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Methods for solving elliptic partial differential equations involving the representation of solutions by way of analytic functions of a complex variable.

The theory of analytic functions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d0319301.png" /></td> </tr></table>

of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d0319302.png" /> is the theory of two real-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d0319303.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d0319304.png" /> satisfying the Cauchy–Riemann system of equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d0319305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d0319306.png" />, which is essentially equivalent to the Laplace equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d0319307.png" /></td> </tr></table>

Since the 1930s, methods of analytic function theory have been used to an increasing extent in the general theory of equations of elliptic type. Thus arose a new branch of analysis, a substantial extension of the classical theory of analytic functions and their applications. The main subject of this branch are representation formulas for all solutions of a very extensive class of equations of elliptic type by analytic functions of one complex variable. For linear equations these representations are realized using certain linear operators, expressed in terms of the coefficients of the equations. These formulas make it possible to extend the properties of analytic functions to solutions of an equation of elliptic type, and important properties such as the uniqueness theorem, the principle of the argument, Liouville's theorem, etc., are often literally preserved. Taylor and Laurent series, the Cauchy integral formula, the compactness principle, the principle of analytic continuation, etc., are extended in a natural way.

The complex representation formulas permit the construction of various families of particular solutions of equations displaying certain properties. For instance, it is possible to construct various classes of so-called elementary solutions with point singularities, which are employed to obtain various integral formulas. So-called complete systems of particular solutions can be constructed; the latter have the property that their linear combinations approximate any solution. Complex representation formulas also make it possible to reduce many boundary value problems to equivalent problems for analytic functions, and to construct Fredholm or singular integral equations which are equivalent to these problems. Boundary value problems of non-Fredholm type may also be studied and a condition of normal solvability and explicit formulas for the index may thus be obtained. See [[Boundary value problem, complex-variable methods|Boundary value problem, complex-variable methods]].

==Elliptic equations with analytic coefficients.==
Let the second-order equation of elliptic type

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d0319308.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

be given, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d0319309.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193011.png" /> are analytic functions in the real variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193013.png" /> in some domain of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193014.png" />-plane, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193015.png" />. Analytic continuation of the coefficients into the domain of the independent complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193017.png" /> yields the following form of equation (1):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

A simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193019.png" /> is said to be a fundamental domain for equation (1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193022.png" /> are analytic functions of two independent variables in the cylindrical domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193024.png" /> denotes the mirror image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193025.png" /> with respect to the real axis.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193026.png" /> is a simply-connected domain, all solutions of equation (1) regular in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193027.png" /> are expressed by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193029.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193030.png" /> is an arbitrary holomorphic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193032.png" /> are arbitrary fixed points; the analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193033.png" /> of four independent complex arguments in the cylindrical domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193034.png" /> is said to be the Riemann function of equation (1). It is the solution of the integral equation of Volterra type:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193036.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193037.png" /></td> </tr></table>

The correspondence between the family of solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193038.png" /> of equation (1) and the family of holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193039.png" /> produced by formula (3) will be a one-to-one correspondence if the values of the imaginary parts of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193040.png" /> are fixed in a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193041.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193042.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193043.png" />, the inversion formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193044.png" /></td> </tr></table>

is valid. Equation (4) can be solved by the method of successive approximation. An approximate expression of the Riemann function can be obtained in this way.

If the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193045.png" /> is multiply connected, formula (3) usually gives multi-valued solutions. In order to obtain all single-valued solutions of equation (1) in this case, multi-valued functions of a certain type must be taken for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193046.png" /> in (3).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193047.png" /> be a doubly-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193048.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193049.png" /> be a bounded continuum completing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193050.png" /> to a simply-connected domain. All solutions of equation (1) which are single-valued in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193051.png" /> are then given by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193052.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193053.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193055.png" /> are fixed points, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193056.png" /> is a multi-valued analytic function of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193057.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193058.png" /></td> </tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193059.png" /> is an arbitrary real constant, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193060.png" /> is an arbitrary holomorphic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193061.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193062.png" /> is any simple, closed, piecewise-smooth curve lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193063.png" /> and enclosing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193064.png" />. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193066.png" /> are expressed by the formulas

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193067.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193068.png" /></td> </tr></table>

Complex representations of the type (3) are also extended to a system of equations written in a vector form (1), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193069.png" /> is a vector with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193070.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193071.png" /> are square matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193072.png" /> whose entries are analytic functions of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193073.png" />.

In a domain in which there is at least one positive solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193074.png" /> for (1), equation (1) may be converted to the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193075.png" /></td> </tr></table>

by substituting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193076.png" />. (Such a solution always exists in a small neighbourhood of any fixed point, and also in any domain where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193077.png" />.) In this case equation (1) is equivalent to the system of equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193078.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

which is a special case of the generalized Cauchy–Riemann system. Introducing the complex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193079.png" />, the system may be written as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193080.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

If the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193081.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193082.png" /> are analytic functions of the complex arguments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193083.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193084.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193086.png" />) in some cylindrical domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193087.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193088.png" /> is a simply-connected domain, then the solution of equation (7) in a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193089.png" /> is given by the formula:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193090.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193091.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193092.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193093.png" /> are analytic functions of their arguments, defined in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193095.png" /> and constructed by the method of successive approximation; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193096.png" /> is an arbitrary analytic function of the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193097.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d03193099.png" /> are entire functions of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930100.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930101.png" />, the representation (8) is valid for any simply-connected domain of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930102.png" />-plane, irrespective of the behaviour of the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930104.png" /> near infinity.

Let the elliptic equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930105.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>

be given, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930106.png" /> are analytic functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930108.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930109.png" /> is a fundamental domain for equation (9), any regular solution of this equation in a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930110.png" /> is expressed by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930111.png" /></td> <td valign="top" style="width:5%;text-align:right;">(10)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930112.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930113.png" /> are arbitrary functions holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930114.png" />, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930115.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930116.png" /> is the complex Riemann function of equation (9) which depends analytically on the complex arguments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930117.png" /> in the cylindrical domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930118.png" />. If the conditions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930119.png" /></td> </tr></table>

are satisfied, formula (10) realizes a one-to-one correspondence between the family of solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930120.png" /> of equation (9) and the family of holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930121.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930122.png" /> is a multiply-connected domain, formula (10) usually yields multi-valued solutions. However, as for equations of the second order, the formula may be modified so that it yields all single-valued solutions of equation (9) in a multiply-connected domain as well. Formula (10) may also be extended to a system of equations such as (9), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930123.png" /> is a vector and the coefficients are matrices.

For several equations of mathematical physics the Riemann function is explicitly given using elementary or special functions.

For the equation of the vibrating membrane,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930124.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930125.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930126.png" /> is the Bessel function of order zero, and the complex plane is taken as fundamental domain. The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930127.png" /> yields the Laplace equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930128.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930129.png" /> and formula (3) assumes the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930130.png" /></td> </tr></table>

For the equation of spherical functions,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930131.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930132.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930133.png" /> is the Legendre function of the first kind; any simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930134.png" /> which satisfies the condition: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930135.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930136.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930137.png" /> (e.g. the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930138.png" />), may be taken as fundamental domain. In the case of the Euler–Darboux equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930139.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930140.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930141.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930142.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930143.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930144.png" /> is the hypergeometric series; the half-planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930145.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930146.png" /> may be taken as fundamental domains.

For the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930147.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930148.png" /></td> </tr></table>

Goursat's formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930149.png" /></td> </tr></table>

is valid.

The method of complex representation of solutions is also applicable to a certain class of non-linear equations. For instance, let the Gauss equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930150.png" /></td> </tr></table>

be given, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930151.png" /> is a given function (this is a well-known equation in differential geometry). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930152.png" /> is any particular solution of this equation, the functions of the type

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930153.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930154.png" /> is an arbitrary analytic function, are also solutions. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930155.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930156.png" /> and all solutions of the Gauss equation may be expressed by the formula:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930157.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930158.png" />, it should be assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930159.png" />.

==Elliptic equations with non-analytic coefficients.==
Let there be given a generalized system of Cauchy–Riemann equations (7) with coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930160.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930161.png" /> defined on the whole complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930162.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930163.png" /> and belonging to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930164.png" />, i.e.

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930165.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930166.png" /></td> </tr></table>

If the coefficients are given in a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930167.png" /> and belong to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930168.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930169.png" />, they will satisfy the above conditions when extended by zero outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930170.png" />. Under these assumptions, equation (7) usually has no solutions in the classical sense. One accordingly considers a so-called generalized solution: A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930171.png" /> is called a solution of equation (7) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930172.png" /> if it has a derivative in the generalized sense (as defined by S.L. Sobolev) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930173.png" /> and satisfies the equation almost-everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930174.png" />.

The theory of functions satisfying equation (7) is a far-reaching generalization of the classical theory of analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930175.png" /> and retains their principal features. For this reason solutions of equations of the type (7) are known as generalized analytic functions (cf. [[Generalized analytic function|Generalized analytic function]]).

Any solution of equation (7) (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930176.png" />) satisfies the integral equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930177.png" /></td> <td valign="top" style="width:5%;text-align:right;">(11)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930178.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930179.png" /> is a holomorphic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930180.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930181.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930182.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930183.png" />, equation (11) has a unique solution, which is expressed by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930184.png" /></td> <td valign="top" style="width:5%;text-align:right;">(12)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930185.png" /></td> </tr></table>

The resolvents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930186.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930187.png" /> depend on the coefficients of equation (7) and are constructed by the method of successive approximation.

Formula (12) gives a general (linear) representation of solutions of (7) by analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930188.png" />. In particular, it permits the construction of the so-called fundamental kernels

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930189.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930190.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930191.png" /> is some fixed point, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930192.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930193.png" /> are solutions of the integral equation (11) corresponding to the functions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930194.png" /></td> </tr></table>

These kernels permit one to state the generalized Cauchy formula:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930195.png" /></td> <td valign="top" style="width:5%;text-align:right;">(13)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930196.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930197.png" />, this formula becomes the classical Cauchy formula. Formula (13) may serve to extend many properties of analytic functions which are usually demonstrated by the Cauchy formula to generalized analytic functions. In particular, it is possible to generalize the classical theorems on analytic continuation, to construct a theory of generalized integrals of Cauchy type to obtain representations of generalized analytic functions in the form of contour integrals with a real density, etc.

The functions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930198.png" /></td> </tr></table>

are the fundamental kernels of the adjoint equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930199.png" /></td> <td valign="top" style="width:5%;text-align:right;">(14)</td></tr></table>

If, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930200.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930202.png" /> satisfy the equations (7) and (14), respectively, and if they are continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930203.png" />, one has the identity (the analogue of the classical Cauchy theorem):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930204.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930205.png" /> is a solution of equation (7) in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930206.png" />, there exists an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930207.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930208.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930209.png" /></td> <td valign="top" style="width:5%;text-align:right;">(15)</td></tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930210.png" /></td> <td valign="top" style="width:5%;text-align:right;">(16)</td></tr></table>

and which belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930211.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930212.png" />; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930213.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930214.png" />.

In particular, this formula permits the extension of fundamental theorems of the classical theory of analytic functions — the uniqueness theorem, Liouville's theorem, the principle of the argument, the compactness principle, etc. — to the solutions of equations such as (7). Formula (16) may be reversed: Given an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930215.png" /> it is possible to find a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930216.png" /> satisfying the non-linear integral equation (16).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930217.png" /> be an analytic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930218.png" />, which may have arbitrary singularities, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930219.png" /> be a fixed point. Then there exists a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930220.png" /> of equation (7) such that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930221.png" /> is continuously extendable onto the whole plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930222.png" />, belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930223.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930224.png" />, does not vanish anywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930225.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930226.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930227.png" /> satisfies the integral equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930228.png" /></td> <td valign="top" style="width:5%;text-align:right;">(17)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930229.png" /></td> </tr></table>

which has a unique solution, while the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930230.png" /> satisfies the non-linear integral equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930231.png" /></td> <td valign="top" style="width:5%;text-align:right;">(18)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930232.png" /></td> </tr></table>

from which, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930233.png" />, the representation (15) is obtained.

The problem of reducing a second-order elliptic equation of general type

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930234.png" /></td> <td valign="top" style="width:5%;text-align:right;">(19)</td></tr></table>

to the form (1) is equivalent to the problem of reducing the positive quadratic form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930235.png" /></td> </tr></table>

to canonical form. This problem is equivalent to finding the homeomorphisms defined by the solution of the Beltrami equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930236.png" /></td> <td valign="top" style="width:5%;text-align:right;">(20)</td></tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930237.png" /></td> </tr></table>

If (19) is a uniformly elliptic equation (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930238.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930239.png" />.

The principal problem in the study of the Beltrami equation is the construction of a solution in a given domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930240.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930241.png" /> is a solution of the Beltrami equation realizing a homeomorphism of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930242.png" /> onto the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930243.png" />, then any other solution of the equation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930244.png" /> has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930245.png" /></td> <td valign="top" style="width:5%;text-align:right;">(21)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930246.png" /> is an arbitrary analytic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930247.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930248.png" /> is measurable, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930249.png" /> outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930250.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930251.png" />, the Beltrami equation has a solution of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930252.png" /></td> <td valign="top" style="width:5%;text-align:right;">(22)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930253.png" /> satisfies the singular integral equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930254.png" /></td> <td valign="top" style="width:5%;text-align:right;">(23)</td></tr></table>

in which the integral is understood in the sense of the Cauchy principal value. This equation has a unique solution in some class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930255.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930256.png" />, and may be obtained by the method of successive approximation. The function (22) belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930257.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930258.png" />, realizes a homeomorphism of the plane onto itself, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930259.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930260.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930261.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930262.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930263.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930264.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930265.png" />.

A first-order uniformly elliptic system of general type in complex notation has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930266.png" /></td> <td valign="top" style="width:5%;text-align:right;">(24)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930267.png" /></td> </tr></table>

It may be reduced to the form (7) with the aid of a homeomorphism defined by a solution of an equation of the type (20).

Any solution of equation (24) in some bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930268.png" />, for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930269.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930270.png" />, can be represented in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930271.png" /></td> <td valign="top" style="width:5%;text-align:right;">(25)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930272.png" /> is some homeomorphism defined by a solution of the Beltrami equation (20) with the coefficient

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930273.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930274.png" /> is an analytic function in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930275.png" />, and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930276.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930277.png" />, is holomorphic outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930278.png" /> and vanishes at infinity. The representation (25) is also valid if the coefficients of the left-hand side of equation (24) depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930279.png" /> and on its derivatives of any order, provided the above conditions hold for the solutions under study. Formula (25) is reversible, as is formula (15).

Formula (25) permits one to transfer several properties of the classical theory of analytic functions to solutions of equation (24): the uniqueness theorem, the principle of the argument, the maximum principle, etc.

A general quasi-conformal mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930280.png" /> is a solution of some uniformly elliptic system of the form (24) (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930281.png" />). The converse proposition is also true. Accordingly, the above results permit one to solve the main problems in the theory of [[Quasi-conformal mapping|quasi-conformal mapping]] by a purely analytical method.

Systems of first-order equations of elliptic type with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930282.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930283.png" />, unknown functions in two independent variables may, with certain natural restrictions, be reduced to the canonical form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930284.png" /></td> <td valign="top" style="width:5%;text-align:right;">(26)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930285.png" /> is the unknown vector with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930286.png" /> complex-valued components, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930287.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930288.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930289.png" /> are square matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930290.png" />. The theory of equations of the form (26) has many features in common with the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930291.png" />, but has also its own special features.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.N. Vekua, "New methods for solving elliptic equations" , North-Holland (1967) (Translated from Russian) {{MR|0212370}} {{ZBL|0146.34301}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.N. Vekua, "Generalized analytic functions" , Pergamon (1962) (Translated from Russian) {{MR|0152665}} {{MR|0150320}} {{MR|0138774}} {{ZBL|0127.03505}} {{ZBL|0100.07603}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.N. Vekua, "Systems of first-order differential equations of elliptic type and boundary value problems, with an application to shell theory" ''Mat. Sb.'' , '''31''' : 2 (1952) pp. 217–314 (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S. Bergman, "Integral operators in the theory of linear partial differential equations" , Springer (1961) {{MR|0141880}} {{ZBL|0093.28701}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> L. Bers, "Theory of pseudo-analytic functions" , New York Univ. Inst. Math. Mech. (1953) {{MR|0057347}} {{ZBL|0051.31603}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> B.V. Boyarskii, "A general representation of solutions of an elliptic system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930292.png" /> equations in the plane" ''Dokl. Akad. Nauk. SSSR'' , '''122''' : 4 (1958) pp. 543–546 (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> B.V. Boyarskii, "Some boundary value problems for systems of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031930/d031930293.png" /> elliptic equations in the plane" ''Dokl. Akad. Nauk. SSSR'' , '''124''' : 1 (1959) pp. 15–18 (In Russian) {{MR|116140}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> T. Carleman, "Sur les systèmes linéaires aux dérivées partielles du premier ordre à deux variables" ''C.R. Acad. Sci.'' , '''197''' (1933) pp. 471–474 {{MR|}} {{ZBL|0007.16202}} {{ZBL|59.0469.03}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian) {{MR|0226183}} {{ZBL|0167.09401}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> Z.I. Khalilov, "On boundary value problems for an elliptic equation" ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''11''' (1947) pp. 345–362 (In Russian) (French summary)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience (1965) (Translated from German) {{MR|0195654}} {{ZBL|}} </TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Tutschke, "Partielle komplexe Differentialgleichungen in einer und mehreren komplexen Variablen" , Deutsch. Verlag Wissenschaft. (1977) {{MR|481388}} {{ZBL|0361.35002}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G.F. Carrier, C.E. Pearson, "Partial differential equations" , Acad. Press (1976) {{MR|0404823}} {{ZBL|0323.35001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> L. Bers, "An outline of the theory of pseudoanalytic functions" ''Bull. Amer. Math. Soc.'' , '''62''' (1956) pp. 291–331 {{MR|0081936}} {{ZBL|0072.07703}} </TD></TR></table>