Cauchy–Riemann equations

In mathematics, the Cauchy–Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations that provides 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 (d'Alembert 1752). Later, Leonhard Euler connected this system to the analytic functions (Euler 1777). Cauchy (1814) then used these equations to construct his theory of functions. Riemann's dissertation (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)    

and

(1b)    

Typically the pair u and v are taken to be the real and imaginary parts of a complex-valued function f(x + iy) = u(x,y) + iv(x,y). Suppose that u and v are continuously differentiable on an open subset of C. Then f = u+iv is holomorphic if and only if the partial derivatives of u and v satisfy the Cauchy-Riemann equations (1a) and (1b).

Contents 1 Interpretation and reformulation 1.1 Conformal mappings 1.2 Complex differentiability 1.3 Independence of the complex conjugate 1.4 Physical interpretation 1.5 Other representations 2 Inhomogeneous equations 3 Generalizations 3.1 Goursat's theorem and its generalizations 3.2 Several variables 4 See also 5 References 6 External links

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 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

Firstly, 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. 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

The Cauchy-Riemann equations are necessary and sufficient conditions for the complex differentiability (or holomorphicity) of a function (Ahlfors 1953, §1.2). 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 Rudin (1966), 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 α and β (which are the partial derivatives of ƒ) such that

f(z) = αx + βy + η(z)z

where z = x + iy and as Since and , the above can be re-written as

Using the two differential operators

the above equality can be written as

For real values of z, we have and for purely imaginary z we have hence f(z) / z 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 conjugate of z, denoted , is defined by

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

(3)    

where the differential operator 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 (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

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

The second Cauchy-Riemann equation (1b) 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 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 (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

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 z = r e^iθ, 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 + iv 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,

for all ζ ∈ D.

Generalizations

Goursat's theorem and its generalizations

Suppose that ƒ = u + iv is a complex-valued function which is differentiable as a function ƒ : R² → R². 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 (Rudin 1966, Theorem 11.2). In particular, continuous differentiability of ƒ need not be assumed (Dieudonné 1969, §9.10, Ex. 1).

The hypotheses of Goursat's theorem can be weakened significantly. If ƒ = u + iv is continuous in an open set Ω and the partial derivatives 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 theorem.

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) = z⁵ / |z|⁴). Similarly, some additional assumption is needed besides the Cauchy-Riemann equations (such as continuity), as the following example illustrates (Looman 1923, p. 107)

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 (Gray & Morris 1978, Theorem 9):

• If ƒ(z) is locally integrable in an open domain Ω ⊂ C, and satisfies the Cauchy-Riemann equations weakly, then ƒ 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

annihilates holomorphic functions. This generalizes most directly the formulation

where

See also

• List of complex analysis topics
• Morera's theorem

References

• Ahlfors, Lars (1953), Complex analysis (3rd ed.), McGraw Hill (published 1979), ISBN 0-07-000657-1 .
• d'Alembert, J. (1752), Essai d'une nouvelle théorie de la résistance des fluides, Paris, http://gallica2.bnf.fr/ark:/12148/bpt6k206036b.modeAffichageimage.f1.langFR.vignettesnaviguer .
• Cauchy, A.L. (1814), Mémoire sur les intégrales définies,, Oeuvres complètes Ser. 1, 1, Paris (published 1882), pp. 319–506 
• Chanson, H. (2007), "Le Potentiel de Vitesse pour les Ecoulements de Fluides Réels: la Contribution de Joseph-Louis Lagrange." ('Velocity Potential in Real Fluid Flows: Joseph-Louis Lagrange's Contribution.')", Journal La Houille Blanche 5: 127–131, doi:10.1051/lhb:2007072, ISSN 0018-6368, http://espace.library.uq.edu.au/view/UQ:119883 .
• Dieudonné, Jean Alexander (1969), Foundations of modern analysis, Academic Press .
• Euler, L. (1797), Nova Acta Acad. Sci. Petrop. 10: 3–19 
• Gray, J. D.; Morris, S. A. (1978), "When is a Function that Satisfies the Cauchy–Riemann Equations Analytic?", The American Mathematical Monthly 85 (4): 246–256, April 1978, http://www.jstor.org/stable/2321164 .
• Looman, H. (1923), "Über die Cauchy–Riemannschen Differeitalgleichungen", Göttinger Nach.: 97–108 .
• Pólya, George; Szegö, Gabor (1978), Problems and theorems in analysis I, Springer, ISBN 3-540-63640-4 
• Riemann, B. (1851), "Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen komplexen Grösse", in H. Weber, Riemann's gesammelte math. Werke, Dover, 1953, pp. 3–48 
• Rudin, Walter (1966), Real and complex analysis (3rd ed.), McGraw Hill (published 1987), ISBN 0-07-054234-1 .
• Solomentsev, E.D. (2001), "Cauchy–Riemann conditions", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 

External links

• Weisstein, Eric W., "Cauchy-Riemann Equations" from MathWorld.
• Cauchy-Riemann Equations Module by John H. Mathews