Cauchy–Riemann equations

Template:No footnotes In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is holomorphic. This system of equations first appeared in the work of Jean le Rond d'Alembert Template:Harv. Later, Leonhard Euler connected this system to the analytic functions Template:Harv. Template:Harvtxt then used these equations to construct his theory of functions. Riemann's dissertation Template:Harv on the theory of functions appeared in 1851.

The Cauchy–Riemann equations on a pair of real-valued functions of two real variables u(x,y) and v(x,y) are the two equations:

(1a)Template:Quad ${\displaystyle {\dfrac {\partial u}{\partial x}}={\dfrac {\partial v}{\partial y}}}$

(1b)Template:Quad${\displaystyle {\dfrac {\partial u}{\partial y}}=-{\dfrac {\partial v}{\partial x}}}$

Typically u and v are taken to be the real and imaginary parts respectively of a complex-valued function of a single complex variable z = x + iy, f(x + iy) = u(x,y) + iv(x,y). Suppose that u and v are real-differentiable at a point in an open subset of C ( C is the set of complex numbers), which can be considered as functions from R2 to R. This implies that the partial derivatives of u and v exist (although they need not be continuous) and we can approximate small variations of f linearly. Then f = u + iv is complex-differentiable at that point if and only if the partial derivatives of u and v satisfy the Cauchy–Riemann equations (1a) and (1b) at that point. The sole existence of partial derivatives satisfying the Cauchy–Riemann equations is not enough to ensure complex differentiability at that point. It is necessary that u and v be real differentiable, which is a stronger condition than the existence of the partial derivatives, but it is not necessary that these partial derivatives be continuous.

Holomorphy is the property of a complex function of being differentiable at every point of an open and connected subset of C (this is called a domain in C). 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 (1a) and (1b) are satisfied throughout the domain we are dealing with.

The reason why Euler and some other authors relate the Cauchy–Riemann equations with analyticity is that a major theorem in complex analysis says that 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.

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: 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

(2)Template:Quad${\displaystyle {i{\dfrac {\partial f}{\partial x}}}={\dfrac {\partial f}{\partial y}}.}$

In this form, the equations correspond structurally to the condition that the Jacobian matrix is of the form

${\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}},}$

where ${\displaystyle \scriptstyle a=\partial u/\partial x=\partial v/\partial y}$ and ${\displaystyle \scriptstyle b=\partial v/\partial x=-\partial u/\partial y}$. 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. 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.

Complex differentiability

Suppose that

${\displaystyle f(z)=u(z)+i\cdot v(z)}$

is a function of a complex number z. Then the complex derivative of f at a point z0 is defined by

${\displaystyle \lim _{\underset {h\in \mathbf {C} }{h\to 0}}{\frac {f(z_{0}+h)-f(z_{0})}{h}}=f'(z_{0})}$

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

${\displaystyle \lim _{\underset {h\in \mathbf {R} }{h\to 0}}{\frac {f(z_{0}+h)-f(z_{0})}{h}}={\frac {\partial f}{\partial x}}(z_{0}).}$

On the other hand, approaching along the imaginary axis,

${\displaystyle \lim _{\underset {h\in \mathbf {R} }{h\to 0}}{\frac {f(z_{0}+ih)-f(z_{0})}{ih}}={\frac {1}{i}}{\frac {\partial f}{\partial y}}(z_{0}).}$

The equality of the derivative of f taken along the two axes is

${\displaystyle i{\frac {\partial f}{\partial x}}(z_{0})={\frac {\partial f}{\partial y}}(z_{0}),}$

which are the Cauchy–Riemann equations (2) at the point z0.

Conversely, if f : C → C is a function which is differentiable when regarded as a function on R2, then f is complex differentiable if and only if the Cauchy–Riemann equations hold. In other words, if u and v are real-differentiable functions of two real variables, obviously u + iv is a (complex-valued) real-differentiable function, but u + iv is complex-differentiable if and only if the Cauchy–Riemann equations hold.

Indeed, following Template:Harvtxt, suppose f is a complex function defined in an open set Ω ⊂ C. Then, writing z = x + iy for every z ∈ Ω, one can also regard Ω as an open subset of R2, and f 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 f(z) = 0, just for notational simplicity – the proof is identical in general case. So assume f 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 α and β (which are the partial derivatives of f) such that we have the linear approximation

${\displaystyle f(z)=\alpha x+\beta y+\eta (z)z\,}$

where z = x + iy and η(z) → 0 as zz0 = 0. Since ${\displaystyle z+{\bar {z}}=2x}$ and ${\displaystyle z-{\bar {z}}=2iy}$, the above can be re-written as

${\displaystyle f(z)={\frac {\alpha -i\beta }{2}}z+{\frac {\alpha +i\beta }{2}}{\bar {z}}+\eta (z)z\,}$

Defining the two Wirtinger derivatives as

${\displaystyle {\frac {\partial }{\partial z}}={\frac {1}{2}}{\Bigl (}{\frac {\partial }{\partial x}}-i{\frac {\partial }{\partial y}}{\Bigr )},\;\;\;{\frac {\partial }{\partial {\bar {z}}}}={\frac {1}{2}}{\Bigl (}{\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}{\Bigr )},}$

the above equality can be written as

${\displaystyle {\frac {f(z)}{z}}=\left({\frac {\partial f}{\partial z}}\right)(0)+\left({\frac {\partial f}{\partial {\bar {z}}}}\right)(0)\cdot {\frac {\bar {z}}{z}}+\eta (z),\;\;\;\;(z\neq 0).}$

For real values of z, we have ${\displaystyle {\bar {z}}/z=1}$ and for purely imaginary z we have ${\displaystyle {\bar {z}}/z=-1}$ hence f(z)/z has a limit at 0 (i.e., f is complex differentiable at 0) if and only if ${\displaystyle \scriptstyle (\partial f/\partial {\bar {z}})(0)=0}$. But this is exactly the Cauchy–Riemann equations, thus f is differentiable 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 conjugate of z, denoted ${\displaystyle {\bar {z}}}$, is defined by

${\displaystyle {\overline {x+iy}}:=x-iy}$

for real x and y. The Cauchy–Riemann equations can then be written as a single equation

(3)Template:Quad${\displaystyle {\dfrac {\partial f}{\partial {\bar {z}}}}=0}$

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 ${\displaystyle {\bar {z}}}$. 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 Template:Harv 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 field

${\displaystyle {\bar {f}}={\begin{bmatrix}u\\-v\end{bmatrix}}}$

regarded as a (real) two-component vector. Then the second Cauchy–Riemann equation (1b) asserts that ${\displaystyle {\bar {f}}}$ is irrotational (its curl is 0):

${\displaystyle {\frac {\partial (-v)}{\partial x}}-{\frac {\partial u}{\partial y}}=0.}$

The first Cauchy–Riemann equation (1a) asserts that the vector field is solenoidal (or divergence-free):

${\displaystyle {\frac {\partial u}{\partial x}}+{\frac {\partial (-v)}{\partial y}}=0.}$

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 Template:Harv. 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.

Other representations

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

${\displaystyle {\frac {\partial u}{\partial n}}={\frac {\partial v}{\partial s}},\quad {\frac {\partial v}{\partial n}}=-{\frac {\partial u}{\partial s}}}$

for any coordinate system (n(x, y), s(x, y)) such that the pair (∇n, ∇s) is orthonormal and positively oriented. As a consequence, in particular, in the system of coordinates given by the polar representation z = re, the equations then take the form

${\displaystyle {\partial u \over \partial r}={1 \over r}{\partial v \over \partial \theta },\quad {\partial v \over \partial r}=-{1 \over r}{\partial u \over \partial \theta }.}$

Combining these into one equation for f gives

${\displaystyle {\partial f \over \partial r}={1 \over ir}{\partial f \over \partial \theta }.}$

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

${\displaystyle {\frac {\partial u}{\partial x}}-{\frac {\partial v}{\partial y}}=\alpha (x,y)}$
${\displaystyle {\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial x}}=\beta (x,y)}$

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

${\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}=\varphi (z,{\bar {z}})}$

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,

${\displaystyle f(\zeta ,{\bar {\zeta }})={\frac {1}{2\pi i}}\iint _{D}\varphi (z,{\bar {z}}){\frac {dz\wedge d{\bar {z}}}{z-\zeta }}}$

for all ζ ∈ D.

Generalizations

Goursat's theorem and its generalizations

Suppose that f = u + iv is a complex-valued function which is differentiable as a function f : R2R2. 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 Template:Harv. In particular, continuous differentiability of f need not be assumed Template:Harv.

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 satisfies 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 Template:Harv

${\displaystyle f(z)={\begin{cases}\exp(-z^{-4})&\mathrm {if\ } z\not =0\\0&\mathrm {if\ } z=0\end{cases}}}$

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 Template:Harv:

• 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. As often formulated, the d-bar operator

${\displaystyle {\bar {\partial }}}$

annihilates holomorphic functions. This generalizes most directly the formulation

${\displaystyle {\partial f \over \partial {\bar {z}}}=0,}$

where

${\displaystyle {\partial f \over \partial {\bar {z}}}={1 \over 2}\left({\partial f \over \partial x}+i{\partial f \over \partial y}\right).}$

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, are of great interest in the theory of solitons and integrable systems.

Definition in Clifford algebra

In Clifford algebra the complex number ${\displaystyle z=x+iy}$ is representaed as ${\displaystyle z\equiv x+Iy}$ where ${\displaystyle I\equiv \sigma _{1}\sigma _{2}}$. The fundamental derivative operator in Clifford algebra of Complex numbers is defined as ${\displaystyle \nabla \equiv \sigma _{1}\partial _{x}+\sigma _{2}\partial _{y}}$. The function ${\displaystyle f=u+Iv}$ is considered analytic ⇔ ${\displaystyle \nabla f=0}$, which can be calculated in following way:

${\displaystyle 0=\nabla f=(\sigma _{1}\partial _{x}+\sigma _{2}\partial _{y})(u+\sigma _{1}\sigma _{2}v)=\sigma _{1}\partial _{x}u+\underbrace {\sigma _{1}\sigma _{1}\sigma _{2}} _{=\sigma _{2}}\partial _{x}v+\sigma _{2}\partial _{y}u+\underbrace {\sigma _{2}\sigma _{1}\sigma _{2}} _{=-\sigma _{1}}\partial _{y}v=0}$
${\displaystyle \nabla f=\sigma _{1}(\partial _{x}u-\partial _{y}v)+\sigma _{2}(\partial _{x}v+\partial _{y}u)=0}$${\displaystyle {\begin{cases}\partial _{x}u-\partial _{y}v=0\\\partial _{x}v+\partial _{y}u=0\end{cases}}}$

Henceforth in traditional notation:

${\displaystyle {\begin{cases}{\dfrac {\partial u}{\partial x}}={\dfrac {\partial v}{\partial y}}\\{\dfrac {\partial u}{\partial y}}=-{\dfrac {\partial v}{\partial x}}\end{cases}}}$

References

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