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| In mathematics, the '''Bochner–Martinelli formula''' is a generalization of the [[Cauchy integral formula]] to functions of [[several complex variables]], introduced by {{harvs|txt|first=Enzo|last=Martinelli| authorlink=Enzo Martinelli|year=1938}} and {{harvs|txt|first=Salomon|last=Bochner|authorlink=Salomon Bochner|year=1943}}.
| | Hello! My name is Alexander. <br>It is a little about myself: I live in Switzerland, my city of Basel. <br>It's called often Northern or cultural capital of . I've married 1 years ago.<br>I have 2 children - a son (Janis) and the daughter (Johnson). We all like Metal detecting.<br><br>My web page :: google; [http://www.Google.com/ http://www.Google.com/], |
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| ==History==
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| {{quote
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| |text= Formula (53) of the present paper and a proof of theorem 5 based on it have just been published by [[Enzo Martinelli]] '''(...)'''.<ref>Bochner refers explicitly to the article {{harv|Martinelli|1942-1943}}, apparently being not aware of the earlier one {{harv|Martinelli|1938}}, which actually contains Martinelli's proof of the formula. However, the earlier article is explicitly cited in the later one, as it can be seen from {{harv|Martinelli|1942-1943|loc=p. 340, footnote 2}}.</ref> The present author may be permitted to state that these results have been presented by him in a [[Princeton University|Princeton]] graduate course in Winter 1940/1941 and were subsequently incorporated, in a Princeton doctorate thesis (June 1941) by Donald C. May, entitled: An integral formula for analytic functions of {{math|''k''}} variables with some applications.
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| |sign=Salomon Bochner
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| |source={{harv|Bochner|1943|loc=p. 652, footnote 1}}.
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| }}
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| {{quote
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| |text= However this author's claim in ''loc. cit.'' footnote 1,<ref>Bochner refers to his claim in {{harv|Bochner|1943|loc=p. 652, footnote 1}}.</ref> that he might have been familiar with the general shape of the formula before Martinelli, was wholly unjustified and is hereby being retracted.
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| |sign=Salomon Bochner
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| |source={{harv|Bochner|1947|loc=p. 15, footnote *}}.
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| }}
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| ==Bochner–Martinelli kernel==
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| For {{math|''ζ''}}, {{math|''z''}} in ℂ<sup>''n''</sup> the Bochner–Martinelli kernel {{math|ω(''ζ'',''z'')}} is a differential form in {{math|''ζ''}} of bidegree {{math|(''n'',''n''−1)}} defined by
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| :<math>\omega(\zeta,z) = \frac{(n-1)!}{(2\pi i)^n}\frac{1}{|z-\zeta|^{2n}}
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| \sum_{1\le j\le n}(\overline\zeta_j-\overline z_j) \, d\overline\zeta_1 \and d\zeta_1 \and \cdots \and d\zeta_j \and \cdots \and d\overline\zeta_n \and d\zeta_n</math>
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| (where the term {{math|''d''{{overline|''ζ''}}<sub>''j''</sub>}} is omitted). | |
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| Suppose that {{math|''f''}} is a continuously differentiable function on the closure of a domain {{math|''D''}} in ℂ<sup>''n''</sup> with piecewise smooth boundary {{math|∂''D''}}. Then the Bochner–Martinelli formula states that if {{math|''z''}} is in the domain {{math|''D''}} then
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| :<math>\displaystyle f(z) = \int_{\partial D}f(\zeta)\omega(\zeta, z) - \int_D\overline\partial f(\zeta)\and\omega(\zeta,z).</math>
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| In particular if {{math|''f''}} is holomorphic the second term vanishes, so
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| :<math>\displaystyle f(z) = \int_{\partial D}f(\zeta)\omega(\zeta, z). </math>
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| ==See also==
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| *[[Bergman–Weil formula]]
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| ==Notes==
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| {{reflist|30em}}
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| ==References==
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| *{{Citation
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| | last = Aizenberg
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| | first = L. A.
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| | author-link = Lev Aizenberg
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| | last2 = Yuzhakov
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| | first2 = A. P.
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| | author2-link = Aleksandr Yuzhakov
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| | title = Integral Representations and Residues in Multidimensional Complex Analysis
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| | url = http://books.google.it/books?id=2ZWsf6ufee8C&printsec=frontcover&hl=en&#v=onepage&q&f=true
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| | place = [[Providence R.I.]]
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| | series = Translations of Mathematical Monographs
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| | volume = 58
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| | publisher = [[American Mathematical Society]]
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| | pages = x+283
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| | year = 1983
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| | origyear = 1979
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| | isbn = 0-8218-4511-X
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| | mr = 0735793
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| | zbl = 0537.32002
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| }}
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| *{{Citation
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| | last1=Bochner
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| | first1=Salomon
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| | author1-link=Salomon Bochner
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| | title=Analytic and meromorphic continuation by means of Green's formula
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| | jstor=1969103
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| | mr=0009206
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| | zbl = 0060.24206
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| | series=Second Series
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| | year=1943
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| | journal=[[Annals of Mathematics]]
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| | issn=0003-486X
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| | volume=44
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| | pages=652–673}}.
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| *{{Citation
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| | last1 = Bochner
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| | first1 = Salomon
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| | author1-link = Salomon Bochner
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| | title = On compact complex manifolds
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| | journal = [[The Journal of the Indian Mathematical Society]]
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| | series = New Series,
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| | volume = 11
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| | pages = 1–21
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| | year = 1947
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| | url =
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| | doi =
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| | mr = 0023919
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| | zbl = 0038.23701
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| }}.
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| *{{eom|id=b/b016720|first=E.M.|last= Chirka|title=Bochner–Martinelli representation formula}}
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| *{{Citation | last1=Krantz | first1=Steven G. | title=Function theory of several complex variables | url=http://books.google.com/books?isbn=9780821827246 | publisher=AMS Chelsea Publishing, Providence, RI | isbn=978-0-8218-2724-6 | mr=1846625 | year=2001}}
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| *{{Citation
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| | last = Kytmanov
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| | first = Alexander M.
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| | author-link = Alexander Kytmanov
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| | title=The Bochner-Martinelli integral and its applications
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| | origyear = 1992
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| | url=http://books.google.com/books?isbn=376435240X
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| | publisher = [[Birkhäuser Verlag]]
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| | pages = xii+305
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| | year = 1995
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| | isbn = 978-3-7643-5240-0
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| | mr = 1409816
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| | zbl = 0834.32001
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| }}.
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| *{{Citation
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| | last = Kytmanov
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| | first = Alexander M.
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| | author-link = Alexander Kytmanov
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| | last2 = Myslivets
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| | first2 = Simona G.
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| | author2-link =
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| | title = Интегральные представления и их приложения в многомерном комплексном анализе
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| | url = http://www.eastview.com/russian/books/product.asp?SKU=930345B&f_locale=_CYR&active_tab=1
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| | place = [[Krasnoyarsk|Красноярск]]
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| | publisher = [[Siberian Federal University|СФУ]]
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| | pages = 389
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| | year = 2010
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| | isbn = 978-5-7638-1990-8
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| | mr =
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| | zbl =
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| }} (English translation of title:"''Integral representations and their application in multidimensional complex analysis''").
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| *{{Citation
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| | last = Martinelli
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| | first = Enzo
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| | author-link = Enzo Martinelli
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| | title = Alcuni teoremi integrali per le funzioni analitiche di più variabili complesse
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| | language = Italian
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| | year = 1938
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| | journal = Memorie della Reale Accademia d'Italia
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| | volume = 9
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| | pages = 269–283
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| | id =
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| | jfm = 64.0322.04
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| | zbl = 0022.24002
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| }}. This is the first paper where the now called Bochner–Martinelli formula is introduced and proved: an English translation of the title reads as:-"''Some integral theorems for analytic functions of several complex variables''".
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| *{{Citation
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| | last = Martinelli
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| | first = Enzo
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| | author-link = Enzo Martinelli
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| | title = Sopra una dimostrazione di R. Fueter per un teorema di Hartogs
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| | language = Italian
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| | journal = [[Commentarii Mathematici Helvetici]]
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| | volume = 15
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| | issue = 1
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| | pages = 340–349
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| | year = 1942-1943
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| | url = http://retro.seals.ch/digbib/en/view?rid=comahe-002:1942-1943:15::26
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| | doi = 10.5169/seals-14896
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| | id =
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| | mr = 0010729
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| | zbl = 0028.15201
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| }}. Available at the [http://retro.seals.ch/digbib/home SEALS Portal]. In this paper Martinelli gave a proof of [[Hartogs' extension theorem]] by using the Bochner–Martinelli formula. An English translation of the title reads as:-"''On a proof of R. Fueter of a theorem of Hartogs''".
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| *{{Citation
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| | last = Martinelli
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| | first = Enzo
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| | author-link = Enzo Martinelli
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| | title = Introduzione elementare alla teoria delle funzioni di variabili complesse con particolare riguardo alle rappresentazioni integrali
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| | language = Italian
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| | place = Rome
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| | publisher = [[Accademia Nazionale dei Lincei]]
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| | year = 1984
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| | series = Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni
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| | volume = 67
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| | pages = 236+II
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| | url = http://www.lincei.it/pubblicazioni/catalogo/volume.php?lg=e&rid=33233
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| | doi =
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| | id =
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| | isbn =
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| }}. The notes of a course, published by the [[Accademia Nazionale dei Lincei]], taught when he was in charge to the academy as a "''Professore Linceo''". An English translation of the title reads as:-"''Elementary introduction to the theory of functions of complex variables with particular regard to integral representations''".
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| {{DEFAULTSORT:Bochner-Martinelli formula}}
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| [[Category:Complex analysis]]
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| [[Category:Several complex variables]]
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Hello! My name is Alexander.
It is a little about myself: I live in Switzerland, my city of Basel.
It's called often Northern or cultural capital of . I've married 1 years ago.
I have 2 children - a son (Janis) and the daughter (Johnson). We all like Metal detecting.
My web page :: google; http://www.Google.com/,