Geometric function theory, potential theory, isoperimetric inequalities in mathematical physics. General symmetrization principles for various symmetrization-like transformation of condensers in n-space have been established for a wide range of capacities induced by functionals dependent on the argument, the function and its first partial derivatives. Generalized condensers having three and more plates on the Riemann sphere have been first introduced and studied. Common properties of these condensers and asymptotic formulas for their capacities under degeneration of some plates have been found. General approach to application of conformal capacity of condensers and their symmetrization in geometric theory of functions of a complex variable has been developed. In particular, new metric properties of plane sets have been found, extremal decomposition theorems with free poles and new covering and distortion theorems for univalent and multivalent functions proved. New properties of polynomials and rational functions have been established using geometric function theory methods. In particular, new Bernstein-type inequalities and some inequalities for zeros, critical points and critical values of polynomials have been proved.
Biography
Graduated with honors as mathematics major from the Far Eastern State University in 1973 and entered postgraduate course at the chair of function theory of the Kuban State University (Krasnodar) the same year. Depended PhD (candidate of Science) thesis entitled “Certain symmetrization-like transformations and covering problems in geometric theory of functions of a complex variable” at the Applied Mathematics and Mechanics Institute of the Ukranian Academy of Sciences (Donetsk) in 1977. Scientific Advisor – professor I.P. Mityuk. Defended habilitation thesis (Doctor of Science) entitled “Symmetrization method in geometric function theory” at the Mathematics Institute of the Siberian Branch of the Russian Academy of Sciences (Novosibirsk). Has been elected correspondent member of the Russian Academy of Sciences (Far Eastern Branch) in 2003. Teaching – 1977-78 – assistant teacher, 1978-83 – senior teacher, 1983-1989 – docent (associate professor), 1989 – present full professor of the chair of mathematical analysis at the Far Eastern State University (at present chair of function theory and functional analysis). The head of the laboratory of mathematical analysis at the Institute of Applied Mathematics of the Far Eastern Branch of the Russian Academy of Sciences since 1991.
Main publications:
V. N. Dubinin, “On the change in harmonic measure under symmetrization”, Math. USSR-Sb., 52:1 (1985), 267–273
V. N. Dubinin, “Capacities and geometric transformations of subsets in n-space”, Geometric and Functional Analysis, 3:4 (1993), 342-369
V. N. Dubinin, “Symmetrization in the geometric theory of functions of a complex variable”, Russian Math. Surveys, 49:1 (1994), 1–79
V. N. Dubinin, “Conformal mappings and inequalities for algebraic polynomials”, St. Petersburg Math. J., 13:5 (2002), 717–737
V. N. Dubinin, “Inequalities for critical values of polynomials”, Sb. Math., 197:8 (2006), 1167–1176
V. N. Dubinin, “Methods of geometric function theory in classical and modern problems for polynomials”, Russian Math. Surveys, 67:4 (2012), 599–684
V. N. Dubinin, “On one extremal problem for complex polynomials with constraints on critical values”, Siberian Math. J., 55:1 (2014), 63–71
V. N. Dubinin, Condenser capacities and symmetrization in geometric function theory, Basel: Birkhauser / Springer, 2014 , xii+344 pp.
V. N. Dubinin, “Circular symmetrization of condensers on Riemann surfaces”, Sb. Math., 206:1 (2015), 61–86
V. N. Dubinin, “Geometric estimates for the Schwarzian derivative”, Russian Math. Surveys, 72:3 (2017), 479–511
V. N. Dubinin, “Variational formulae for conformal capacity”, Sb. Math., 215:1 (2024), 90–100
2.
V. N. Dubinin, “Blaschke products and Smale's conjecture on complex polynomials”, Math. Notes, 116:2 (2024), 252–257
3.
V. N. Dubinin, “Teichmüller's modulsatz and the variation of the Dirichlet integral”, Sibirsk. Mat. Zh., 65:2 (2024), 288–294
4.
V. N. Dubinin, “Bounded holomorphic functions in a circular annulus”, Algebra i Analiz, 36:6 (2024), 30–46
5.
V. N. Dubinin, “Green energy of discrete signed measure on concentric circles”, Izv. Math., 87:2 (2023), 265–283
6.
V. N. Dubinin, “Boundary Distortion and the Schwarzian Derivative of a Univalent Function in a Circular Annulus”, Math. Notes, 113:6 (2023), 776–783
7.
V. N. Dubinin, “On the dissymmetrization theorem”, Siberian Electronic Mathematical Reports, 20:1 (2023), 477–485
8.
V. N. Dubinin, “An asymptotic formula for the capacity of a condenser when all its plates are degenerate”, Dal'nevost. Mat. Zh., 23:2 (2023), 184–189
9.
V. N. Dubinin, “An upper bound for the least critical values of finite Blaschke products”, Sb. Math., 213:6 (2022), 744–751
10.
V. N. Dubinin, V. Yu. Kim, “On the condensers with variable plates, potential levels and domain of definition”, Dal'nevost. Mat. Zh., 22:1 (2022), 55–60
11.
V. N. Dubinin, “On Holomorphic Coverings of Planar Domains”, Math. Notes, 112:5 (2022), 674–684
12.
V.N.Dubinin, E.G.Prilepkina, “Optimal Green energy points on the circles in d-space”, Journal of Mathematical Analysis and Applications, 499:2 (2021) (Article 125055)
V. N. Dubinin, “Some unsolved problems about condenser capacities on the plane”, Complex Analysis and Dynamical Systems, Ser.V: Trends in Mathematics. Editors: M.Agranovsky, A.Golberg,F.Jacobzon, D.Shoikhet, L.Zalcman, Birhauser, 2018, 81-92
V. N. Dubinin, “Geometric estimates for the Schwarzian derivative”, Russian Math. Surveys, 72:3 (2017), 479–511
29.
V. N. Dubinin, “The Kraus Inequality for Multivalent Functions”, Math. Notes, 102:4 (2017), 516–520
30.
V. N. Dubinin, A. S. Afanaseva-Grigoreva, “On the lemniscates of rational functions”, Dal'nevost. Mat. Zh., 17:2 (2017), 201–209
31.
V. N. Dubinin, “Lemniscate Zone and Distortion Theorems for Multivalent Functions”, Journal of Mathematical Sciences (United States), 234:5 (2018), 598–607
32.
V. N. Dubinin, “An Extremal Problem for the Derivative of a Rational Function”, Math. Notes, 100:5 (2016), 714–719
33.
V. N. Dubinin, “Critical values and moduli of derivative of a complex polynomial at its zeros”, J. Math. Sci. (N. Y.), 225:6 (2017), 877–882
34.
V. N. Dubinin, “The logarithmic energy of zeros and poles of a rational function”, Siberian Math. J., 57:6 (2016), 981–986
35.
V. N. Dubinin, “Circular symmetrization of condensers on Riemann surfaces”, Sb. Math., 206:1 (2015), 61–86
36.
V. N. Dubinin, “Schwarzian Derivative and Covering Arcs of a Pencil of Circles by Holomorphic Functions”, Math. Notes, 98:6 (2015), 920–925
37.
V. N. Dubinin, “Distortion theorems for circumferentially mean $p$-valent functions”, J. Math. Sci. (N. Y.), 217:1 (2016), 28–36
38.
V. N. Dubinin, “Inequalities for harmonic measures with respect to non-overlapping domains”, Izv. Math., 79:5 (2015), 902–918
39.
V. N. Dubinin, V. Y. Kim, “Covering theorem for p-valent functions with Montel's normalization”, Journal of Mathematical Analysis and Applications, 428:1 (2015), 502-507
40.
V. N. Dubinin, “Two-point boundary distortion estimate for Schwarzian derivative of holomorphic function”, FEMJ, 14:2 (2014), 191–199
41.
V. N. Dubinin, “Asymptotic Behavior of the Capacity of a Condenser as Some of Its Plates Contract to Points”, Math. Notes, 96:2 (2014), 187–198
42.
V. N. Dubinin, “On the reduced modulus of the complex sphere”, Siberian Math. J., 55:5 (2014), 882–892
43.
V. N. Dubinin, “On one extremal problem for complex polynomials with constraints on critical values”, Siberian Math. J., 55:1 (2014), 63–71
44.
V. N. Dubinin, “Inequalities for moduli of the circumferentially mean $p$-valent functions”, J. Math. Sci. (N. Y.), 207:6 (2015), 832–838
45.
V. N. Dubinin, “Bounded holomorphic functions covering no concentric circles”, J. Math. Sci. (N. Y.), 207:6 (2015), 825–831
46.
V. N. Dubinin, “Four-point distortion theorem for complex polynomials”, Complex Variables and Elliptic Equations, 59:1 (2014), 59–66
V. N. Dubinin, M. Vuorinen, “Ahlfors-Beurling conformal invariant and relative capacity of compact sets”, Proceedings of the Amer.Math.Soc., 142:11 (2014), 3865–3879
V. N. Dubinin, D. A. Kirillova, “Some applications of extremal decompositions in the geometric function theory”, FEMJ, 10:2 (2010), 130–152
65.
V. N. Dubinin, “Steiner symmetrization and the initial coefficients of univalent functions”, Izv. Math., 74:4 (2010), 735–742
66.
V. N. Dubinin, “On the Finite-Increment Theorem for Complex Polynomials”, Math. Notes, 88:5 (2010), 647–654
67.
V. N. Dubinin, “Lower bounds for the half-plane capacity of compact sets and symmetrization”, Sb. Math., 201:11 (2010), 1635–1646
68.
V. N. Dubinin, E. G. Prilepkina, “Distortion theorems for univalent meromorphic functions on an annulus”, Siberian Math. J., 51:2 (2010), 229–243
69.
V. N. Dubinin, “On the components of the lemniscate containing no critical points of a polynomial other than its zeros”, J. Math. Sci. (N. Y.), 178:2 (2011), 158–162
70.
V. N. Dubinin, “Geometric versions of Schwarz lemma and symmetrization”, J. Math. Sci. (N. Y.), 178:2 (2011), 150–157
71.
V. N. Dubinin, M. Vuorinen, “Robin functions and distortion theorems for regular mappings”, Math. Nachr., 283:11 (2010), 1589–1602
V. N. Dubinin, “Condenser capacity and symmetrization in the extremal decomposition problems”, FEMJ, 9:1-2 (2009), 84–93
73.
V. N. Dubinin, “Quadratic forms involving Green's and Robin functions”, Sb. Math., 200:10 (2009), 1439–1452
74.
V. N. Dubinin, “On I. P. Mityuk's results on the the behavior of the inner radius of a domain and the condenser's capacity under regular mappings”, J. Math. Sci. (N. Y.), 166:2 (2010), 145–154
75.
V. N. Dubinin, M. Vuorinen, “On conformal moduli of polygonal quadrilaters”, Israel J.Math., 171 (2009), 111 – 125
V. N. Dubinin, D. B. Karp, V. A. Shlyk, “Selected problems of geometrical theory of functions and potential theory”, FEMJ, 8:1 (2008), 46–95
78.
V. N. Dubinin, “Majorization Principles for Meromorphic Functions”, Math. Notes, 84:6 (2008), 751–755
79.
V. N. Dubinin, “Condenser capacities and majorization principles in the geometric function theory of a complex variable”, Sib. Èlektron. Mat. Izv., 5 (2008), 465–482
80.
V. V. Vasin, V. N. Dubinin, V. G. Romanov, “Concluding scientific report for the project of Siberian Division of RAS: “A development of theory and computational technology for solving inverse and extremal problems with an application to mathematical physics and gravity-magneto-prospecting””, Sib. Èlektron. Mat. Izv., 5 (2008), 427–439
81.
V. N. Dubinin, D. A. Kirillova, “On extremal decomposition problems”, J. Math. Sci. (N. Y.), 157:4 (2009), 573–583
82.
V. N. Dubinin, D. B. Karp, “Capacities of certain plane condensers and sets under simple geometric transformations”, Complex Variables and Elliptic Equations, 53:6 (2008), 607 – 622
V. N. Dubinin, V. Yu. Kim, “On the covering of radial segments under $p$-valent mappings of a disk and an annulus”, FEMJ, 7:1-2 (2007), 40–47
84.
V. N. Dubinin, S. I. Kalmykov, “A majoration principle for meromorphic functions”, Sb. Math., 198:12 (2007), 1737–1745
85.
V. N. Dubinin, V. Yu. Kim, “Distortion theorems for bounded regular functions in the disk”, J. Math. Sci. (N. Y.), 150:3 (2008), 2018–2026
86.
V. N. Dubinin, D. B. Karp, “Generalized condensers and distortion theorems for conformal mappings of planar domains”, In: The Interaction of Analysis and Geometry, Edited by: V.I.Burenkov, T.Iwaniec, and S.K.Vodopyanov, AMS Contemporary Mathematics, 424, AMER MATHEMATICAL SOC, P.O. BOX 6248, PROVIDENCE, RI 02940 USA, 2007, 33-51
V. N. Dubinin, E. G. Prilepkina, “On variational principles of conformal mappings”, St. Petersburg Math. J., 18:3 (2007), 373–389
88.
V. N. Dubinin, “Lemniscates and Inequalities for the Logarithmic Capacities of Continua”, Math. Notes, 80:1 (2006), 31–35
89.
V. N. Dubinin, “Inequalities for critical values of polynomials”, Sb. Math., 197:8 (2006), 1167–1176
90.
V. N. Dubinin, “Applications of the Schwarz lemma to inequalities for entire functions with constraints on zeros”, J. Math. Sci. (N. Y.), 143:3 (2007), 3069–3076
91.
V. N. Dubinin, “Capacities of condensers, generalizations of Grötzsch Lemmas, and symmetrization”, J. Math. Sci. (N. Y.), 143:3 (2007), 3053–3068
92.
V. N. Dubinin, E. G. Prilepkina, “On the preservation of generalized reduced modulus under some geometric transformations of domains in the plane”, FEMJ, 6:1-2 (2005), 39–56
93.
V. N. Dubinin, “Polynomials with Critical Values on Intervals”, Math. Notes, 78:6 (2005), 768–772
94.
V. N. Dubinin, “Schwarz's lemma and estimates of coefficients for regular functions with free domain of definition”, Sb. Math., 196:11 (2005), 1605–1625
95.
V. N. Dubinin, S. I. Kalmykov, “Extremal properties of Chebyshev polynomials”, FEMJ, 5:2 (2004), 169–177
96.
V. N. Dubinin, N. V. Eyrikh, “Some applications of generalized condensers to analytic function theory”, J. Math. Sci. (N. Y.), 133:6 (2006), 1634–1647
97.
V. N. Dubinin, “Generalized condensers and the asymptotics of their capacities under degeneration of some plates”, J. Math. Sci. (N. Y.), 129:3 (2005), 3835–3842
98.
V. N. Dubinin, “Conformal mappings and inequalities for algebraic polynomials. II”, J. Math. Sci. (N. Y.), 129:3 (2005), 3823–3834
99.
V. N. Dubinin, “Symmetrization methods in the geometric function theory”, Trudy po geometrii i analizu, Iz-vo Mat. Inst., Novosibirsk, 2003, 126–150
100.
V. N. Dubinin, N. V. Eyrikh, “The generalized reduced modulus”, FEMJ, 3:2 (2002), 150–164
101.
V. N. Dubinin, E. G. Prilepkina, “On the uniqueness theorems for the transformations of sets and condensers in the plane”, FEMJ, 3:2 (2002), 137–149
102.
V. N. Dubinin, “On an application of conformal maps to inequalities for rational functions”, Izv. Math., 66:2 (2002), 285–297
103.
V. N. Dubinin, A. V. Olesov, “Application of conformal mappings to the inequalities for polynomials”, J. Math. Sci. (N. Y.), 122:6 (2004), 3630–3640
104.
V. N. Dubinin, “On the Shwarz inequality on the boundary for functions regular in the disk”, J. Math. Sci. (N. Y.), 122:6 (2004), 3623–3629
105.
V. N. Dubinin, “Conformal mappings and inequalities for algebraic polynomials”, St. Petersburg Math. J., 13:5 (2002), 717–737
106.
V. N. Dubinin, V. Yu. Kim, “Averaging transformation of sets and functions on Riemann surfaces”, Russian Math. (Iz. VUZ), 45:5 (2001), 19–27
107.
V. N. Dubinin, E. V. Kostyuchenko, “Extremal problems in the function theory associated with the $n$-fold symmetry”, J. Math. Sci. (N. Y.), 118:1 (2003), 4778–4794
108.
V. N. Dubinin, “Distortion theorems for polynomials on a circle”, Sb. Math., 191:12 (2000), 1797–1807
109.
V. N. Dubinin, V. Yu. Kim, “Reduced moduli and inequalities for the polynomials”, J. Math. Sci. (New York), 110:6 (2002), 3070–3077
110.
E. G. Akhmedzyanova, V. N. Dubinin, “Radial transformations of sets, and inequalities for the transfinite diameter”, Russian Math. (Iz. VUZ), 43:4 (1999), 1–6
111.
V. N. Dubinin, “The majorization principle for $p$-valent functions”, Math. Notes, 65:4 (1999), 447–453
112.
V. N. Dubinin, E. V. Kostyuchenko, “The Teichm\ddot uller extremal problem and distortion theorems in the theory of univalent functions”, Siberian Math. J., 40:2 (1999), 258–261
113.
V. N. Dubinin, E. G. Prilepkina, “On extremal decomposition of $n$-space domains”, J. Math. Sci. (New York), 105:4 (2001), 2180–2189
114.
V. N. Dubinin, L. V. Kovalev, “The reduced modulus of the complex sphere”, J. Math. Sci. (New York), 105:4 (2001), 2165–2179
115.
V. N. Dubinin, “Reduced moduli of open sets in the theory of analytic functions”, Dokl. Math., 58:3 (1998), 458-461
116.
V. N. Dubinin, “The asymptotics of module of a degenerating condenser and some of its applications”, J. Math. Sci. (New York), 95:3 (1999), 2209–2220
117.
V. N. Dubinin, “Symmetrization, Green's function, and conformal mappings”, J. Math. Sci. (New York), 89:1 (1998), 967–975
118.
V. N. Dubinin, “Symmetrization in the geometric theory of functions of a complex variable”, Russian Math. Surveys, 49:1 (1994), 1–79
119.
V. N. Dubinin, “Some properties of the reduced inner modulus”, Siberian Math. J., 35:4 (1994), 689–705
120.
V. N. Dubinin, “Capacities and geometric transformations of subsets in n-space”, Geometric and Functional Analysis, 3:4 (1993), 342-369
V. N. Dubinin, “Transformations of condensers in $n$-space”, J. Math. Sci., 70:6 (1994), 2085–2096
122.
V. N. Dubinin, “On the question of construction of symmetrization transformations”, Mathematical physics and mathematical simulation in ecology, 1, Far-Eastern Branch of the USSR Acad. Sci., Vladivostok, 1990, 55-69
123.
V. N. Dubinin, “On the covering of radial segments under conformal mapping of a disk”, Siberian Math. J., 31:2 (1990), 230–242
124.
V. N. Dubinin, “Distortion theorem in the theory of conformal mappings”, Math. Notes, 44:3 (1988), 668–674
125.
V. N. Dubinin, “A separating transformation of domains, and problems on extremal decomposition”, J. Soviet Math., 53:3 (1991), 252–263
126.
V. N. Dubinin, “Theorems on the covering of segments under a conformal mapping of an annulus”, Soviet Math. (Iz. VUZ), 31:9 (1987), 50–60
127.
V. N. Dubinin, “Transformation of condensers in space”, Soviet Math. Dokl., 36:2 (1988), 217-219
128.
V. N. Dubinin, “An extremal problem for functions that are meromorphic in multiply connected domains”, Soviet Math. (Iz. VUZ), 30:10 (1986), 72–77
129.
V. N. Dubinin, “On covering theorems in the theory of analytic functions”, J. Soviet Math., 43:4 (1988), 2553–2558
V. N. Dubinin, “Transformation of functions and Dirichlet's principle”, Math. Notes, 38:1 (1985), 539–542
132.
V. N. Dubinin, “The symmetrization method in problems on nonoverlapping domains”, Math. USSR-Sb., 56:1 (1987), 107–119
133.
V. N. Dubinin, “On the change in harmonic measure under symmetrization”, Math. USSR-Sb., 52:1 (1985), 267–273
134.
V. N. Dubinin, “Coverings of vertical segments under a conformal mapping”, Math. Notes, 28:1 (1980), 476–480
135.
V. N. Dubinin, “Some covering theorems in the theory of analytic functions”, Differential equations and functional analysis, Far-Eastern Science Centre, Acad. Sci. USSR, Vladivostok, 1980, 98-109
136.
V. N. Dubinin, “The product of internal radii of “partially nonoverlapping” domains”, Questions in the metric theory of mappings and its application (Proc. Fifth Colloq. Quasiconformal Mappings, Generalizations Appl., Donetsk, 1976), Naukova Dumka, Kiev, 1978, 24-31
137.
V. N. Dubinin, “Some symmetrization transformations and their applications to covering theorems”, Mathematical analysis, No. 3. Kuban. Gos. Univ. Nauchn. Trudy, 217, 1976, 17-25
138.
V. N. Dubinin, “On Hayman’s conjecture”, Mathematical analysis, No. 3. Kuban Gos. Univ. Nauchn. Trudy, 217, 1976, 26-34
139.
V. N. Dubinin, “A certain symmetrization method”, Mathematical analysis, No. 2. Kuban. Gos. Univ. Naučn. Trudy, 180, Kuban. Gos. Univ., 1974, 50-64
Books
140.
V. N. Dubinin, Condenser capacities and symmetrization in geometric function theory, Basel: Birkhauser / Springer, 2014 , xii+344 pp.
V. N. Dubinin, The symmetrization method in geometric function theory, D. Sc. dissertation, Vladivostok, 1988
144.
V. N. Dubinin, Some symmetrization transformations and covering theorems in the geometric theory of functions of a complex variable, Candidate’s dissertation, Krasnodar, 1977
Personalia
145.
L. A. Aksent'ev, A. I. Aptekarev, A. M. Bikchentaev, V. V. Goryainov, V. N. Dubinin, A. M. Elizarov, I. R. Kayumov, A. Laptev, S. R. Nasyrov, D. V. Prokhorov, A. G. Sergeev, V. D. Stepanov, “Farit Gabidinovich Avkhadiev (on his 70th birthday)”, Russian Math. Surveys, 73:1 (2018), 181–185
146.
A. I. Aptekarev, V. K. Beloshapka, V. I. Buslaev, V. V. Goryainov, V. N. Dubinin, V. A. Zorich, N. G. Kruzhilin, S. Yu. Nemirovski, S. Yu. Orevkov, P. V. Paramonov, S. I. Pinchuk, A. S. Sadullaev, A. G. Sergeev, S. P. Suetin, A. B. Sukhov, K. Yu. Fedorovskiy, A. K. Tsikh, “Evgenii Mikhailovich Chirka (on his 75th birthday)”, Russian Math. Surveys, 73:6 (2018), 1137–1144
147.
V. M. Buchstaber, V. N. Dubinin, V. A. Kaliaguine, B. S. Kashin, V. N. Sorokin, S. P. Suetin, D. N. Tulyakov, B. N. Chetverushkin, E. M. Chirka, A. A. Shkalikov, “Alexander Ivanovich Aptekarev (on his 60th birthday)”, Russian Math. Surveys, 70:5 (2015), 965–973
148.
F. G. Avkhadiev, V. A. Botvinnik, S. K. Vodop'yanov, M. Vuorinen, V. M. Gol'dstein, V. V. Goryainov, A. A. Grigor'yan, V. N. Dubinin, I. V. Zhuravlev, V. A. Zorich, V. M. Kesel'man, V. A. Klyachin, T. G. Latfullin, A. V. Loboda, A. G. Losev, O. Martio, V. I. Pelikh, S. I. Pinchuk, Yu. G. Reshetnyak, A. S. Romanov, A. G. Sergeev, V. G. Tkachev, E. M. Chirka, “Vladimir Mikhailovich Miklyukov (obituary)”, Russian Math. Surveys, 69:3 (2014), 565–568
149.
V. N. Dubinin, G. V. Kuz'mina, “Igor Petrovich Mityuk (1928–1995). On the occasion of the 80 anniversary of birthday”, Analytical theory of numbers and theory of functions. Part 24, Zap. Nauchn. Sem. POMI, 371, POMI, St. Petersburg, 2009, 5
150.
L. A. Aksent'ev, I. A. Aleksandrov, G. K. Antonyuk, V. A. Babeshko, V. Ya. Gutlyanskii, V. N. Dubinin, B. E. Levitsii, Yu. A. Mitropol'skii, Yu. G. Reshetnyak, Yu. E. Hohlov, “Igor Petrovich Mityuk (obituary)”, Russian Math. Surveys, 51:2 (1996), 315–316;
A capacity look at the Schwarzian derivative V. N. Dubinin The 27th International Conference on Finite and Infinite Dimensional Complex Analysis and Applications August 14, 2019 10:30