Cauchy-Riemann equations

In mathematics, the Cauchy-Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two partial differential equations which provide a necessary and sufficient condition for a differentiable function to be holomorphic in an open set. This system of equations first appeared in the work of Jean le Rond d'Alembert harv|d'Alembert|1752. Later, Leonhard Euler connected this system to the analytic functions harv|Euler|1777. harvtxt|Cauchy|1814 then used these equations to construct his theory of functions. Riemann's dissertation harv|Riemann|1851 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:

:(1a)quad $${ partial u over partial x } = { partial v over partial y }

and

:(1b)quad$${ partial u over partial y } = -{ partial v over partial x } .

Typically the pair "u" and "v" are taken to be the real and imaginary parts of a complex-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

Conformal mappings

The Cauchy-Riemann equations are often reformulated in a variety of ways. Firstly, they may be written in complex form

:(2)quad$${ i { partial f over partial x } } = { partial f over partial y } .

In this form, the equations correspond structurally to the condition that the Jacobian matrix is of the form:$$egin{pmatrix} a & -b \ b & ;; a end{pmatrix},

where $$scriptstyle a=partial u/partial x=partial v/partial y and $$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.

Independence of the complex conjugate

Let $$ar{z} denote the complex conjugate of "z", defined by:$$overline{x + iy} := x - iyfor real "x" and "y". The Cauchy-Riemann equations are sometimes written as a single equation

:(3)quad$$frac{partial f}{partialar{z = 0

where the differential operator $$frac{partial}{partialar{z is defined by

:$$frac{partial}{partialar{z = frac{1}{2}left(frac{partial}{partial x} + ifrac{partial}{partial y} ight).

In this form, the Cauchy-Riemann equations can be interpreted as the statement that "f" is independent of the variable $$ar{z}.

Complex differentiability

The Cauchy-Riemann equations are necessary and sufficient conditions for the complex differentiability (or holomorphicity) of a function harv|Ahlfors|1953|loc=§1.2. Specifically, suppose that

:$$f(z) = u(z) + i v(z)

is a function of a complex number "z"∈C. Then the complex derivative of "f" at a point "z"₀ is defined by

:$$lim_{underset{hinmathbb{C{h o 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

:$$lim_{underset{hinmathbb{R{h o 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,

:$$lim_{underset{hin mathbb{R{h o 0 frac{f(z_0+ih)-f(z_0)}{ih} =lim_{underset{hin mathbb{R{h o 0 -ifrac{f(z_0+ih)-f(z_0)}{h} =-ifrac{partial f}{partial y}(z_0).

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

:$$frac{partial f}{partial x}(z_0)=-ifrac{partial f}{partial y}(z_0),

which are the Cauchy-Riemann equations (2) at the point "z"₀.

Conversely, if "f":C → C is a function which is differentiable when regarded as a function into R², then "f" is complex differentiable if and only if the Cauchy-Riemann equations hold.

Physical interpretation

One interpretation of the Cauchy-Riemann equations harv|Pólya|Szegö|1978 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

:$$ar{f} = egin{bmatrix}u\ -vend{bmatrix}

regarded as a (real) two-component vector. Then the first Cauchy-Riemann equation (1a) asserts that $$ar{f} is irrotational:

:$$frac{partial v}{partial x} - frac{partial u}{partial y} = 0.

The second Cauchy-Riemann equation (1b) asserts that the vector field is solenoidal (or divergence-free):

:$$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 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 theorem.) In fluid dynamics, such a vector field is a potential flow harv|Chanson|2000. 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 continuously differentiable pair of functions "u" and "v", then so do:$$frac{partial u}{partial s} = frac{partial u}{partial n},quad frac{partial u}{partial n} = -frac{partial u}{partial s}for any coordinates ("n"("x","y"), "s"("x","y")) such that the pair $$scriptstyle ( abla n, abla s) is orthonormal and positively oriented. As a consequence, in particular,in the system of coordinates given by the polar representation "z"="re"^iθ the equations then take the form

:$${ partial u over partial r } = {1 over r}{ partial v over partial heta},quad{ partial v over partial r } = -{1 over r}{ partial u over partial heta}.

Combining these into one equation for "f" gives

:$${partial f over partial r} = {1 over i r}{partial f over partial heta}.

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

:$$frac{partial u}{partial x}-frac{partial v}{partial y} = alpha(x,y)

:$$frac{partial u}{partial y}+frac{partial v}{partial x} = eta(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

:$$frac{partial f}{partialar{z = phi(z,ar{z})

where "f"="u"+i"v" and φ=(α+iβ)/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,

:$$f(zeta,ar{zeta}) = frac{1}{2pi i}iint_D phi(z,ar{z})frac{dzwedge dar{z{z-zeta}

for all ζ∈"D".

Generalizations

Goursat's theorem and its generalizations

Suppose that "f" = "u"+i"v" 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 harv|Rudin|1966|loc=Theorem 11.2. In particular, continuous differentiability of "f" need not be assumed harv|Dieudonné|1969|loc=§9.10, Ex. 1.

The hypotheses of Goursat's theorem can be weakened significantly. If "f"="u"+i"v" is continuous in an open set Ω and the partial derivatives of "f" with respect to "x" and "y" exist in 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") = "z"⁵/|z|⁴). Similarly, some additional assumption is needed besides the Cauchy-Riemann equations (such as continuity), as the following example illustrates harv|Looman|1923|p=107:$$f(z) = egin{cases}exp(-z^{-4})&mathrm{if }z ot=0\0&mathrm{if }z=0end{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 harv|Gray|Morris|1978|loc=Theorem 9:
* 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.

everal 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"

:$$ar{partial}

annihilates holomorphic functions. This generalizes most directly the formulation

:$${partial f over partial ar z} = 0,

where

:$${partial f over partial ar z} = {1 over 2}left({partial f over partial x} - {1 over i}{partial f over partial y} ight).

ee also

*List of complex analysis topics

References

*citation|first=Lars|last=Ahlfors|authorlink=Lars Ahlfors|title=Complex analysis|publisher=McGraw Hill|year=1953|publication-date=1979|edition=3rd|isbn=0-07-000657-1.
*citation|first=J.|last=d'Alembert|authorlink=Jean le Rond d'Alembert|title=Essai d'une nouvelle théorie de la résistance des fluides|url=http://gallica2.bnf.fr/ark:/12148/bpt6k206036b.modeAffichageimage.f1.langFR.vignettesnaviguer|publication-place=Paris|year=1752.
*citation|first=A.L.|last=Cauchy|authorlink=Augustin Cauchy|title=Mémoire sur les intégrales définies, |series=Oeuvres complètes Ser. 1|volume=1|publication-place=Paris|year=1814|publication-date=1882|pages=319–506
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*citation|first=L.|last=Euler|authorlink=Leonhard Euler|journal=Nova Acta Acad. Sci. Petrop.|volume=10|year=1797|pages=3–19
*citation|title=When is a Function that Satisfies the Cauchy-Riemann Equations Analytic?|first1=J. D.|last1=Gray|first2=S. A.|last2=Morris|journal=The American Mathematical Monthly|volume=85|number=4|year=1978|publication-date=April 1978|pages=246-256|url=http://www.jstor.org/stable/2321164.
*citation|first=H.|last=Looman|title=Über die Cauchy-Riemannschen Differeitalgleichungen|journal=Göttinger Nach.|year=1923|pages=97-108.
*
*citation|last=Riemann|first=B.|authorlink=Bernhard Riemann|contribution=Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen komplexen Grösse|editor=H. Weber|title=Riemann's gesammelte math. Werke|publisher=Dover|publication-date=1953|pages=3–48|year=1851
*.
*

External links

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* [http://math.fullerton.edu/mathews/c2003/CauchyRiemannMod.html Cauchy-Riemann Equations Module by John H. Mathews]