Inverse problems in integral formulas

293 It is well known that the values of an analytic func tion are closely interrelated, since its value along a closed contour Γ completely determine its value inside Γ (Cauchy formula); i.e., the values on Γ are boundary controls [1]. A broad class of ill posed problems arising in phys ics, engineering, and other fields consists of inverse problems (see, e.g., [2–4]). An inverse problem for the Cauchy integral formula in the case of a circle was solved in [5]. Let D be a star shaped domain with respect to the origin in the plane of a complex variable z, f(z) be a holomorphic function in D, and

It is well known that the values of an analytic func tion are closely interrelated, since its value along a closed contour Γ completely determine its value inside Γ (Cauchy formula); i.e., the values on Γ are boundary controls [1].
A broad class of ill posed problems arising in phys ics, engineering, and other fields consists of inverse problems (see, e.g., [2][3][4]). An inverse problem for the Cauchy integral formula in the case of a circle was solved in [5].
Let D be a star shaped domain with respect to the origin in the plane of a complex variable z, f(z) be a holomorphic function in D, and The following result hold true [6]. Theorem 1. If f(z) is holomorphic in D and continu ous, together with its derivative f '(z), in the closed domain , then, for z ∈ D, (2) (τ is real), where the integrals are taken along the con tour Γ in the positive direction.
Let K r be the disk |z| < r and C r be its boundary |z| = r (0 < r ≤ R).
where 0 < r < R and C r is traversed in the positive direc tion.
Proof. Let z be an arbitrary point of K R that does not belong to the circle C r , and let C ρ be a circle cen tered at the origin of radius ρ (ρ < R) such that z and C r lie inside C ρ . Consider a closed doubly connected domain whose boundary is the composite contour Γ = C ρ + . In this domain, the function is an analytic function of ξ (with z and t being con stant). Therefore, by the second mean value theorem, where 0 < α < +∞. However, Since < 1 (n = 1, 2, …) for 0 ≤ t < +∞, the series under the integral sign converges uniformly (for constant z and ξ such that |z| < |ξ|) with respect to t ∈ [0, α], where α < +∞. Integrating this series with respect to t on the interval [0, α], we obtain The last series converges uniformly (for constant z and ξ such that |z| < |ξ|) with respect to α for 0 ≤ α ≤ +∞, since ≤ 1 and, hence, On the other hand, for |z| < |ξ| = ρ, we have Then, passing in (6) to the limit as α → +∞, we obtain or, by virtue of formula (1), Formula (4) is proved in a similar manner to (3) with the only difference being that function (5) is replaced by · .
Remark. Under the condition of Theorem 2, for mulas (3) and (4) remain valid if C r is an arbitrary closed piecewise smooth curve enclosing the point О and lying inside the boundary C R of the disk K R .
Theorem 3. If f(z) is holomorphic in the disk K R and continuous, together with its derivative f '(z), in the closed disk : |z| ≤ R, then formulas (3) and Proof. Formulas (3) and (4) in K R \C r , where 0 < r < R, were proved in Theorem 2. By virtue of the condition of Theorem 3, formulas (1) and (2) hold in K R or, equivalently, in K R \C R , we have the formulas