利用者:ShuBraque/sandbox/コーシー・リーマンの方程式

数学の複素解析の分野において、コーシー・リーマンの方程式（英: Cauchy–Riemann equations）とは、オーギュスタン＝ルイ・コーシーおよびベルンハルト・リーマンの両者にちなんで名付けられた等式系であり、2つの偏微分方程式からなる系である。コーシーリーマンの関係式とも呼ばれる。複素関数が複素微分可能であるための必要十分条件であり、言い換えれば正則関数であるための必要十分条件である。また、コーシー・リーマンの方程式は複素関数が連続であるかどうか、微分可能であるかどうかの基準としてもしばしば使われる。この等式系を最初に言及したのはJean le Rond d'Alembertの著作である(d'Alembert 1752)。後に、レオンハルト・オイラーはこの等式系を解析関数と結びつけた(Euler 1797)。Cauchy (1814)はさらにコーシー・リーマンの方程式を彼の関数論を構築するために用いた。この理論に関するリーマンの論文(Riemann 1851)はは1851年に発表された。

2つの実変数u(x,y) および v(x,y)の2つの実数値関数に関するコーシー・リーマンの方程式は次の2つの等式である。

${\begin{aligned}(1a)\qquad &{\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}\\[6pt](1b)\qquad &{\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}\end{aligned}}$

通常、u が単一複素変数 z = x + iy、f(x + iy) = u(x,y) + iv(x,y) の複素数値関数の実部、v が虚部として扱われる。u および v はC の開部分集合の一点において実微分可能（real differentiable）であると仮定する（Cは複素数の集合である）。すると、これはR²からRへの関数であると考えることができる。これは u および v の偏導関数が存在することを示唆し（ただしこの導関数が連続であるとは限らない）、また f の小変化を線形に近づけることができることを示唆する。するとこの点において u および v の偏導関数がコーシー・リーマンの方程式(1a)および(1b)を満たすことを必要十分条件として、この点でf = u + ivが複素微分可能であるということができる。単にコーシー・リーマンの方程式を満たす偏導関数が存在するだけではその点において複素微分が可能であることを保証しない。u および v は実微分可能である必要があり、これは単なる偏導関数の存在よりも強い条件であるが、これらの偏導関数は必ずしも連続である必要はない。

正則性(Holomorphy) は複素関数の性質であり、Cの連結開部分集合すべての点で微分可能だという性質である（これはCの領域と呼ばれる）。従って、扱われている領域すべてで等式(1a)および(1b)を満たすことを必要十分条件として、実部 u 、虚部 v が実微分可能関数な複素関数 f は正則であると考えることができる。正則関数は解析的である。また、解析的な関数は正則関数であるとも言うことができる。これはすなわち、複素解析では、領域のすべての点で複素微分可能な（正則な）関数は、解析関数である。ただし、実微分可能な関数に関しては真ではない。

実際の用法としては、この性質の対偶を用いてある関数 f (z)が微分不可能であることを示すことが多い。すなわち、関数 f (z)に対して点 z においてコーシー・リーマンの方程式が成り立たないとき、その点において微分不可能であることがわかる。

具体例

z = x + i yとすると、複素関数 f(z) = z²は z 平面上の全ての点で微分可能である。

$f(z)=(x+iy)^{2}=x^{2}-y^{2}+2ixy$

このとき、f(z)の実部と虚部は

$u(x,y)=x^{2}-y^{2}$

$v(x,y)=2xy$

偏導関数は次のようになる。

${\dfrac {\partial u}{\partial x}}=2x$

${\dfrac {\partial u}{\partial y}}=-2y$

${\dfrac {\partial v}{\partial x}}=2y$

${\dfrac {\partial v}{\partial y}}=2x$

これは

${\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}$

${\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}$

であるから、

${\begin{aligned}(1a)\qquad &{\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}\\[6pt](1b)\qquad &{\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}\end{aligned}}$

のコーシー・リーマンの方程式を満たしている。

解釈および再定式化

先述の等式は複素解析の文脈においてある関数が微分可能であるかの条件を示す一つの方法であった。言い換えれば、ひとつだけの複素変数を持つ関数（複素関数）の概念を、伝統的な微分法を用いて包括するものである。この概念を表すメジャーな方法は他にも幾つかあるが、しばしば他の言葉への言い換えが必要となる。

等角写像

まず、コーシー・リーマンの方程式は複素形式に書くことができる。

(2)${i{\dfrac {\partial f}{\partial x}}}={\dfrac {\partial f}{\partial y}}.$

この形式において、コーシー・リーマンの方程式は構造的にヤコビ行列が次の形式のものになる条件に等しい。

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

ただし、 $\scriptstyle a=\partial u/\partial x=\partial v/\partial y$ および $\scriptstyle b=\partial v/\partial x=-\partial u/\partial y$。この形式の行列は複素数の行列表現である。幾何学的には、そのような行列は常に相似拡大を伴う回転の合成写像であり、特に角度を保存する。関数 f(z) のヤコビアンはzにおいて2曲線の交差する点において無限小の線分を持ち、それらを f(z) の対応部分に回転する。従って、ゼロではない導関数を持つコーシー・リーマンの方程式を満たす関数は平面において曲線間の角度を保存する。すなわち、コーシー・リーマンの方程式はある関数が等角（写像）であるための条件となる。

さらに、等角写像同士の合成もまた等角写像となることから、等角写像を伴うコーシー・リーマンの方程式の解の合成は、それ自体がコーシー・リーマンの方程式の解となる必要がある。よって、等角的に不変である。

Complex differentiability

Suppose that

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

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

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

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

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

On the other hand, approaching along the imaginary axis,

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

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

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

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 Rudin (1966), 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 = z₀. So assume f is differentiable at z₀, as a function of two real variables from Ω to C. This is equivalent to the existence of the following linear approximation

$f(z_{0}+\Delta z)-f(z_{0})=f_{x}\,\Delta x+f_{y}\,\Delta y+\eta (\Delta z)\,\Delta z\,$

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

$\Delta f(z_{0})={\frac {f_{x}-if_{y}}{2}}\,\Delta z+{\frac {f_{x}+if_{y}}{2}}\,\Delta {\bar {z}}+\eta (\Delta z)\,\Delta z\,$

Defining the two Wirtinger derivatives as

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

in the limit $\Delta z\rightarrow 0,\Delta {\bar {z}}\rightarrow 0$ the above equality can be written as

$\left.{\frac {df}{dz}}\right|_{z=z_{0}}=\left.{\frac {\partial f}{\partial z}}\right|_{z=z_{0}}+\left.{\frac {\partial f}{\partial {\bar {z}}}}\right|_{z=z_{0}}\cdot {\frac {\bar {dz}}{dz}}+\eta (\Delta z),\;\;\;\;(\Delta z\neq 0).$

For real values of z, we have ${\bar {dz}}/dz=1$ and for purely imaginary z we have ${\bar {dz}}/dz=-1$. Similarly, when approaching z₀ from different directions in the complex plane, the value of ${\bar {dz}}/dz$ is different. But since for complex differentiability the derivative should be the same, approaching from any direction, hence f is complex differentiable at z₀ if and only if $(\partial f/\partial {\bar {z}})=0$ at $z=z_{0}$. But this is exactly the Cauchy–Riemann equations, thus f is differentiable at z₀ if and only if the Cauchy–Riemann equations hold at z₀.

Independence of the complex conjugate

The above proof suggests another interpretation of the Cauchy–Riemann equations. The complex conjugate of z, denoted ${\bar {z}}$, is defined by

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

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

(3)    ${\dfrac {\partial f}{\partial {\bar {z}}}}=0$

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 ${\bar {z}}$. 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

Contour plot of a pair u and v satisfying the Cauchy–Riemann equations. Streamlines (v = const, red) are perpendicular to equipotentials (u = const, blue). The point (0,0) is a stationary point of the potential flow, with six streamlines meeting, and six equipotentials also meeting and bisecting the angles formed by the streamlines.

A standard physical interpretation of the Cauchy–Riemann equations going back to Riemann's work on function theory (see Klein 1893) 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 $u,v$ 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

$\nabla u={\frac {\partial u}{\partial x}}\mathbf {i} +{\frac {\partial u}{\partial y}}\mathbf {j}$

By differentiating the Cauchy–Riemann equations a second time, one shows that u solves Laplace's equation:

${\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0.$

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 $\nabla u\cdot \nabla v=0$. This implies that the gradient of u must point along the $v={\text{const}}$ curves; so these are the streamlines of the flow. The $u={\text{const}}$ curves are the equipotential curves of the flow.

A holomorphic function can therefore be visualized by plotting the two families of level curves $u={\text{const}}$ and $v={\text{const}}$. 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 $\nabla u=0$, the stationary points of the flow, the equipotential curves of $u={\text{const}}$ 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 (1978). Suppose that u and v satisfy the Cauchy–Riemann equations in an open subset of R², and consider the vector field

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

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

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

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

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

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 (Chanson 2007). 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.

This interpretation can equivalently be restated in the language of differential forms. The pair u,v satisfy the Cauchy–Riemann equations if and only if the one-form $v\,dx+u\,dy$ 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 by

$J={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}.$

This 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),v(x,y) are two functions in the plane, put

$f(x,y)={\begin{bmatrix}u(x,y)\\v(x,y)\end{bmatrix}}.$

The Jacobian matrix of f is the matrix of partial derivatives

$Df(x,y)={\begin{bmatrix}{\dfrac {\partial u}{\partial x}}&{\dfrac {\partial u}{\partial y}}\\[12pt]{\dfrac {\partial v}{\partial x}}&{\dfrac {\partial v}{\partial y}}\end{bmatrix}}$

Then the pair of functions u, v satisfies the Cauchy–Riemann equations if and only if the 2×2 matrix Df commutes with J (Kobayashi & Nomizu 1969, Proposition IX.2.2)

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 (1a) and (1b) hold for a differentiable pair of functions u and v, then so do

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

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 = r e^iθ, the equations then take the form

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

Combining these into one equation for f gives

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

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

${\begin{aligned}&{\frac {\partial u}{\partial x}}-{\frac {\partial v}{\partial y}}=\alpha (x,y)\\[4pt]&{\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial x}}=\beta (x,y)\end{aligned}}$

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

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

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,

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

for all ζ ∈ D.

Generalizations

Goursat's theorem and its generalizations

「Cauchy–Goursat theorem」も参照

Suppose that f = u + iv 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 (Rudin 1966, Theorem 11.2). In particular, continuous differentiability of f need not be assumed (Dieudonné 1969, §9.10, Ex. 1).

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 (Looman 1923, p. 107)

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

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 (Gray & Morris 1978, 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.

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

${\bar {\partial }}$

annihilates holomorphic functions. This generalizes most directly the formulation

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

where

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

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

${\begin{aligned}0&=\nabla f=(\sigma _{1}\partial _{x}+\sigma _{2}\partial _{y})(u+\sigma _{1}\sigma _{2}v)\\[4pt]&=\sigma _{1}\partial _{x}u+\underbrace {\sigma _{1}\sigma _{1}\sigma _{2}} _{=\sigma _{2}}\partial _{x}v+\sigma _{2}\partial _{y}u+\underbrace {\sigma _{2}\sigma _{1}\sigma _{2}} _{=-\sigma _{1}}\partial _{y}v=0\end{aligned}}$

Grouping by $\sigma _{1}$ and $\sigma _{2}$:

$\nabla f=\sigma _{1}(\partial _{x}u-\partial _{y}v)+\sigma _{2}(\partial _{x}v+\partial _{y}u)=0\Leftrightarrow {\begin{cases}\partial _{x}u-\partial _{y}v=0\\[4pt]\partial _{x}v+\partial _{y}u=0\end{cases}}$

Henceforth in traditional notation:

${\begin{cases}{\dfrac {\partial u}{\partial x}}={\dfrac {\partial v}{\partial y}}\\[12pt]{\dfrac {\partial u}{\partial y}}=-{\dfrac {\partial v}{\partial x}}\end{cases}}$

Conformal mappings in higher dimensions

Let Ω be an open set in the Euclidean space Rⁿ. The equation for an orientation-preserving mapping $f:\Omega \to \mathbf {R} ^{n}$ to be a conformal mapping (that is, angle-preserving) is that

$Df^{T}Df=(\det(Df))^{2/n}I$

where Df is the Jacobian matrix, with transpose $Df^{T}$, and I denotes the identity matrix (Iwaniec & Martin 2001, p. 32). For n = 2, this system is equivalent to the standard Cauchy–Riemann equations of complex variables, and the solutions are holomorphic functions. In dimension n > 2, 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

References

• Ahlfors, Lars (1953), Complex analysis (3rd ed.), McGraw Hill (1979発行), 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 (1882発行), 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, doi:10.2307/2321164, JSTOR 2321164, https://jstor.org/stable/2321164 .
• Klein, Felix (1893), On Riemann's theory of algebraic functions and their integrals, Cambridge: MacMillan and Bowes ; translated by Frances Hardcastle.
• Iwaniec, T; Martin, G (2001), Geometric function theory and non-linear analysis, Oxford .
• Looman, H. (1923), “Über die Cauchy–Riemannschen Differeitalgleichungen”, Göttinger Nachrichten: 97–108 .
• Kobayashi, S; Nomizu, K (1969), Foundations of differential geometry, volume 2, Wiley .
• Pólya, George; Szegő, Gábor (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 (1987発行), ISBN 0-07-054234-1 .
• Solomentsev, E.D. (2001), “Cauchy–Riemann conditions”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1-55608-010-4, http://eom.springer.de/c/c020970.htm 
• Stewart, Ian; Tall, David (1983), Complex Analysis (1st ed.), CUP (1984発行), ISBN 0-521-28763-4 .

External links

• Weisstein, Eric W. "Cauchy–Riemann Equations". mathworld.wolfram.com (英語).
• Cauchy–Riemann Equations Module by John H. Mathews