# American Institute of Mathematical Sciences

May  2013, 7(2): 491-498. doi: 10.3934/ipi.2013.7.491

## A note on analyticity properties of far field patterns

 1 Mathematisches Institut, Universität Leipzig, 04009 Leipzig, Germany 2 Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, FI-00076 Aalto 3 Aalto University, Department of Mathematics and Systems Analysis, FI-00076 Aalto, Finland

Received  October 2012 Revised  December 2012 Published  May 2013

In scattering theory the far field pattern describes the directional dependence of a time-harmonic wave scattered by an obstacle or inhomogeneous medium, when observed sufficiently far away from these objects. Considering plane wave excitations, the far field pattern can be written as a function of two variables, namely the direction of propagation of the incident plane wave and the observation direction, and it is well-known to be separately real analytic with respect to each of them. We show that the far field pattern is in fact a jointly real analytic function of these two variables.
Citation: Roland Griesmaier, Nuutti Hyvönen, Otto Seiskari. A note on analyticity properties of far field patterns. Inverse Problems and Imaging, 2013, 7 (2) : 491-498. doi: 10.3934/ipi.2013.7.491
##### References:
 [1] F. E. Browder, Real analytic functions on product spaces and separate analyticity, Canad. J. Math., 13 (1961), 650-656. doi: 10.4153/CJM-1961-054-1. [2] A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33. doi: 10.1515/jiip.2008.002. [3] D. L. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," John Wiley & Sons, New York, 1983. [4] D. L. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," $2^{nd}$ edition, Springer-Verlag, Berlin, 1998. [5] F. Hartogs, Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten, Math. Ann., 62 (1906), 1-88. doi: 10.1007/BF01448415. [6] L. Hörmander, "An Introduction to Complex Analysis in Several Variables," $3^{rd}$ edition, North-Holland, Amsterdam, 1990. [7] N. Hyvönen, P. Piiroinen and O. Seiskari, Point measurements for a Neumann-to-Dirichlet map and the Calderón problem in the plane, SIAM J. Math. Anal., 44 (2012), 3526-3536. doi: 10.1137/120872164. [8] A. Kirsch, The MUSIC algorithm and the factorization method in inverse scattering theory for inhomogeneous media, Inverse Problems, 18 (2002), 1025-1040. doi: 10.1088/0266-5611/18/4/306. [9] A. Kirsch, "An Introduction to the Mathematical Theory of Inverse Problems," $2^{nd}$ edition, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4419-8474-6. [10] A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford University Press, Oxford, 2008. [11] S. G. Krantz, "Function Theory of Several Complex Variables," $2^{nd}$ edition, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. [12] J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications," I, Springer-Verlag, New York, 1972. [13] A. I. Nachman, Reconstructions from boundary measurements, Ann. of Math., 128 (1988), 531-576. doi: 10.2307/1971435. [14] R. G. Novikov, A multidimensional inverse spectral problem for the equation $-\Delta \psi + (v(x)-Eu(x))\psi=0$, translation in Funct. Anal. Appl., 22 (1988), 263-272. doi: 10.1007/BF01077418. [15] A. G. Ramm, Recovery of the potential from fixed-energy scattering data, Inverse Problems, 4 (1988), 877-886. doi: 10.1088/0266-5611/4/3/020.

show all references

##### References:
 [1] F. E. Browder, Real analytic functions on product spaces and separate analyticity, Canad. J. Math., 13 (1961), 650-656. doi: 10.4153/CJM-1961-054-1. [2] A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33. doi: 10.1515/jiip.2008.002. [3] D. L. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," John Wiley & Sons, New York, 1983. [4] D. L. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," $2^{nd}$ edition, Springer-Verlag, Berlin, 1998. [5] F. Hartogs, Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten, Math. Ann., 62 (1906), 1-88. doi: 10.1007/BF01448415. [6] L. Hörmander, "An Introduction to Complex Analysis in Several Variables," $3^{rd}$ edition, North-Holland, Amsterdam, 1990. [7] N. Hyvönen, P. Piiroinen and O. Seiskari, Point measurements for a Neumann-to-Dirichlet map and the Calderón problem in the plane, SIAM J. Math. Anal., 44 (2012), 3526-3536. doi: 10.1137/120872164. [8] A. Kirsch, The MUSIC algorithm and the factorization method in inverse scattering theory for inhomogeneous media, Inverse Problems, 18 (2002), 1025-1040. doi: 10.1088/0266-5611/18/4/306. [9] A. Kirsch, "An Introduction to the Mathematical Theory of Inverse Problems," $2^{nd}$ edition, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4419-8474-6. [10] A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford University Press, Oxford, 2008. [11] S. G. Krantz, "Function Theory of Several Complex Variables," $2^{nd}$ edition, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. [12] J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications," I, Springer-Verlag, New York, 1972. [13] A. I. Nachman, Reconstructions from boundary measurements, Ann. of Math., 128 (1988), 531-576. doi: 10.2307/1971435. [14] R. G. Novikov, A multidimensional inverse spectral problem for the equation $-\Delta \psi + (v(x)-Eu(x))\psi=0$, translation in Funct. Anal. Appl., 22 (1988), 263-272. doi: 10.1007/BF01077418. [15] A. G. Ramm, Recovery of the potential from fixed-energy scattering data, Inverse Problems, 4 (1988), 877-886. doi: 10.1088/0266-5611/4/3/020.
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