View source for Bergman integral operator

A complex [[Integral transform|integral transform]] which maps holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b1103001.png" /> of a complex variable onto complex solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b1103002.png" /> of linear second-order partial differential equations in the plane, originally of elliptic type (cf. also [[Analytic function|Analytic function]]; [[Elliptic partial differential equation|Elliptic partial differential equation]]). To apply the transform, complex (independent) variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b1103003.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b1103004.png" /> are introduced instead of the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b1103005.png" /> in the plane; note that the Laplace potential operator (cf. [[Laplace operator|Laplace operator]]) is

<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/b/b110/b110300/b1103006.png" /></td> </tr></table>

Then the transform is realized by a complex line integral (cf. [[Curvilinear integral|Curvilinear integral]]) 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/b/b110/b110300/b1103007.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

(Bergman type) or

<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/b/b110/b110300/b1103008.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

(Vekua type). Here, the generating kernels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b1103009.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b11030010.png" />) (the so-called complex Riemann function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b11030011.png" /> a complex parameter) depend on the differential equation to be solved and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b11030012.png" /> is an arbitrary holomorphic function. The path of integration may be (e.g.) a straight line.

For example, the complex solutions of the [[Helmholtz equation|Helmholtz equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b11030013.png" /> are given by (a1) (respectively, by (a2)) with

<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/b/b110/b110300/b11030014.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/b/b110/b110300/b11030015.png" /></td> </tr></table>

(where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b11030016.png" /> is a Bessel function, cf. [[Bessel functions|Bessel functions]]).

The analytic properties of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b11030017.png" /> are closely related to the analytic properties of the holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b11030018.png" />, e.g., location and type of singularities. Further, complete sets of solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b11030019.png" /> can be found as transforms of complete sets of holomorphic functions, such as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b11030020.png" />. These (and similar) properties are the basis of applications in mathematical physics. However, there is no simple relation between the boundary values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b11030021.png" /> and those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b11030022.png" /> on a prescribed boundary.

The method has been generalized widely: to equations of higher dimensions, higher order, and of other (parabolic, mixed, composite) type. Here, holomorphic functions of two (or more) complex variables are mapped onto solutions, using integral transforms. E.g., the solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b11030023.png" /> of a parabolic equation in two spacial variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b11030024.png" /> and time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b11030025.png" /> may be found (and studied) by integral transforms

<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/b/b110/b110300/b11030026.png" /></td> </tr></table>

where, in this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b11030027.png" /> is a holomorphic function of two variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110300/b11030028.png" />. In this way a unified method of explicitly constructing solutions of equations of different types and different dimensions has been established.

====References====
<table><TR><TD valign="top">[a1]</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">[a2]</TD> <TD valign="top"> D.L. Colton, "Partial differential equations in the complex domain" , Pitman (1975) {{MR|0481427}} {{ZBL|0323.35003}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R.P. Gilbert, "Constructive methods for elliptic equations" , Springer (1974) {{MR|0447784}} {{ZBL|0302.35001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M.W. Kracht, E. Kreyszig, "Methods of complex analysis in partial differential equations (with applications)" , Wiley (1988) {{MR|941372}} {{ZBL|0644.35005}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> E. Lanckau, "Complex integral operators in mathematical physics" , W. Barth (1993) {{MR|1224234}} {{ZBL|0803.35001}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> I.N. Vekua, "New methods for solving elliptic equations" , Wiley (1967) (In Russian) {{MR|0212370}} {{ZBL|0146.34301}} </TD></TR></table>