Inverse problems in integral formulas

I. I. Bavrin

Čebyševskij sbornik, Tome 16 (2015) no. 3, pp. 70-77.

Résumé

In complex analysis an integral representations are one of the powerful tools of research. The theory of analytic functions of complex variables is largely built on the basis of Cauchy's integral formula [1]. An important class of ill-posed problems arising in physics, engineering and other fields, are so-called inverse problems [2–4]. In [5, 6] author sets an integral formula for the function $f(z)$, holomorphic in the circle $K_R:\, |z|$, (it is in the introduction of this article as formula (1)) is the solution of the inverse problem for Cauchy's integral formula in the circle $K_R$. Equation (1), unlike the Cauchy formula for the values of the function $f(z)$ on any circumference $C_r\!:\, |z|=r \, (0$ lying in a circle $K_R$, or an arbitrary closed piecewise smooth lines covering the origin and contained within a circle circle $C_R$ — the circle border $K_R$ expresses its values at all other points in the range of $K_R$. In [5] the solution of inverse problems for Poisson's formula [1] and Schwartz' formula [7] and the formulas for derivatives of Cauchy formula [1] in [5, 6] are obtained. The inverse problem for the Poisson integral formula is used in [8] for generalization of Poisson–Jensen's formula [7] from which Poisson–Jensen's formula and Jensen's formula follow as a special cases. Similarly, [9] and the inverse problem are used for the generalization of the Schwartz–Jensen's formula [7]. In the case of ring $D$: $r|z|$ to [10] the integral representation is set (in [10] is a formula (1)) for a holomorphic function in $D$ D which, unlike the Cauchy formula for the ring, according to the values on an arbitrary closed piecewise smooth lines hugging origin and contained within the ring $D$, expresses its values at all other points of the rings, in [10] the inverse problem for the Cauchy formula in the case of ring $D$ is solved. In the article [11] for the case of the circle $K_R$ the solution of inverse problems for integral formulas is found and given in [12] (in [12] these are formulas (3) and (4)) are valid for functions holomorphic in a star domain with respect to the origin. The Cauchy formula holds in the case of several complex variables (see., Eg, [13]). In the article [14] for the case of polydisc $$E_R=E(R_1,\ldots,R_n)=\{z=(z_1,\ldots,z_n):\, |z_1|,\ldots,|z_n|\}$$ the inverse problems for the Cauchy formula and by deriving from its are resolved (analogous to the Poisson formulas for the case of one complex variable). The inverse problems [15] in the case of integral Temlyakov's formulas are solved (these are formulas, see., Eg, [16]). Finally, in this article, in the case of a convex domain and circle (respectively Theorem 2 and 3) new integral representations (3) and (5) are set, of which (3) is an integral representation for holomorphic functions in a convex domain, and (5) is a solution of the inverse problem for the integral representation (3) in a circle $K_R$. Bibliography: 16 titles.

Détail

Keywords: holomorphic function, Cauchy integral formula, Poisson integral formula, Schwarz integral formula, Poisson–Jensen formula, Schwartz–Jensen formula, star domain, convex domain, polydisc.

Bibliographie

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[4] Marchuk G. I., “On the formulation of inverse problems”, Dokl. Akad. Nauk SSSR, 156:3 (1964), 503–506 (in Russian) | MR | Zbl

[5] Bavrin I. I., “Integral representations of functions analytic and harmonic in a disk”, Dokl. Math., 78:1 (2008), 522–524 | DOI | MR | Zbl

[6] Bavrin I. I., “Inverse problems for the Cauchy integral formula and for formulas for the derivatives of the Cauchy integral”, Dokl. Math., 78:2 (2008), 679–680 | DOI | MR | Zbl

[7] Markushevich A. I., The theory of analytic functions, Nauka, M., 1968, 624 pp. (in Russian)

[8] Bavrin I. I., “A generalization of the Poisson–Jensen formula”, Dokl. Math., 81:2 (2010), 193–195 | DOI | MR | Zbl

[9] Bavrin I. I., “A generalization of the Schwartz–Jensen formula”, Dokl. Math., 82:1 (2010), 566–567 | DOI | MR | Zbl

[10] Bavrin I. I., “The inverse problem for the Cauchy integral formula in an annulus”, Dokl. Math., 80:2 (2009), 660–661 | DOI | MR | Zbl

[11] Bavrin I. I., “Inverse problems in integral formulas”, Dokl. Math., 87:3 (2013), 293–295 | DOI | MR | Zbl

[12] Bavrin I. I., “Integral representations in starlike domains”, Dokl. Math., 86:3 (2012), 799–800 | DOI | MR | Zbl

[13] Vladimirov V. S., Methods in the theory of functions of several complex variables, With a Foreword by N. N. Bogoljubov, Nauka, M., 1964, 411 pp. (in Russian) | MR

[14] Bavrin I. I., “Inverse problems for Cauchy, Schwarz, and Poisson integral formulas in a polydisk”, Dokl. Math., 82:2 (2010), 787–789 (in Russian) | Zbl

[15] Bavrin I. I., “Integral representations in multicircular domains: inverse problems”, Dokl. Math., 84:3 (2011), 837–840 (in Russian) | Zbl

[16] Bavrin I. I., The operator method in complex analysis, Prometei, M., 1991, 200 pp. (in Russian) | MR