View source for Generalized analytic function

A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g0437701.png" /> satisfying a system

<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/g/g043/g043770/g0437702.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

with real coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g0437703.png" /> that are functions of the real variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g0437704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g0437705.png" />. Using the notations

<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/g/g043/g043770/g0437706.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/g/g043/g043770/g0437707.png" /></td> </tr></table>

the original system can be written 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/g/g043/g043770/g0437708.png" /></td> </tr></table>

If the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g0437709.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377010.png" /> of the system (1) belong to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377012.png" />, in the whole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377013.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377014.png" />, then in any domain of this plane every generalized analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377015.png" /> satisfying (1) 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/g/g043/g043770/g04377016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</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/g/g043/g043770/g04377017.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377018.png" /> is a well-defined analytic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377019.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377020.png" />.

The relation between a generalized analytic function and an analytic function, given by formula (2), is non-linear if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377021.png" />. In terms of a given analytic function, the generalized analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377022.png" /> is uniquely determined by the non-linear integral equation (2).

There exists a linear operator

<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/g/g043/g043770/g04377023.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/g/g043/g043770/g04377024.png" /></td> </tr></table>

that establishes a one-to-one correspondence between the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377025.png" /> that are analytic in a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377026.png" /> and continuous on the closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377027.png" />, and the set of generalized analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377028.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377031.png" /> are well-defined functions which can be expressed in terms of the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377033.png" /> of the system (1).

Formula (3) leads to various integral representations for generalized analytic functions, generalizing the Cauchy integral formula for analytic functions. The representation of a generalized analytic function in the form (3) turns out to be useful in the investigation of boundary value problems for generalized analytic functions.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377035.png" /> are analytic functions of the real variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377036.png" />, then one has the following representation of the generalized analytic functions defined in a simply-connected domain:

<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/g/g043/g043770/g04377037.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/g/g043/g043770/g04377038.png" /></td> </tr></table>

in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377040.png" /> are analytic functions of their arguments, expressible in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377042.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377043.png" /> is an arbitrary analytic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377044.png" />. (Formula (4) is not a special case of formula (3).)

In particular, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377046.png" /> are entire functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377048.png" />, then (4) is valid for any simply-connected domain in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377049.png" />-plane.

The problem of reducing a general second-order 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/g/g043/g043770/g04377050.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

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/g/g043/g043770/g04377051.png" /></td> </tr></table>

is equivalent to the problem of reducing the positive quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377052.png" /> to canonical form. The latter problem, in turn, reduces to that of finding homeomorphisms defined by solutions of the [[Beltrami equation|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/g/g043/g043770/g04377053.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</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/g/g043/g043770/g04377054.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/g/g043/g043770/g04377055.png" /></td> </tr></table>

If (5) is a uniformly-elliptic equation (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377056.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377057.png" />.

The basic problem in the study of the Beltrami equation is the construction of a solution for a given domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377058.png" />. This follows from the following assertion: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377059.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/g/g043/g043770/g04377060.png" /> onto the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377061.png" />, then every other solution in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377062.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/g/g043/g043770/g04377063.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377064.png" /> is an arbitrary analytic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377065.png" />.

When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377066.png" /> is measurable, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377067.png" /> outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377069.png" />, then a single-valued solution of the Beltrami equation (6) is given by the function

<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/g/g043/g043770/g04377070.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377071.png" /> satisfies the [[Singular integral equation|singular integral equation]] (the integral is understood as a Cauchy principal value)

<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/g/g043/g043770/g04377072.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>

This equation has a unique solution in some class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377074.png" />. It can be obtained by, for example, the method of successive approximation (cf. [[Sequential approximation, method of|Sequential approximation, method of]]). The function (8) belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377076.png" />, and realizes a topological mapping of the plane onto itself, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377078.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377079.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377082.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377083.png" />.

A uniformly-elliptic system consisting of two general first-order elliptic equations has, in complex notation, 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/g/g043/g043770/g04377084.png" /></td> <td valign="top" style="width:5%;text-align:right;">(10)</td></tr></table>

By means of a homeomorphism defined by a solution of a certain equation of the form (6), the system (10) can be reduced to the form (1). But it can also be studied directly, avoiding thereby certain additional restrictions.

Consider equation (10) in some bounded domain with the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377086.png" />. Then every solution of (10) 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/g/g043/g043770/g04377087.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/g/g043/g043770/g04377088.png" /> is a homeomorphism defined by a solution of the Beltrami equation (6) with 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/g/g043/g043770/g04377089.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377090.png" /> is an analytic function in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377091.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377092.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377093.png" />, is holomorphic outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377094.png" /> and vanishes at infinity. The representation (11) also holds when the coefficients at the left-hand side of (10) depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377095.png" /> and its derivatives of any order, provided that the conditions given above are satisfied for the solutions considered. As in (2), formula (11) can be inverted.

Formula (11) allows one to transfer a whole of series of properties of the classical theory of analytic functions to solutions of (10): the uniqueness theorem (cf. [[Uniqueness properties of analytic functions|Uniqueness properties of analytic functions]]), the principle of the argument (cf. [[Argument, principle of the|Argument, principle of the]]), the [[Maximum principle|maximum principle]], etc.

A general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377097.png" />-quasi-conformal mapping is a solution of some uniformly-elliptic system of the form (10) (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377098.png" />). The converse is also valid. Hence the results stated above enable one to solve basic problems on [[Quasi-conformal mapping|quasi-conformal mapping]] by purely analytic means.

The theory of generalized analytic functions has made an exhaustive investigation of a generalized Riemann–Hilbert problem possible (cf. also [[Riemann–Hilbert problem (analytic functions)|Riemann–Hilbert problem (analytic functions)]]). The problem is to find a solution of (1) continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g04377099.png" />, with boundary condition

<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/g/g043/g043770/g043770100.png" /></td> <td valign="top" style="width:5%;text-align:right;">(12)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770101.png" /> are given real-valued functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770103.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770104.png" />. In general, the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770105.png" /> is multiply connected.

Problem (12) can be reduced to an equivalent singular integral equation, and a complete qualitative analysis of the boundary value problem (12) can be obtained.

Let the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770106.png" /> of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770107.png" /> consist of a finite number of simple closed curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770108.png" />, satisfying the Lyapunov conditions (cf. [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]). Since the form of the equation and the boundary conditions remain unchanged under conformal mapping, one may assume without loss of generality that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770109.png" /> is the unit circle with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770110.png" /> lying in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770111.png" /> considered, and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770112.png" /> are circles lying inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770113.png" />.

The index of the problem (12) is the integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770114.png" /> equal to the change in

<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/g/g043/g043770/g043770115.png" /></td> </tr></table>

when the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770116.png" /> goes round <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770117.png" /> once in the positive direction. The boundary condition can be reduced to the simpler 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/g/g043/g043770/g043770118.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770119.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770120.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770121.png" />, and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770122.png" /> are certain real parameters which can be uniquely expressed in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770123.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770124.png" />.

For the adjoint problem:

<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/g/g043/g043770/g043770125.png" /></td> <td valign="top" style="width:5%;text-align:right;">(13)</td></tr></table>

the index is given by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770126.png" />.

The basic results for the problem (12) can be formulated as follows.

1) Problem (12) has a solution if and only if

<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/g/g043/g043770/g043770127.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770128.png" /> is an arbitrary solution of the adjoint problem.

2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770129.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770130.png" /> be the numbers of linearly independent solutions of the homogeneous problems (12) and (13), respectively. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770131.png" />.

3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770132.png" />, then the homogeneous problem (12) has no non-trivial solutions.

4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770133.png" />, then the homogeneous problem (12) has exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770134.png" /> linearly independent solutions, and the inhomogeneous problem (12) has a (unique) solution if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770135.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770136.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770137.png" />; and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770138.png" /> is a complete system of solutions of the homogeneous problem (13).

5) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770139.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770140.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770141.png" /> and all solutions of the homogeneous problem have the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770142.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770143.png" /> is a real constant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770144.png" /> is a continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770145.png" />.

The above results completely characterize the problem (12) in the simply-connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770146.png" /> and multiply-connected (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770147.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770148.png" />) cases. The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770149.png" /> requires special consideration.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.N. Vekua,   "Generalized analytic functions" , Pergamon  (1962)  (Translated from Russian)</TD></TR></table>



====Comments====
Of course, an [[Analytic function|analytic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770150.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770151.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043770/g043770152.png" />), whence the name  "generalized analytic function" , sometimes also pseudo-analytic function (cf. [[#References|[a1]]], [[#References|[a2]]]), for a function satisfying (1).

The system (1) is usually called the Carleman system or Bers–Vekua system.

For the Riemann–Hilbert problem see also [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Bers,   "An outline of the theory of pseudo-analytic functions"  ''Bull. Amer. Math. Soc.'' , '''62'''  (1956)  pp. 291–331</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Bers,   "Theory of pseudo-analytic functions" , New York Univ. Inst. Math. Mech.  (1953)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  Yu.L. Rodin,   "Generalized analytic functions on Riemann surfaces" , Springer  (1987)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  Yu.L. Rodin,   "The Riemann boundary problem on Riemann surfaces" , Reidel  (1988)  (Translated from Russian)</TD></TR></table>