Bergman integral operator

A complex 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; 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) 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) 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 $\Delta u+4k^2u=0$ 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).

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

[a1]	S. Bergman, "Integral operators in the theory of linear partial differential equations" , Springer (1961) MR0141880 Zbl 0093.28701
[a2]	D.L. Colton, "Partial differential equations in the complex domain" , Pitman (1975) MR0481427 Zbl 0323.35003
[a3]	R.P. Gilbert, "Constructive methods for elliptic equations" , Springer (1974) MR0447784 Zbl 0302.35001
[a4]	M.W. Kracht, E. Kreyszig, "Methods of complex analysis in partial differential equations (with applications)" , Wiley (1988) MR941372 Zbl 0644.35005
[a5]	E. Lanckau, "Complex integral operators in mathematical physics" , W. Barth (1993) MR1224234 Zbl 0803.35001
[a6]	I.N. Vekua, "New methods for solving elliptic equations" , Wiley (1967) (In Russian) MR0212370 Zbl 0146.34301