Generalized analytic function

A function $ w ( z) = u ( x , y ) + i v ( x , y ) $ satisfying a system

$$ \tag{1 } \frac{\partial u }{\partial x } - \frac{\partial v }{\partial y } + a u + b v = 0 ,\ \ \frac{\partial u }{\partial y } + \frac{\partial v }{\partial x } + c u + d v = 0 $$

with real coefficients $ a , b , c , d $ that are functions of the real variables $ x $ and $ y $. Using the notations

$$ 2 \frac \partial {\partial \overline{z} } = \ \frac \partial {\partial x } + i \frac \partial {\partial y } , $$

$$ 4 A = a + b + i ( c - b ) ,\ 4 B = a - d + i ( c + b ) , $$

the original system can be written in the form

$$ \frac{\partial w }{\partial \overline{z} } + A w + B \overline{w} = 0 . $$

If the coefficients $ A $ and $ B $ of the system (1) belong to the class $ L _ {p,2} $, $ p > 2 $, in the whole $ z $-plane $ E $, then in any domain of this plane every generalized analytic function $ w ( z) $ satisfying (1) can be represented in the form

$$ \tag{2 } w ( z) = \Phi ( z) e ^ {\omega ( z) } , $$

where

$$ \omega ( z) = \frac{1} \pi {\int\limits \int\limits } _ { D } \frac{A ( \zeta ) + B ( \zeta ) {\overline{w} } / {w } }{\zeta - z } \ d \xi d \eta ,\ \ \zeta = \xi + i \eta , $$

and $ \Phi ( z) $ is a well-defined analytic function of $ z $ in $ D $.

The relation between a generalized analytic function and an analytic function, given by formula (2), is non-linear if $ B \neq 0 $. In terms of a given analytic function, the generalized analytic function $ w ( z) $ is uniquely determined by the non-linear integral equation (2).

There exists a linear operator

$$ \tag{3 } w ( z) = $$

$$ = \ \Phi ( z) + {\int\limits \int\limits } _ { D } \Gamma _ {1} ( z , \zeta ) \Phi ( \zeta ) d \xi \ d \eta + {\int\limits \int\limits } _ { D } \Gamma _ {2} ( z , \zeta ) \overline{ {\Phi ( \zeta ) }} d \xi d \eta $$

that establishes a one-to-one correspondence between the set of functions $ \Phi ( z) $ that are analytic in a bounded domain $ D $ and continuous on the closed domain $ D \cup S $, and the set of generalized analytic functions $ w ( z) $ on $ D $, where $ \Gamma _ {1} ( z , \zeta ) $ and $ \Gamma _ {2} ( z , \zeta ) $ are well-defined functions which can be expressed in terms of the coefficients $ A $ and $ B $ 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 $ A $ and $ B $ are analytic functions of the real variables $ x , y $, then one has the following representation of the generalized analytic functions defined in a simply-connected domain:

$$ \tag{4 } w ( z) = \mathop{\rm exp} \ \left ( \int\limits _ {z _ {0} } ^ { z } A ( z , \tau ) d \tau \right ) + $$

$$ + \left \{ \Phi ( z) + \int\limits _ {z _ {0} } ^ { z } \Gamma tilde _ {1} ( z , \overline{z} , t ) \Phi ( t) d t + \int\limits _ {z _ {0} } ^ { {z } bar } \widetilde \Gamma _ {2} ( z , \overline{z}, t ) \overline{ {\Phi ( t) }} d t \right \} , $$

in which $ \widetilde \Gamma _ {1} $ and $ \widetilde \Gamma _ {2} $ are analytic functions of their arguments, expressible in terms of $ A $ and $ B $, and $ \Phi ( z) $ is an arbitrary analytic function of $ z $. (Formula (4) is not a special case of formula (3).)

In particular, when $ A $ and $ B $ are entire functions of $ x $ and $ y $, then (4) is valid for any simply-connected domain in the $ z $-plane.

The problem of reducing a general second-order elliptic equation

$$ \tag{5 } a \frac{\partial ^ {2} u }{\partial x ^ {2} } + 2 b \frac{\partial ^ {2} u }{\partial x \partial y } + c \frac{\partial ^ {2} u }{\partial y ^ {2} } + d \frac{\partial u }{\partial x } + e \frac{\partial u }{\partial y } + f u = 0 $$

to the form

$$ \frac{\partial ^ {2} u }{\partial x ^ {2} } + \frac{\partial ^ {2} u }{\partial y ^ {2} } + A \frac{\partial u }{\partial x } + B \frac{\partial u }{\partial y } + C u = 0 $$

is equivalent to the problem of reducing the positive quadratic form $ a d x ^ {2} + 2 b d x d y + c d y ^ {2} $ to canonical form. The latter problem, in turn, reduces to that of finding homeomorphisms defined by solutions of the Beltrami equation

$$ \tag{6 } \frac{\partial w }{\partial \overline{z} } - q ( z) \frac{\partial w }{\partial z } = 0 ,\ \ w = u + i v , $$

where

$$ q ( z) = \ \frac{( a - \sqrt \Delta - ib ) }{( a + \sqrt \Delta + ib ) } , $$

$$ \Delta = a c - b ^ {2} ,\ | q ( z) | < 1 . $$

If (5) is a uniformly-elliptic equation ( $ \Delta \geq \Delta _ {0} = \textrm{ const } > 0 $), then $ | q ( z) | < q _ {0} = \textrm{ const } < 1 $.

The basic problem in the study of the Beltrami equation is the construction of a solution for a given domain $ D $. This follows from the following assertion: If $ \omega ( z) $ is a solution of the Beltrami equation, realizing a homeomorphism of the domain $ D $ onto the domain $ \omega ( D) $, then every other solution in $ D $ has the form

$$ \tag{7 } w ( z) = \Phi [ \omega ( z) ] , $$

where $ \Phi $ is an arbitrary analytic function in $ \omega ( D) $.

When $ q ( z) $ is measurable, $ q ( z) = 0 $ outside $ D $ and $ | q ( z) | \leq q _ {0} < 1 $, then a single-valued solution of the Beltrami equation (6) is given by the function

$$ \tag{8 } w ( z) = z - \frac{1} \pi {\int\limits \int\limits } _ { E } \frac{\rho ( \zeta ) d \xi d \eta }{\zeta - z } , $$

where $ \rho $ satisfies the singular integral equation (the integral is understood as a Cauchy principal value)

$$ \tag{9 } \rho ( z) - \frac{q ( z) } \pi {\int\limits \int\limits } _ { E } \frac{\rho ( \zeta ) d \xi d \eta }{( \zeta - z ) ^ {2} } = \ q ( z) . $$

This equation has a unique solution in some class $ L _ {p} ( E) $, $ p > 2 $. It can be obtained by, for example, the method of successive approximation (cf. Sequential approximation, method of). The function (8) belongs to the class $ C _ \alpha ( E) $, $ \alpha = ( p - 2 ) / 2 $, and realizes a topological mapping of the plane onto itself, with $ w ( \infty ) = \infty $, $ z ^ {-1} w \rightarrow 1 $ as $ z + \infty $. If $ q \in C _ \alpha ^ {m} ( E) $, $ 0 < \alpha < 1 $, $ m \geq 0 $, then $ w ( z) \in C _ \alpha ^ {m+1} $.

A uniformly-elliptic system consisting of two general first-order elliptic equations has, in complex notation, the form

$$ \tag{10 } \frac{\partial w }{\partial \overline{z} } - q _ {1} ( z) \frac{\partial w }{\partial z } - q _ {2} ( z) \frac{\partial \overline{w} }{\partial \overline{z} } + A w + B \overline{w} = 0 . $$

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 $ A , B \in L _ {p} ( D) $, $ p > 2 $. Then every solution of (10) can be represented in the form

$$ \tag{11 } w ( z) = \Phi [ \omega ( z) ] e ^ {\phi ( z) } , $$

where $ \omega ( z) $ is a homeomorphism defined by a solution of the Beltrami equation (6) with coefficient

$$ q ( z) = q _ {1} ( z) + q _ {2} ( z) \frac{ {\partial w / \partial z } bar }{\partial w / \partial z } , $$

$ \Phi ( \omega ) $ is an analytic function in the domain $ \omega ( D) $; and $ \phi ( z) \in C _ \alpha ( E) $, $ \alpha = ( p - 2 ) / 2 $, is holomorphic outside $ D \cup S $ and vanishes at infinity. The representation (11) also holds when the coefficients at the left-hand side of (10) depend on $ w $ 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), the principle of the argument (cf. Argument, principle of the), the maximum principle, etc.

A general $ Q $-quasi-conformal mapping is a solution of some uniformly-elliptic system of the form (10) (with $ A = B = 0 $). The converse is also valid. Hence the results stated above enable one to solve basic problems on 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)). The problem is to find a solution of (1) continuous on $ D \cup S $, with boundary condition

$$ \tag{12 } \mathop{\rm Re} [ \overline{ {\lambda ( z) }} w ( z) ] = \ \alpha u + \beta v = \gamma ,\ \ z \in S , $$

where $ \alpha , \beta , \gamma $ are given real-valued functions in $ C _ \alpha ( S) $, $ 0 < \alpha < 1 $, and $ \alpha ^ {2} + \beta ^ {2} = 1 $. In general, the domain $ D $ 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 $ S $ of the domain $ D $ consist of a finite number of simple closed curves $ S _ {0} \dots S _ {m} $, satisfying the Lyapunov conditions (cf. 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 $ S _ {0} $ is the unit circle with centre at $ z = 0 $ lying in the domain $ D $ considered, and that $ S _ {1} \dots S _ {m} $ are circles lying inside $ S _ {0} $.

The index of the problem (12) is the integer $ n $ equal to the change in

$$ \frac{1}{2 \pi } \mathop{\rm arg} \ [ \alpha ( \zeta ) + i \beta ( \zeta ) ] , $$

when the point $ \zeta $ goes round $ S $ once in the positive direction. The boundary condition can be reduced to the simpler form

$$ \mathop{\rm Re} [ z ^ {-n} e ^ {iC(z)} w ( z) ] = \gamma ,\ z \in S , $$

where $ C ( z) = C _ {j} $ on $ S _ {j} $, with $ C _ {0} = 0 $, and where $ C _ {1} \dots C _ {m} $ are certain real parameters which can be uniquely expressed in terms of $ \alpha $ and $ \beta $.

For the adjoint problem:

$$ \tag{13 } \left . \begin{array}{c} \frac{\partial w ^ {*} }{\partial \overline{z} } - A w ^ {*} - \overline{B} w ^ {*} = 0 ,\ \ z \in D , \\ \mathop{\rm Re} \left [ ( \alpha + i \beta ) \frac{d \overline{z} }{d s } w ^ {*} ( z) \right ] = 0 ,\ \ z \in S , \\ \end{array} \right \} $$

the index is given by the formula $ n ^ \prime = m - n - 1 $.

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

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

$$ \int\limits _ { S } ( \alpha + i \beta ) w ^ {*} \gamma d s = 0 , $$

where $ w ^ {*} $ is an arbitrary solution of the adjoint problem.

2) Let $ l $ and $ l ^ \prime $ be the numbers of linearly independent solutions of the homogeneous problems (12) and (13), respectively. Then $ l - l ^ \prime = n - n ^ \prime = 2 n - m + 1 $.

3) If $ n < 0 $, then the homogeneous problem (12) has no non-trivial solutions.

4) If $ n > m - 1 $, then the homogeneous problem (12) has exactly $ l = 2 n + m - 1 $ linearly independent solutions, and the inhomogeneous problem (12) has a (unique) solution if and only if $ \int _ {S} ( \alpha + i \beta ) w _ {j} ^ {*} \gamma d s = 0 $; $ j = 1 \dots l ^ \prime $; $ l ^ \prime = m- 2n + 1 $; and where $ w _ {j} $ is a complete system of solutions of the homogeneous problem (13).

5) If $ m = 0 $ and $ n = 0 $, then $ l = 1 $ and all solutions of the homogeneous problem have the form $ w ( z) = i c e ^ {\omega _ {0} ( z) } $, where $ c $ is a real constant and $ \omega _ {0} $ is a continuous function on $ D \cup S $.

The above results completely characterize the problem (12) in the simply-connected $ ( m = 0 ) $ and multiply-connected ( $ n < 0 $, $ n > m - 1 $) cases. The case $ 0 \leq n \leq m - 1 $ requires special consideration.

References

[1]	I.N. Vekua, "Generalized analytic functions" , Pergamon (1962) (Translated from Russian)

Comments

Of course, an analytic function $ w $ satisfies $ \partial w / \partial \overline{z} = 0 $ (i.e. $ A = B = 0 $), whence the name "generalized analytic function" , sometimes also pseudo-analytic function (cf. [a1], [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 [a4].

References

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