Cauchy-Riemann partial differential equations

The Cauchy-Riemann partial differential equations (also: Cauchy-Riemann differential equations or Cauchy-Riemann equations ) in the mathematical subfield of function theory are a system of two partial differential equations of two real-valued functions . They build a bridge from the real-differentiable functions to the complex-differentiable ones of the (complex) function theory . ${\displaystyle \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {C} \rightarrow \mathbb {C} }$

They first appear at d'Alembert in 1752 . Euler linked this system with the analytical functions in 1777. In a purely function-theoretical context, they appear in Cauchy in 1814 and in Riemann's dissertation in 1851 .

definition

The Cauchy-Riemann differential equations (CRDG) are the system of two differential equations of two real-valued functions in two real variables : ${\displaystyle u,v\colon \mathbb {R} ^{2}\rightarrow \mathbb {R} }$   ${\displaystyle x,\,y}$

${\displaystyle \left.{\begin{aligned}{\frac {\partial u}{\partial x}}(x,y)&={\frac {\partial v}{\partial y}}(x,y)&\qquad \\{\frac {\partial u}{\partial y}}(x,y)&=-{\frac {\partial v}{\partial x}}(x,y)&\qquad \end{aligned}}\right\}}$ (CRDG)

Relationship to the holomorphic functions

Compare also the section Explanations in the article about holomorphic functions .

Isomorphism between the real plane and the complex numbers

${\displaystyle \mathbb {C} }$is naturally a two-dimensional real vector space with the canonical base . This gives rise to a natural identification . A point has the real Cartesian coordinates , or short . A complex-valued function on an open subset of can therefore be understood as a -valent function of two real variables by breaking it down into its real and imaginary parts . ${\displaystyle (1,\mathrm {i} )}$ ${\displaystyle \mathbb {C} \simeq \mathbb {R} ^{2}}$ ${\displaystyle z\in \mathbb {C} }$ ${\displaystyle x:=\operatorname {Re} (z),\;y:=\operatorname {Im} (z)\in \mathbb {R} }$ ${\displaystyle z=x+\mathrm {i} y}$ ${\displaystyle f:U\subset \mathbb {C} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle f\left(x+\mathrm {i} y\right)=u\left(x,y\right)+\mathrm {i} v\left(x,y\right)}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle (x,y)\in {\tilde {U}}:=\{(a,b)\in \mathbb {R} ^{2}\ |\ a+\mathrm {i} b\in U\}}$

Complex differentiability

An important elementary result of the theory of functions is the relationship between the solutions of the Cauchy-Riemann differential equation and the holomorphic (that is, the functions that are complexly differentiable on an open set ). ${\displaystyle U\subseteq \mathbb {C} }$

A function is complex differentiable if and only if its correspondence is (real) differentiable and the functions and the Cauchy-Riemann differential equations satisfy. In this case . ${\displaystyle f}$ ${\displaystyle U}$ ${\displaystyle {\tilde {f}}(x,y)=f(x+\mathrm {i} y)=:u(x,y)+\mathrm {i} v(x,y)}$ ${\displaystyle {\tilde {U}}}$ ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle f'={\tfrac {\partial f}{\partial x}}=-\mathrm {i} {\tfrac {\partial f}{\partial y}}}$

In particular, this statement clarifies the connection between complex and real differentiability of mappings of the plane into the plane. Furthermore it can even be shown that the terms holomorphic and analytical are equivalent. For more equivalent characterizations see Holomorphic Function # Equivalent Properties of Holomorphic Functions of a Variable .

Derivation

If in is differentiable in complex, then exists ${\displaystyle f}$ ${\displaystyle U}$

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

for each . Solving for results in ${\displaystyle z_{0}\in U}$ ${\displaystyle f(z_{0}+h)}$

${\displaystyle f(z_{0}+h)=f(z_{0})+f'(z_{0})\cdot h+r(h)\quad {\text{mit}}\;\lim _{h\rightarrow 0}{\frac {r(h)}{|h|}}=0.}$

If you dismantle and , you get ${\displaystyle f'(z_{0})=:a+\mathrm {i} b}$ ${\displaystyle h=\Delta x+\mathrm {i} \Delta y}$

${\displaystyle f(x_{0}+\Delta x+\mathrm {i} (y_{0}+\Delta y))=f(x_{0}+\mathrm {i} y_{0})+(a+\mathrm {i} b)\Delta x+(-b+\mathrm {i} a)\Delta y+r(h).}$

This shows that is totally differentiable and that the partial derivatives of are given by ${\displaystyle f=u+\mathrm {i} v}$ ${\displaystyle u,v}$

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

example

The function , is holomorphic because their real part and its imaginary are real-differentiable and it is ${\displaystyle f\colon \mathbb {C} \to \mathbb {C} }$ ${\displaystyle f(x+\mathrm {i} y)=x^{2}-y^{2}+2\mathrm {i} {xy}}$ ${\displaystyle u(x,y)=x^{2}-y^{2}}$ ${\displaystyle v(x,y)=2xy}$

${\displaystyle {\frac {\partial u}{\partial x}}(x,y)=2x={\frac {\partial v}{\partial y}}(x,y)}$,

${\displaystyle {\frac {\partial u}{\partial y}}(x,y)=-2y=-{\frac {\partial v}{\partial x}}(x,y)}$.

Other properties

Polar coordinates

The Cauchy-Riemann differential equations can also be represented in other coordinates than the Cartesian ones. The representation in polar coordinates is explained below. A representation of a complex number in polar form is . This means that one has to consider the partial derivatives of to or . For this applies ${\displaystyle z=r\mathrm {e} ^{\mathrm {i} \phi }}$ ${\displaystyle f}$ ${\displaystyle r}$ ${\displaystyle \phi }$

${\displaystyle {\frac {\partial f}{\partial r}}={\frac {\partial z}{\partial r}}{\frac {\partial f}{\partial z}}=\mathrm {e} ^{\mathrm {i} \phi }f'\,\quad {\frac {\partial f}{\partial \phi }}={\frac {\partial z}{\partial \phi }}{\frac {\partial f}{\partial z}}=\mathrm {i} r\mathrm {e} ^{\mathrm {i} \phi }f'.}$

It follows with : ${\displaystyle f=u+\mathrm {i} v}$

${\displaystyle 0={\frac {\partial f}{\partial r}}+{\frac {\mathrm {i} }{r}}{\frac {\partial f}{\partial \phi }}={\frac {\partial u}{\partial r}}+{\frac {\mathrm {i} }{r}}{\frac {\partial u}{\partial \phi }}+\mathrm {i} {\frac {\partial v}{\partial r}}+{\frac {\mathrm {i} ^{2}}{r}}{\frac {\partial v}{\partial \phi }}=\left({\frac {\partial u}{\partial r}}-{\frac {1}{r}}{\frac {\partial v}{\partial \phi }}\right)+\mathrm {i} \left({\frac {1}{r}}{\frac {\partial u}{\partial \phi }}+{\frac {\partial v}{\partial r}}\right).}$

Since both brackets have to disappear, the following applies:

${\displaystyle {\frac {\partial u}{\partial r}}={\frac {1}{r}}{\frac {\partial v}{\partial \phi }}}$

and

${\displaystyle {\frac {\partial v}{\partial r}}=-{\frac {1}{r}}{\frac {\partial u}{\partial \phi }}.}$

These are the Cauchy-Riemann differential equations in polar coordinates.

Relationship to the conforming images

→ Main articles : conformal mapping  and biholomorphic mapping

The complex representation of the Cauchy-Riemann differential equations is

${\displaystyle 0=f'+\mathrm {i} ^{2}f'={\frac {\partial f}{\partial x}}+\mathrm {i} {\frac {\partial f}{\partial y}}\quad \Rightarrow \quad \mathrm {i} {\frac {\partial f}{\partial x}}={\frac {\partial f}{\partial y}}}$

This form of the equation corresponds to the requirement that the Jacobi matrix has the following structure in the matrix representation of the complex numbers

${\displaystyle \left[{\begin{array}{lr}a&-b\\b&a\end{array}}\right]}$     With     ${\displaystyle a={\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}\ ,\quad b={\frac {\partial v}{\partial x}}=-{\frac {\partial u}{\partial y}}}$

The associated with these matrices linear maps are provided and are not both zero, turning stretches in space , is , and , where the scale factor and the rotation angle is. This mapping is therefore angular and orientation scatter; that is, the (oriented) angle between two curves in the plane is retained. Functions that satisfy the Cauchy-Riemann differential equations and whose derivative does not vanish at any point are thus conformal. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle a=r\,\cos(\phi )}$ ${\displaystyle b=r\,\sin(\phi )}$ ${\displaystyle r\neq 0}$ ${\displaystyle \phi }$

Representation by the Cauchy-Riemann operator

→ Main article : Wirtinger calculus

In this section, a more compact notation of the Cauchy-Riemann differential equations is shown. It becomes clear that in holomorphic functions must be independent of the complex conjugate . ${\displaystyle z}$ ${\displaystyle {\bar {z}}}$

A complex number and its complex conjugate are related to the real part and the imaginary part by means of the equations ${\displaystyle z}$ ${\displaystyle {\bar {z}}}$ ${\displaystyle x}$ ${\displaystyle y}$

${\displaystyle {\begin{aligned}z&=x+\mathrm {i} y\ ,\ &{\bar {z}}&=x-\mathrm {i} y\\x&={\frac {z+{\bar {z}}}{2}}\ ,\ &y&={\frac {z-{\bar {z}}}{2\mathrm {i} }}\end{aligned}}}$

together.

Because of this relationship, it makes sense to use the differential operators

${\displaystyle {\begin{aligned}\partial :={\frac {\partial }{\partial z}}&={\frac {\partial x}{\partial z}}{\frac {\partial }{\partial x}}+{\frac {\partial y}{\partial z}}{\frac {\partial }{\partial y}}={\frac {1}{2}}{\Bigl (}{\frac {\partial }{\partial x}}-\mathrm {i} {\frac {\partial }{\partial y}}{\Bigr )}\\{\bar {\partial }}:={\frac {\partial }{\partial {\bar {z}}}}&={\frac {\partial x}{\partial {\bar {z}}}}{\frac {\partial }{\partial x}}+{\frac {\partial y}{\partial {\bar {z}}}}{\frac {\partial }{\partial y}}={\frac {1}{2}}{\Bigl (}{\frac {\partial }{\partial x}}+\mathrm {i} {\frac {\partial }{\partial y}}{\Bigr )}\end{aligned}}}$

define. The operator is called the Cauchy-Riemann operator , and the calculus of these operators is called the Wirtinger calculus . The equation is obtained with the complex representation of the Cauchy-Riemann differential equations from the previous section ${\displaystyle {\bar {\partial }}}$

${\displaystyle 0={\frac {\partial f}{\partial x}}+\mathrm {i} {\frac {\partial f}{\partial y}}=2{\frac {\partial f}{\partial {\bar {z}}}}=2{\bar {\partial }}f\,.}$

Here the partial derivative according to the complex conjugate variable could be identified. the equation

${\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}=0}$   or.   ${\displaystyle {\bar {\partial }}f=0}$

is an alternative representation of the Cauchy-Riemann differential equations and means that if is holomorphic, it must be independent of . Thus, analytic functions can be viewed as real functions of one complex variable rather than a complex function of two real variables. ${\displaystyle f}$ ${\displaystyle {\bar {z}}}$

Relationship to the harmonic functions

Be and functions as in the section "Isomorphism between the real plane and the complex numbers" . Then and are harmonic functions if is holomorphic. Then, namely and twice continuously differentiable (they are even smooth ) and satisfy the Cauchy-Riemann equations. For example for then follows with Black's theorem ${\displaystyle u,v}$ ${\displaystyle f}$ ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle f=u+\mathrm {i} v}$ ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle u}$

${\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}={\frac {\partial }{\partial x}}\left({\frac {\partial v}{\partial y}}\right)={\frac {\partial }{\partial y}}\left({\frac {\partial v}{\partial x}}\right)=-{\frac {\partial ^{2}u}{\partial y^{2}}}}$,

so with the Laplace operator . A similar calculation applies to and results . ${\displaystyle \Delta u=0}$ ${\displaystyle \Delta }$ ${\displaystyle v}$ ${\displaystyle \Delta v=0}$

From the lemma of Weyl follows that any distribution , the Cauchy-Riemann equations in triggers distributional sense , regularly needs to be. Therefore, distributional solutions of the Cauchy-Riemann differential equations are also holomorphic functions. ${\displaystyle T\in {\mathcal {D}}'(\Omega )}$

Physical interpretation

This interpretation does not directly use complex variables. Let there be a function with . The scalar fields and should fulfill the Cauchy-Riemann differential equations (note other sign convention): ${\displaystyle f}$ ${\displaystyle f=u-\mathrm {i} v}$ ${\displaystyle u}$ ${\displaystyle v}$

${\displaystyle {\frac {\partial u}{\partial x}}=-{\frac {\partial v}{\partial y}}\ ,\quad {\frac {\partial v}{\partial x}}={\frac {\partial u}{\partial y}}\;.}$

Now consider the vector field as a real three-component vector: ${\displaystyle {\vec {f}}}$

${\displaystyle {\vec {f}}={\begin{bmatrix}u\\v\\0\end{bmatrix}}\;.}$

Then the first Cauchy-Riemann differential equation describes the freedom of sources :

${\displaystyle 0={\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=\operatorname {div} {\vec {f}}}$

and the second equation describes the freedom of rotation :

${\displaystyle 0={\frac {\partial v}{\partial x}}-{\frac {\partial u}{\partial y}}=\left[\operatorname {rot} {\vec {f}}\right]_{3}\;.}$

It is therefore source-free and has a potential. In fluid mechanics , such a field describes a two-dimensional potential flow . ${\displaystyle {\vec {f}}}$

Inhomogeneous Cauchy-Riemann differential equation in a variable

definition

The inhomogeneous Cauchy-Riemann differential equation has the representation

${\displaystyle {\bar {\partial }}u=f,}$

where is the Cauchy-Riemann operator , is a given function and is the solution we are looking for. The fact that the homogeneous Cauchy-Riemann differential equations defined above corresponds to is already mentioned above in the article. The theory of the inhomogeneous Cauchy-Riemann differential equation is different for solutions in from solutions in with and is touched on here in two different sections. ${\displaystyle {\bar {\partial }}}$ ${\displaystyle f}$ ${\displaystyle u}$ ${\displaystyle {\bar {\partial }}u=0}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle n>1}$

Fundamental solution

For dimension , the fundamental solution of the Cauchy-Riemann operator is given by. That is, the distribution produced by the function solves the equation , where is the delta distribution . If there is a smooth test function with a compact carrier, then you can see the validity of the statement based on it ${\displaystyle n=1}$ ${\displaystyle \textstyle {\frac {\partial }{\partial {\overline {z}}}}}$ ${\displaystyle \textstyle {\frac {1}{\pi z}}}$ ${\displaystyle \textstyle u(z)={\frac {1}{\pi z}}}$ ${\displaystyle \textstyle {\frac {\partial }{\partial {\overline {z}}}}u(z)=\delta }$ ${\displaystyle \delta }$ ${\displaystyle \textstyle \phi \in C_{c}^{\infty }(\mathbb {C} )}$

${\displaystyle {\begin{aligned}\left({\frac {\partial }{\partial {\overline {z}}}}{\frac {1}{z}},\phi \right)_{{\mathcal {D}}\times {\mathcal {D}}'}&=-{\frac {1}{2\mathrm {i} }}\int _{\mathbb {C} }{\frac {1}{z}}\,{\frac {\partial }{\partial {\overline {z}}}}\phi (z)\mathrm {d} {\overline {z}}\mathrm {d} z\\&=-\lim _{\epsilon \to 0}{\frac {1}{2\mathrm {i} }}\int _{\mathbb {C} \backslash B_{\epsilon }}\left({\frac {1}{z}}\,{\frac {\partial }{\partial {\overline {z}}}}\phi (z)+\phi (z)\,{\frac {\partial }{\partial {\overline {z}}}}{\frac {1}{z}}\right)\mathrm {d} {\overline {z}}\mathrm {d} z\\&=-\lim _{\epsilon \to 0}{\frac {1}{2\mathrm {i} }}\int _{\mathbb {C} \backslash B_{\epsilon }}{\frac {\partial }{\partial {\overline {z}}}}{\frac {\phi (z)}{z}}\mathrm {d} {\overline {z}}\mathrm {d} z\\&=\lim _{\epsilon \to 0}{\frac {1}{2\mathrm {i} }}{\frac {2\mathrm {i} }{2\mathrm {i} }}\int _{\mathbb {R} ^{2}\backslash B_{\epsilon }}\left(i{\frac {\partial }{\partial x}}{\frac {\phi (x+\mathrm {i} y)}{x+\mathrm {i} y}}-{\frac {\partial }{\partial y}}{\frac {\phi (x+\mathrm {i} y)}{x+\mathrm {i} y}}\right)\mathrm {d} x\mathrm {d} y\\&=\lim _{\epsilon \to 0}{\frac {1}{2\mathrm {i} }}\int _{\partial B_{\epsilon }}\left({\frac {\phi (x+\mathrm {i} y)}{x+\mathrm {i} y}}\mathrm {d} x+\mathrm {i} {\frac {\phi (x+\mathrm {i} y)}{x+\mathrm {i} y}}\mathrm {d} y\right)\\&=\lim _{\epsilon \to 0}{\frac {1}{2\mathrm {i} }}\int _{\partial B_{\epsilon }}{\frac {\phi (z)}{z}}\mathrm {d} z\\&=\pi \phi (0).\end{aligned}}}$

Integral representation

For with you get with ${\displaystyle f\in C^{k}(\mathbb {C} )}$ ${\displaystyle k\geq 1}$

${\displaystyle u(\zeta )={\frac {1}{2\pi \mathrm {i} }}\int _{\mathbb {C} }{\frac {f(z)}{z-\zeta }}\mathrm {d} z\mathrm {d} {\bar {z}}}$

a solution of the inhomogeneous Cauchy-Riemann differential equation with . ${\displaystyle {\bar {\partial }}u=f}$ ${\displaystyle u\in C^{k}(\mathbb {C} )}$

Inhomogeneous Cauchy-Riemann differential equation in several variables

The following is the dimension of the underlying space or the number of components of a function. ${\displaystyle n\in \mathbb {N} }$

definition

The inhomogeneous Cauchy-Riemann differential equation is also represented in several variables

${\displaystyle {\bar {\partial }}u=f,}$

here is the Dolbeault transverse operator , is a given - complex differential form with a compact carrier and is the solution we are looking for. Explicitly this means that the system ${\displaystyle {\bar {\partial }}}$ ${\displaystyle f=(f_{1},\ldots ,f_{n})}$ ${\displaystyle (0,1)}$ ${\displaystyle u}$

${\displaystyle {\frac {\partial u}{\partial {\bar {z}}_{j}}}=f_{j}}$

of partial differential equations for must be solved. The differential operator is the Cauchy-Riemann operator . ${\displaystyle j=1,\ldots ,n}$ ${\displaystyle {\tfrac {\partial }{\partial {\bar {z}}_{j}}}}$

Necessary condition

For the requirement is necessary. You can see this if you apply the Dolbeault transverse operator on both sides of the equation. In this way we get that since for the Dolbeault operator we have differential forms , must have. Since is a (0,1) -form does not mean that it is a holomorphic differential form , because only (p, 0) -forms that satisfy this equation are called holomorphic. ${\displaystyle n>1}$ ${\displaystyle {\bar {\partial }}f=0}$ ${\displaystyle {\bar {\partial }}{\bar {\partial }}u={\bar {\partial }}f}$ ${\displaystyle {\bar {\partial }}{\bar {\partial }}=0}$ ${\displaystyle {\bar {\partial }}f=0}$ ${\displaystyle f}$ ${\displaystyle {\bar {\partial }}f=0}$ ${\displaystyle f}$

Existence statement

Let be a (0,1) -form with and . Then there exists a function such that the Cauchy-Riemann differential equation is satisfied. ${\displaystyle f=(f_{1},\ldots ,f_{n})}$ ${\displaystyle {\bar {\partial }}f=0}$ ${\displaystyle f_{j}\in C_{c}^{k}(\mathbb {C} ^{n})}$ ${\displaystyle u\in C_{c}^{k}(\mathbb {C} ^{n})}$ ${\displaystyle {\bar {\partial }}u=f}$

literature

• Eberhard Freitag , Rolf Busam: Function theory. 1st volume . 3rd revised and expanded edition. Springer, Berlin a. a. 2000, ISBN 3-540-67641-4 ( Springer textbook ).
• Lars Hörmander : An Introduction to Complex Analysis in Several Variables. 2nd revised edition. North Holland Pub. Co. u. a., Amsterdam a. a. 1973, ISBN 0-7204-2450-X ( North-Holland mathematical Library , 7).

Individual evidence

1. ^ J. d'Alembert : Essai d'une nouvelle théorie de la resistance des fluides . In: gallica . 1752.
2. ^ L. Euler : Ulterior disquisitio de formulis integralibus imaginariis . In: Nova Acta Acad. Sci. Petrop. . 10, 1797, pp. 3-19.
3. ^ AL Cauchy: Mémoire sur les intégrales définies . In: Oeuvres complètes Ser. 1 . 1, 1814, pp. 319-506.
4. ↑ B. Riemann : Basics for a general theory of the functions of a variable complex quantity . In: pdf . August.
5. ↑ Otto Forster : Riemann surfaces (= Heidelberger Taschenbücher 184). Springer, Berlin a. a. 1977, ISBN 3-540-08034-1 , p. 174.