Lectures Introductory to the Theory of Functions of Two Complex Variables Deliv

Cover Lectures Introductory to the Theory of Functions of Two Complex Variables Deliv
Lectures Introductory to the Theory of Functions of Two Complex Variables Deliv
Forsyth, Andrew Russell, 1858-1942
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f It is established for the case of n variables, Weierstrass, Ges. Werke, t. Ii, pp. 62 — 64.
124] PERIODS 215 Next, suppose that our uniform analytic function is not expressible in a form F(az + a'z') for any constants a and a' whatever; and consider a region in the field of variation where the function f(z, z) is regular. No relation oz dz for non-vanishing values of a and a, is satisfied over the whole of this region ; hence we can take places z x and */, z 2 and z 2 within the region, such
...that I Jia |, where "12 — 9 /Qi, -Q d/Pi, V ) a*! ' a*/ a/p 2, V) a/p 2, «/) t/io dz 2 dz 2 is finite and not zero. Also when we take places z 1 + u x and z/ 4- «/, z* + u^ and */ + u 2, z x + v^ and z( + v x ', z 2 + v 2 and ^ 2 ' + v 2, where all the quantities I Mi 1 1 I ^/ I » I ^2 1, I V I, I v 1 1, | v/ 1, | v 2 1, | v 2 ' | are infinitesimally small, the quantity j J l2, | where dfjZt + Ui, Zi + t> s/ + Vi) dZi dzi df(z 2 4- u 2, z 2 + m 2 ') 9/p 2 + p 2, z 2 '+v 2 ) dz 2 dz 2 differs from j J 12 j only infinitesimally, and therefore its modulus is finite and not zero.

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