In the field of
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
in
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Cauchy–Riemann equations, named after
Augustin Cauchy and
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
, consist of a
system
A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purpose and express ...
of two
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...
s which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a
complex function to be
holomorphic (complex differentiable). This system of equations first appeared in the work of
Jean le Rond d'Alembert
Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the '' Encyclopéd ...
. Later,
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
connected this system to the
analytic functions. Cauchy then used these equations to construct his theory of functions. Riemann's dissertation on the theory of functions appeared in 1851.
The Cauchy–Riemann equations on a pair of real-valued functions of two real variables and are the two equations:
Typically ''u'' and ''v'' are taken to be the
real and
imaginary parts respectively of a
complex-valued function of a single complex variable , . Suppose that and are real-
differentiable at a point in an
open subset
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
of , which can be considered as functions from to . This implies that the partial derivatives of and exist (although they need not be continuous), so we can approximate small variations of linearly. Then is complex-
differentiable, at that point if and only if the partial derivatives of and satisfy the Cauchy–Riemann equations () and () at that point. The existence of partial derivatives satisfying the Cauchy–Riemann equations there doesn't ensure complex differentiability: and must be real differentiable, which is a stronger condition than the existence of the partial derivatives, but in general, weaker than continuous differentiability.
Holomorphy is the property of a complex function of being differentiable at every point of an open and connected subset of (this is called a
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
in ). Consequently, we can assert that a complex function ''f'', whose real and imaginary parts ''u'' and ''v'' are real-differentiable functions, is
holomorphic if and only if, equations () and () are satisfied throughout the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
we are dealing with.
Holomorphic functions are analytic and vice versa. This means that, in complex analysis, a function that is complex-differentiable in a whole domain (holomorphic) is the same as an analytic function. This is not true for real differentiable functions.
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 equations are one way of looking at the condition on a function to be differentiable in the sense of
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
: in other words they encapsulate the notion of
function of a complex variable
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
by means of conventional
differential calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve ...
. 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
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
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
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
with a
scaling, and in particular preserves
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
s. The Jacobian of a function takes infinitesimal line segments at the intersection of two curves in z 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
Suppose that
is a function of a complex number
. Then the complex derivative of
at a point
is defined by
provided this limit exists.
If this limit exists, then it may be computed by taking the limit as
along the real axis or imaginary axis; in either case it should give the same result. Approaching along the real axis, one finds
On the other hand, approaching along the imaginary axis,
The equality of the derivative of taken along the two axes is
which are the Cauchy–Riemann equations (2) at the point .
Conversely, if is a function which is differentiable when regarded as a function on , then ''f'' is complex differentiable if and only if the Cauchy–Riemann equations hold. In other words, if and are real-differentiable functions of two real variables, obviously is a (complex-valued) real-differentiable function, but is complex-differentiable if and only if the Cauchy–Riemann equations hold.
Indeed, following Rudin, suppose ''f'' is a complex function defined in an open set . Then, writing for every , one can also regard Ω as an open subset of R
2, and ''f'' as a function of two real variables ''x'' and ''y'', which maps to C. We consider the Cauchy–Riemann equations at . So assume ''f'' is differentiable at , as a function of two real variables from Ω to C. This is equivalent to the existence of the following linear approximation
where and as . Since
and
, the above can be re-written as
Defining the two
Wirtinger derivatives as
in the limit
the above equality can be written as
Now consider the potential values of
when the limit is taken at the origin. For ''z'' along the real line,
so that
. Similarly for purely imaginary ''z'' we have
so that the value of
is not well defined at the origin. It's easy to verify that
is not well defined at any complex ''z'', hence ''f'' is complex differentiable at ''z''
0 if and only if
at
. But this is exactly the Cauchy–Riemann equations, thus ''f'' is differentiable at ''z''
0 if and only if the Cauchy–Riemann equations hold at ''z''
0.
Independence of the complex conjugate
The above proof suggests another interpretation of the Cauchy–Riemann equations. The
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of ''z'', denoted
, is defined by
for real ''x'' and ''y''. The Cauchy–Riemann equations can then be written as a single equation
by using the
Wirtinger derivative with respect to the conjugate variable. In this form, the Cauchy–Riemann equations can be interpreted as the statement that ''f'' is independent of the variable
. As such, we can view analytic functions as true functions of ''one'' complex variable as opposed to complex functions of ''two'' real variables.
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
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
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of ''u'', defined by
By differentiating the Cauchy–Riemann equations a second time, one shows that ''u'' solves
Laplace's equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \na ...
:
That is, ''u'' is a
harmonic function. This means that the
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
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
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
. 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
In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
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 R
2, and consider the
vector field
regarded as a (real) two-component vector. Then the second Cauchy–Riemann equation () asserts that
is
irrotational (its
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was ...
is 0):
The first Cauchy–Riemann equation () asserts that the vector field is
solenoidal (or
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
-free):
Owing respectively to
Green's theorem
In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem.
Theorem
Let be a positively orie ...
and the
divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the ...
, such a field is necessarily a
conservative
Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization in ...
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
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
, such a vector field is a
potential flow. In
magnetostatics, such vector fields model static
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
s on a region of the plane containing no current. In
electrostatics
Electrostatics is a branch of physics that studies electric charges at rest ( static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for a ...
, 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 form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s. The pair ''u'',''v'' satisfy the Cauchy–Riemann equations if and only if the
one-form is both
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
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''),''v''(''x'',''y'') are two functions in the plane, put
The
Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
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
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...
, 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 system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
s. If (1a) and (1b) hold for a differentiable pair of functions ''u'' and ''v'', then so do
for any coordinate system such that the pair (∇''n'', ∇''s'') is
orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of ...
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 R
2. These equations are usually combined into a single equation
where ''f'' = ''u'' + i''v'' and ''𝜑'' = (''α'' + i''β'')/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,
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.
The hypotheses of Goursat's theorem can be weakened significantly. If is continuous in an open set Ω and the
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s 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 In the mathematical field of complex analysis, the Looman–Menchoff theorem states that a continuous complex-valued function defined in an open set of the complex plane is holomorphic if and only if it satisfies the Cauchy–Riemann equations. It ...
.
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:
: 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 In the theory of partial differential equations, a partial differential operator P defined on an open subset
:U \subset^n
is called hypoelliptic if for every distribution u defined on an open subset V \subset U such that Pu is C^\infty (smoo ...
partial differential equations.
Several variables
There are Cauchy–Riemann equations, appropriately generalized, in the theory of
several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variable ...
. They form a significant
overdetermined system of PDEs. This is done using a straightforward generalization of the
Wirtinger derivative
In complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operators), named after Wilhelm Wirtinger who introduced them in 1927 in the course of his studies on the theory of functions of se ...
, 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''
annihilates holomorphic functions. This generalizes most directly the formulation
where
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
The sine-Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally introduced by in the course of study of surf ...
, are of great interest in the theory of
soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the me ...
s and
integrable system
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
s.
Definition in Clifford algebra
In
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperco ...
the complex number
is represented as
where
. The fundamental derivative operator in Clifford algebra of
Complex numbers 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 R
''n''. 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. 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
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad' ...
.
See also
*
List of complex analysis topics
*
Morera's theorem
*
Wirtinger derivatives
References
Sources
*
*
*
Further reading
*
*
*
External links
*
Cauchy–Riemann Equations Module by John H. Mathews
{{DEFAULTSORT:Cauchy-Riemann equations
Partial differential equations
Complex analysis
Harmonic functions
Bernhard Riemann
Augustin-Louis Cauchy