Cauchy–Riemann equations explained

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 form a necessary and sufficient condition for a complex function of a complex variable to be complex differentiable.

These equations are andwhere and are real differentiable bivariate functions.

Typically, and are respectively the real and imaginary parts of a complex-valued function of a single complex variable where and are real variables; and are real differentiable functions of the real variables. Then is complex differentiable at a complex point if and only if the partial derivatives of and satisfy the Cauchy–Riemann equations at that point.

A holomorphic function is a complex function that is differentiable at every point of some open subset of the complex plane . It has been proved that holomorphic functions are analytic and analytic complex functions are complex-differentiable. In particular, holomorphic functions are infinitely complex-differentiable.

This equivalence between differentiability and analyticity is the starting point of all complex analysis.

History

The Cauchy–Riemann equations first appeared in the work of Jean le Rond d'Alembert. Later, Leonhard Euler 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.

Simple example

Suppose that

z = x + iy . The complex-valued function f(z) = z ^{2} is differentiable at any point in the complex plane. $$f(z)\; =\; (x\; +\; iy)^2\; =\; x^2\; -\; y^2\; +\; 2ixy$$The real part u(x,y) and the imaginary part v(x, y) areand their partial derivatives are$$u\_x\; =\; 2x;\backslash quad\; u\_y\; =\; -2y;\backslash quad\; v\_x\; =\; 2y;\backslash quad\; v\_y\; =\; 2x$$

We see that indeed the Cauchy–Riemann equations are satisfied,

u _{x} = v _{y} and u _{y} = - v _{x} .

Interpretation and reformulation

The Cauchy-Riemann equations are one way of looking at the condition for 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

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

a = \partial u/\partial x = \partial v/\partial y and 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. 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

Let$$f(z)\; =\; u(z)\; +\; i\; \backslash cdot\; v(z)$$where $u$ and

v are real-valued functions, be a complex-valued function of a complex variable $z\; =\; x\; +\; i\; y$ where $x$ and $y$ are real variables. $f(z)\; =\; f(x\; +\; iy)\; =\; f(x,y)$ so the function can also be regarded as a function of real variables $x$ and $y$. Then, the complex-derivative of $f$ at a point $z\_0=x\_0+iy\_0$ is defined by$$f\text{'}(z\_0)\; =\backslash lim\_\; \backslash frac$$provided this limit exists (that is, the limit exists along every path approaching $z\_$, and does not depend on the chosen path).

A fundamental result of complex analysis is that

f is complex differentiable at z _{0} (that is, it has a complex-derivative), if and only if the bivariate real functions u(x + iy) and v(x + iy) are differentiable at (x _{0,y} _{0),} and satisfy the Cauchy–Riemann equations at this point.^[1]

In fact, if the complex derivative exists at $z\_0$, then it may be computed by taking the limit at $z\_0$ along the real axis and the imaginary axis, and the two limits must be equal. Along the real axis, the limit is $$\backslash lim\_\; \backslash frac\; =\; \backslash left.\; \backslash frac\; \backslash right\; \backslash vert\_$$and along the imaginary axis, the limit is$$\backslash lim\_\; \backslash frac\; =\; \backslash left.\; \backslash frac\backslash frac\; \backslash right\; \backslash vert\; \_.$$

So, the equality of the derivatives implies $$i\; \backslash left.\; \backslash frac\; \backslash right\; \backslash vert\; \_\; =\; \backslash left.\; \backslash frac\; \backslash right\; \backslash vert\; \_$$which is the complex form of Cauchy–Riemann equations at $z\_0$.

(Note that if

f is complex differentiable at z _{0} , it is also real differentiable and the Jacobian of f at z _{0} is the complex scalar f'(z _{0)} , regarded as a real-linear map of C , since the limit |f(z) - f(z _{0)-f'(z} _{0)(z-z} _{0)|/|z-z} _{0|\to} 0 as z\to z _{0} .)

Conversely, if is differentiable at $z\_$ (in the real sense) and satisfies the Cauchy-Riemann equations there, then it is complex-differentiable at this point. Assume that as a function of two real variables and is differentiable at (real differentiable). This is equivalent to the existence of the following linear approximation $$\backslash Delta\; f(z\_0)\; =\; f(z\_0\; +\; \backslash Delta\; z)\; -\; f(z\_0)\; =\; f\_x\; \backslash ,\backslash Delta\; x\; +\; f\_y\; \backslash ,\backslash Delta\; y\; +\; \backslash eta(\backslash Delta\; z)$$where $f\_x\; =\; \backslash left.\; \backslash frac\backslash right\; \backslash vert\; \_$, $f\_y\; =\; \backslash left.\; \backslash frac\; \backslash right\; \backslash vert\; \_$,, and $\backslash eta(\backslash Delta\; z)\; /\; |\backslash Delta\; z|\; \backslash to\; 0$ as .

Since $\backslash Delta\; z\; +\; \backslash Delta\; \backslash bar=\; 2\; \backslash ,\; \backslash Delta\; x$ and $\backslash Delta\; z\; -\; \backslash Delta\; \backslash bar=2i\; \backslash ,\; \backslash Delta\; y$, the above can be re-written as

$$\backslash Delta\; f(z\_0)\; =\; \backslash frac\; \backslash ,\; \backslash Delta\; z\; +\; \backslash frac\; \backslash ,\; \backslash Delta\; \backslash bar\; +\; \backslash eta(\backslash Delta\; z)\backslash ,$$$$\backslash frac\; =\; \backslash frac+\; \backslash frac\backslash cdot\; \backslash frac\; +\; \backslash frac,\; \backslash ;\backslash ;\backslash ;\backslash ;(\backslash Delta\; z\; \backslash neq\; 0).$$

Now, if $\backslash Delta\; z$ is real, $\backslash Delta\backslash bar\; z/\backslash Delta\; z\; =\; 1$, while if it is imaginary, then $\backslash Delta\backslash bar\; z/\backslash Delta\; z=-1$. Therefore, the second term is independent of the path of the limit $\backslash Delta\; z\backslash to\; 0$ when (and only when) it vanishes identically: $f\_x\; +\; i\; f\_y=0$, which is precisely the Cauchy–Riemann equations in the complex form. This proof also shows that, in that case,$$\backslash left.\backslash frac\backslash right|\_\; =\; \backslash lim\_\backslash frac\; =\; \backslash frac.$$

Note that the hypothesis of real differentiability at the point

z _{0} is essential and cannot be dispensed with. For example,^[2] the function f(x,y) = \sqrt{|xy|} , regarded as a complex function with imaginary part identically zero, has both partial derivatives at (x _{0,y} _{0)=(0,0)} , and it moreover satisfies the Cauchy–Riemann equations at that point, but it is not differentiable in the sense of real functions (of several variables), and so the first condition, that of real differentiability, is not met. Therefore, this function is not complex differentiable.

Some sources^{[3] [4]} state a sufficient condition for the complex differentiability at a point

z _{0} as, in addition to the Cauchy–Riemann equations, the partial derivatives of u and v be continuous at the point because this continuity condition ensures the existence of the aforementioned linear approximation. Note that it is not a necessary condition for the complex differentiability. For example, the function f(z) = z ^{2e} ^{i/|z|} is complex differentiable at 0, but its real and imaginary parts have discontinuous partial derivatives there. Since complex differentiability is usually considered in an open set, where it in fact implies continuity of all partial derivatives (see below), this distinction is often elided in the literature.

Independence of the complex conjugate

The above proof suggests another interpretation of the Cauchy–Riemann equations. The complex conjugate of

z , denoted $\backslash bar$, is defined by$$\backslash overline\; :=\; x\; -\; iy$$for real variables x and y . Defining the two Wirtinger derivatives as$$\backslash frac\; =\; \backslash frac\; \backslash left(\backslash frac\; -\; i\; \backslash frac\; \backslash right),\; \backslash ;\backslash ;\backslash ;\; \backslash frac\; =\; \backslash frac\; \backslash left(\backslash frac\; +\; i\; \backslash frac\; \backslash right),$$the Cauchy–Riemann equations can then be written as a single equation$$\backslash frac\; =\; 0,$$and the complex derivative of $f$ in that case is $\backslash frac=\backslash frac.$ In this form, the Cauchy–Riemann equations can be interpreted as the statement that a complex function $f$ of a complex variable $z$ is independent of the variable $\backslash bar$. As such, we can view analytic functions as true functions of one complex variable ($z$) instead of complex functions of two real variables ($x$ and $y$).

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 u and v 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 of u, defined by$$\backslash nabla\; u\; =\; \backslash frac\backslash mathbf\; i\; +\; \backslash frac\backslash mathbf\; j.$$

By differentiating the Cauchy–Riemann equations for the functions u and v, with the symmetry of second derivatives, one shows that u solves Laplace's equation:$$\backslash frac\; +\; \backslash frac\; =\; 0.$$That is, u is a harmonic function. This means that the divergence 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 $\backslash nabla\; u\backslash cdot\backslash nabla\; v\; =\; 0$ ($\backslash nabla\; u\backslash cdot\backslash nabla\; v\; =\; \backslash frac\; \backslash cdot\; \backslash frac\; +\; \backslash frac\; \backslash cdot\; \backslash frac\; =\; \backslash frac\; \backslash cdot\; \backslash frac\; -\; \backslash frac\; \backslash cdot\; \backslash frac\; =\; 0$), i.e., the direction of the maximum slope of u and that of v are orthogonal to each other. This implies that the gradient of u must point along the $v\; =\; \backslash text$ curves; so these are the streamlines of the flow. The $u\; =\; \backslash text$ curves are the equipotential curves of the flow.

A holomorphic function can therefore be visualized by plotting the two families of level curves $u=\backslash text$ and $v=\backslash text$. Near points where the gradient of u (or, equivalently, v) is not zero, these families form an orthogonal family of curves. At the points where $\backslash nabla\; u=0$, the stationary points of the flow, the equipotential curves of $u=\backslash text$ 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 R², and consider the vector field$$\backslash bar\; =\; \backslash begin\; u\backslash \backslash \; -v\; \backslash end$$regarded as a (real) two-component vector. Then the second Cauchy–Riemann equation asserts that

\bar{f} is irrotational (its curl is 0):$$\backslash frac\; -\; \backslash frac\; =\; 0.$$

The first Cauchy–Riemann equation asserts that the vector field is solenoidal (or divergence-free):$$\backslash frac\; +\; \backslash frac=0.$$

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. 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.

vdx + u dy is both closed 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 byThis is a complex structure in the sense that the square of J is the negative of the 2×2 identity matrix:

J ^{2} = - I . As above, if u(x,y) and v(x,y) are two functions in the plane, put

$$f(x,y)\; =\; \backslash beginu(x,y)\backslash \backslash v(x,y)\backslash end.$$

The Jacobian matrix 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, 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 systems. If and hold for a differentiable pair of functions u and v, then so do$$\backslash frac\; =\; \backslash frac,\backslash quad\; \backslash frac\; =\; -\backslash frac$$

for any coordinate system such that the pair $(\backslash nabla\; n,\backslash nabla\; s)$ is orthonormal and positively oriented. As a consequence, in particular, in the system of coordinates given by the polar representation

z = r e ^{i\theta} , the equations then take the form$$=,\backslash quad\; =\; -.$$

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 R². These equations are usually combined into a single equation$$\backslash frac\; =\; \backslash varphi(z,\backslash bar)$$where f = u + iv and ' = (α + iβ)/2.

If ' is C^k, then the inhomogeneous equation is explicitly solvable in any bounded domain D, provided ' is continuous on the closure of D. Indeed, by the Cauchy integral formula,$$f\backslash left(\backslash zeta,\; \backslash bar\backslash right)\; =\; \backslash frac\; \backslash iint\_D\; \backslash varphi\backslash left(z,\; \backslash bar\backslash right)\; \backslash ,\; \backslash frac$$for all ζ ∈ D.

Generalizations

Goursat's theorem and its generalizations

See also: Cauchy–Goursat theorem.

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 derivatives 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.

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 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. This is done using a straightforward generalization of the Wirtinger derivative, 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$$\backslash bar$$annihilates holomorphic functions. This generalizes most directly the formulation$$=\; 0,$$where$$=\; \backslash left(+\; i\backslash right).$$

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

C\ell( 2 ) , the complex number z = x + iy is represented as z \equiv x + J y where J \equiv \sigma _{1} \sigma _{2} , (\sigma ^{2=1 ,} _{2} \sigma _{1} \sigma _{2} + \sigma _{2} \sigma _{1} = 0 , so J ^{2=-1} ). The Dirac operator in this Clifford algebra is defined as \nabla \equiv \sigma _{1} \partial _{x} + \sigma _{2\partial} _{y} . The function f = u + J v is considered analytic if and only if \nabla f = 0 , which can be calculated in the following way:

Grouping by

\sigma _{1} and \sigma _{2} :

$$\backslash nabla\; f\; =\; \backslash sigma\_1\; (\backslash partial\_x\; u\; -\; \backslash partial\_y\; v)\; +\; \backslash sigma\_2\; (\backslash partial\_x\; v\; +\; \backslash partial\_y\; u)\; =\; 0\; \backslash Leftrightarrow\; \backslash begin\backslash partial\_x\; u\; -\; \backslash partial\_y\; v\; =\; 0\backslash \backslash [4pt]\backslash partial\_x\; v\; +\; \backslash partial\_y\; u\; =\; 0\backslash end$$

Hence, in traditional notation:

$$\backslash begin\backslash dfrac\; =\; \backslash dfrac\backslash \backslash [12pt]\backslash dfrac\; =\; -\backslash dfrac\backslash end$$

Conformal mappings in higher dimensions

Let Ω be an open set in the Euclidean space Rⁿ. The equation for an orientation-preserving mapping

f:\Omega\to R ^{n} to be a conformal mapping (that is, angle-preserving) is that$$Df^\backslash mathsf\; Df\; =\; (\backslash det(Df))^I$$

where Df is the Jacobian matrix, with transpose

Df ^{T} , 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.

See also

• List of complex analysis topics
• Morera's theorem
• Wirtinger derivatives

Sources

• Gray . J. D. . Morris . S. A. . April 1978 . When is a Function that Satisfies the Cauchy - Riemann Equations Analytic? . The American Mathematical Monthly . 85 . 4 . 246–256 . 10.2307/2321164 . 2321164 .
• Looman . H. . 1923 . Über die Cauchy - Riemannschen Differentialgleichungen . Göttinger Nachrichten . 97–108 . de.
• Book: Marsden. A. Hoffman. M. Basic complex analysis. 1973. W. H. Freeman.
• Book: Rudin, Walter . Walter Rudin . 1966 . Real and complex analysis . McGraw Hill . 1987 . 3rd . 0-07-054234-1.

Further reading

• Book: Ahlfors, Lars . Lars Ahlfors . 1953 . 1979 . Complex analysis . 3rd . McGraw Hill . 0-07-000657-1.
  • Book: Stewart . Ian . Tall . David . Ian Stewart (mathematician) . 1983 . 1984 . Complex Analysis . 1st . CUP . 0-521-28763-4.

External links

• • Cauchy–Riemann Equations Module by John H. Mathews

Notes and References

1. Book: Markushevich, A.I.. Theory of functions of a complex variable 1. Chelsea. 1977., p. 110-112 (Translated from Russian)
2. Book: Titchmarsh, E. The theory of functions. 1939. Oxford University Press., 2.14
3. Book: Arfken . George B. . Mathematical Methods for Physicists: A Comprehensive Guide . Weber . Hans J. . Harris . Frank E. . Academic Press . 2013 . 978-0-12-384654-9 . 7th . 471–472 . English . 11.2 CAUCHY-RIEMANN CONDITIONS.
4. Book: Hassani, Sadri . Mathematical Physics: A Modern Introduction to Its Foundations . Springer . 2013 . 978-3-319-01195-0 . 2nd . 300–301 . English . 10.2 Analytic Functions.