View source for Bergman integral operator

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A complex [[Integral transform|integral transform]] which maps holomorphic functions $f$ of a complex variable onto complex solutions $u$ 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 $z=x+iy$, $z^*=x-iy$ are introduced instead of the coordinates $x,y$ in the plane; note that the Laplace potential operator (cf. [[Laplace operator|Laplace operator]]) is

$$\Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}=4\frac{\partial^2}{\partial z\partial z^*}$$

Then the transform is realized by a complex line integral (cf. [[Curvilinear integral|Curvilinear integral]]) as

$$T[f](z,z^*)=\int\limits_{-\pi/2}^{\pi/2}E(z,z^*,\sin\phi)f(z\cos^2\phi)\,d\phi\label{a1}\tag{a1}$$

(Bergman type) or

$$T[f](z,z^*)=\int\limits_{z_0}^zR(z,z^*,t,t^*)f(t)\,dt\label{a2}\tag{a2}$$

(Vekua type). Here, the generating kernels $E=E(z,z^*,t)$ (respectively, $R=R(z,z^*,t,t^*)$) (the so-called complex Riemann function, $t^*$ a complex parameter) depend on the differential equation to be solved and $f=f(z)$ 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]] $\Delta u+4k^2u=0$ are given by \eqref{a1} (respectively, by \eqref{a2}) with

$$E(z,z^*,\tau)=\cos(k\sqrt{zz^*}t),$$

$$R(z,z^*,t,t^*)=J_0(2k\sqrt{(z-t)(z^*-t^*))}$$

(where $J_0$ is a Bessel function, cf. [[Bessel functions|Bessel functions]]).

The analytic properties of the solution $u=u(z,z^*)=T[f]$ are closely related to the analytic properties of the holomorphic function $f=f(z)$, e.g., location and type of singularities. Further, complete sets of solutions $T[f(z)](z,z^*)$ can be found as transforms of complete sets of holomorphic functions, such as $\{z^n\}_{n=0,1,\dots}$. These (and similar) properties are the basis of applications in mathematical physics. However, there is no simple relation between the boundary values of $f$ and those of $u$ 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 $u$ of a parabolic equation in two spacial variables $z,z^*$ and time $\tau$ may be found (and studied) by integral transforms

$$u(z,z^*,\tau)=T[f(z,\tau)],$$

where, in this case, $f=f(z,t)$ is a holomorphic function of two variables $z,\tau$. 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>