In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, the
Cauchy–Riemann differential equations in
complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
, named after
Augustin CauchyAugustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner. He also gave several important theorems in complex analysis and initiated the study of permutation...
and
Bernhard Riemannwas an influential German mathematician who made contributions to analysis and differential geometry, some of them enabling the later development of general relativity.-Early life:...
, consist of a system of two
partial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables...
s that provides a
necessary and sufficientIn logic, the words necessity and sufficiency refer to the implicational relationships between statements. The assertion that one statement is a necessary and sufficient condition of another means that the former statement is true if and only if the latter is true.* A necessary condition of a...
condition for a differentiable function to be
holomorphicIn mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex-differentiable in a neighborhood of every point in its domain...
in an
open setIn mathematics, more specifically point-set topology and metric topology, the notion of an open set provides a fundamental way to speak of distance in a topological space, without explicitly defining a metric on the space...
. This system of equations first appeared in the work of
Jean le Rond d'AlembertJean le Rond d'Alembert was a French mathematician, mechanician, physicist and philosopher. He was also co-editor with Denis Diderot of the Encyclopédie...
. Later,
Leonhard EulerLeonhard Paul Euler was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany. His surname is in English ; the common English pronunciation is incorrect....
connected this system to the analytic functions . 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
u(
x,
y) and
v(
x,
y) are the two equations:
and
Typically the pair
u and
v are taken to be the
realIn mathematics, the real part of a complex number , is the first element of the ordered pair of real numbers representing , i.e. if , or equivalently, , then the real part of is . It is denoted by or , where is a capital R in the Fraktur typeface...
and
imaginary partIn mathematics, the imaginary part of a complex number , is the second element of the ordered pair of real numbers representing i.e. if , or equivalently, , then the imaginary part of is . It is denoted by or , where is a capital I in the Fraktur typeface....
s of a
complexA complex number, in mathematics, is a number comprising a real number and an imaginary number; it can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit, having the property that i
2 = −1...
-valued function
f(
x + i
y) =
u(
x,
y) + i
v(
x,
y). Suppose that
u and
v are continuously differentiable on an open subset of
C. Then
f =
u+i
v is holomorphic if and only if the partial derivatives of
u and
v satisfy the Cauchy-Riemann equations (1a) and (1b).
Interpretation and reformulation
The equations are one way of looking at the condition on a function to be differentiable (holomorphic) in the sense of
complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
: in other words they encapsulate the notion of function of a complex variable by means of conventional
differential calculusDifferential calculus, a field in mathematics, is the study of how functions change when their inputs change. The primary object of study in differential calculus is the derivative. A closely related notion is the differential. The derivative of a function at a chosen input value describes the...
. 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
Firstly, 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
compositionIn mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
of a
rotationA rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A three-dimensional object rotates around a line called an axis. If the axis of rotation is within the body, the body is said to rotate upon itself, or spin—which implies...
with a scaling, and in particular preserves
angleIn geometry and trigonometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle...
s. 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
conformalIn mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map...
.
Complex differentiability
The Cauchy-Riemann equations are necessary and sufficient conditions for the complex
differentiabilityIn calculus the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point; for example, the derivative of the position of a vehicle with respect to time is the instantaneous velocity...
(or
holomorphicityIn mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex-differentiable in a neighborhood of every point in its domain...
) of a function . Specifically, suppose that
is a function of a complex number
z ∈
C. Then the complex derivative of
ƒ at a point
z0 is defined by
provided this limit exists.
If this limit exists, then it may be computed by taking the limit as
h → 0 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
ƒ taken along the two axes is
which are the Cauchy-Riemann equations (2) at the point
z0.
Conversely, if
ƒ :
C →
C is a function which is differentiable when regarded as a function on
R2, then
ƒ is complex differentiable if and only if the Cauchy-Riemann equations hold.
Indeed, following , suppose
ƒ is a complex function defined in an open set Ω ⊂
C. Then, writing
z =
x + i
y for every
z ∈ Ω, one can also regard Ω as an open subset of
R2, and
ƒ as a function of two real variables
x and
y, which maps Ω ⊂
R2 to
C. We consider the Cauchy-Riemann equations at
z = 0 assuming
ƒ(
z) = 0, just for notational simplicity – the proof is identical in general case. So assume
ƒ is differentiable at 0, as a function of two real variables from Ω to
C. This is equivalent to the existence of two complex numbers and (which are the partial derivatives of
ƒ) such that
where and as Since and , the above can be re-written as
Using the two
differential operatorIn mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .There are certainly reasons not to restrict...
s
the above equality can be written as
For real values of , we have and for purely imaginary we have hence has a limit at 0 (
i.e.,
ƒ is complex differentiable at 0) if and only if . But this is exactly the Cauchy-Riemann equations, thus
ƒ is analytic at 0 if and only if the Cauchy-Riemann equations hold at 0.
Independence of the complex conjugate
The above proof suggests another interpretation of the Cauchy-Riemann equations. The
complex conjugateAs found in mathematics, a complex conjugate is most simply defined as one of a pair of complex numbers, each having the same real parts but with imaginary parts that differ in sign; e.g. 3 + 4i and 3 - 4i are complex conjugates...
of
z, denoted , is defined by
for real
x and
y. The Cauchy-Riemann equations can then be written as a single equation
where the
differential operatorIn mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .There are certainly reasons not to restrict...
is defined by
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
One interpretation of the Cauchy-Riemann equations does not involve complex variables directly. Suppose that
u and
v satisfy the Cauchy-Riemann equations in an open subset of
R2, and consider the
vector fieldIn mathematics a vector field is a construction in vector calculus which associates a vector to every point in a subset of Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of...
regarded as a (real) two-component vector. Then the first Cauchy-Riemann equation (1a) asserts that is
irrotationalIn vector calculus a conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential. Conservative vector fields have the property that the line integral from one point to another is independent of the choice of path connecting the two...
:
The second Cauchy-Riemann equation (1b) asserts that the vector field is
solenoidalIn vector calculus a solenoidal vector field is a vector field v with divergence zero:...
(or
divergenceIn vector calculus, the divergence is an operator that measures the magnitude of a vector field's source or sink at a given point; the divergence of a vector field is a scalar. For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the...
-free):
Owing respectively to
Green's theoremIn physics and mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C...
and the
divergence theoremIn vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...
, such a field is necessarily conserved and free from sources or sinks, having net flux equal to zero through any open domain. (These two observations combine as real and imaginary parts in
Cauchy's integral theoremIn mathematics, the Cauchy integral theorem in complex analysis, named after Augustin-Louis Cauchy, is an important statement about line integrals for holomorphic functions in the complex plane...
.) In
fluid dynamicsIn physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
, such a vector field is a
potential flowIn fluid dynamics, potential flow describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications...
. In
magnetostaticsMagnetostatics is the study of static magnetic fields. In electrostatics, the charges are stationary, whereas here, the currents are stationary or dc...
, such vector fields model static
magnetic fieldMagnetic fields surround magnetic materials and electric currents and are detected by the force they exert on other magnetic materials and moving electric charges...
s on a region of the plane containing no current. In
electrostaticsElectrostatics is the branch of science that deals with the phenomena arising from stationary or slow-moving electric charges.Since classical antiquity it was known that some materials such as amber attract light particles after rubbing. The Greek word for amber, ήλεκτρον , was the source of the...
, they model static electric fields in a region of the plane containing no electric charge.
Other representations
Other representations of the Cauchy-Riemann equations occasionally arise in other
coordinate systemIn mathematics and its applications, a coordinate system is a system for assigning an n-tuple of numbers or scalars to each point in an n-dimensional space. This concept is part of the theory of manifolds. "Scalars" in many cases means real numbers, but, depending on context, can mean complex...
s. If (1a) and (1b) hold for a continuously differentiable pair of functions
u and
v, then so do
for any coordinate system such that the pair is orthonormal and
positively orientedIn mathematics, an orientation on a real vector space is a choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional Euclidean space, the two possible basis orientations are called right-handed and left-handed , respectively...
. 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
ƒ gives
Inhomogeneous equations
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 + i
v 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
closureIn mathematics, the closure of a subset S in a topological space consists of all points which are intuitively "close to S". A point which is in the closure of S is a point of closure of S...
of
D. Indeed, by the Cauchy integral formula,
for all
ζ ∈
D.
Goursat's theorem and its generalizations
Suppose that is a complex-valued function which is
differentiableIn mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in the calculus of variations. Intuitively, it generalizes the idea of linear approximation from...
as a function . Then
Goursat's theorem asserts that
ƒ 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
ƒ 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 derivativeIn 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...
s of
ƒ with respect to
x and
y exist in Ω, and satisfies the Cauchy-Riemann equations throughout Ω, then
ƒ is holomorphic (and thus analytic). This result is the
Looman–Menchoff theoremIn 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...
.
The hypothesis that
ƒ 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.,
ƒ(
z) = . 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 senseIn mathematics, a weak derivative is a generalization of the concept of the derivative of a function for functions not assumed differentiable, but only integrable, i.e. to lie in the Lebesgue space . See distributions for an even more general definition.- Definition :Let be a function in the...
, then the function is analytic. More precisely :
- If ƒ(z) is locally integrable in an open domain Ω ⊂ C, and satisfies the Cauchy-Riemann equations weakly, then ƒ agrees almost everywhere
In measure theory , one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
with an analytic function in Ω.
This is in fact a special case of a more general result on the regularity of solutions of
hypoellipticIn mathematics, more specifically in the theory of partial differential equations, a partial differential operator defined on an open subsetis called hypoelliptic if for every distribution defined on an open subset such that is , must also be ....
partial differential equations.
Several variables
There are Cauchy-Riemann equations, appropriately generalized, in the theory of
several complex variablesThe theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...
. They form a significant
overdetermined systemIn mathematics, a system of linear equations is considered overdetermined if there are more equations than unknowns. The terminology can be described in terms of the concept of counting constants. Each unknown can be seen as an available degree of freedom...
of PDEs. As often formulated, the
d-bar operator
annihilates holomorphic functions. This generalizes most directly the formulation
where
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