator in this Clifford algebra is defined as . The function is considered analytic if and only if , which can be calculated in the following way:
Grouping by and :
Hence, in traditional notation:
Conformal mappings in higher dimensions
Let Ω be an open set in the Euclidean space Rn. The equation for an orientation-preserving mapping to be a
conformal mapping (that is, angle-preserving) is that
where Df is the Jacobian matrix, with transpose , and I denotes the identity matrix.[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.
^Cauchy, Augustin L. (1814). Mémoire sur les intégrales définies. Oeuvres complètes Ser. 1. Vol. 1. Paris (published 1882). pp. 319–506.
^Riemann, Bernhard (1851). "Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen komplexen Grösse". In H. Weber (ed.). Riemann's gesammelte math. Werke (in German). Dover (published 1953). pp. 3–48.
^Markushevich, A.I. (1977). Theory of functions of a complex variable 1. Chelsea., p. 110-112 (Translated from Russian)
^Titchmarsh, E (1939). The theory of functions. Oxford University Press., 2.14
^Arfken, George B.; Weber, Hans J.; Harris, Frank E. (2013). "11.2 CAUCHY-RIEMANN CONDITIONS". Mathematical Methods for Physicists: A Comprehensive Guide (7th ed.). Academic Press. pp. 471–472.
ISBN978-0-12-384654-9.
^Hassani, Sadri (2013). "10.2 Analytic Functions". Mathematical Physics: A Modern Introduction to Its Foundations (2nd ed.). Springer. pp. 300–301.
ISBN978-3-319-01195-0.
^See
Klein, Felix (1893). On Riemann's theory of algebraic functions and their integrals. Translated by Frances Hardcastle. Cambridge: MacMillan and Bowes.
^Iwaniec, T.; Martin, G. (2001). Geometric function theory and non-linear analysis. Oxford. p. 32.
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.
JSTOR2321164.
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.