By H. F. Baker
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Extra info for An Introduction To The Theory Of Multiply Periodic Functions
M x m y*" the function is in fact a constant, and the function (v ' vanishes to the order at first and m 1 (a. ,) ( 2 ). 4,), (4 a ), (A,), (A 4 ) exceed its values ; left sides at the right sides respectively by inversion theorem. ART. 2 > m ) be single-valued on the dissected surface, and analytic, and values on the two sides of the loops (-4,), (A s ) will be the same. Va), we set out to prove. ), the proposition is (p. 7, (1)). J are determinable uniquely, so that, M M and integers , on the dissected surface v/>- m > m + vf ' = e* + M.
Loop. 4,) for v^""' and zero towards |J/ for vtCt towards l we ; \M - at . x e opposite signs reach the other side of a cut ; M respectively, M l M ' ' in the two sheets, we hiive where the integrals on the right are evaluated on the lower sheet, and those these are the equations stated. left on the upper sheet of the surface on the It is ; convenient to denote these equations, for the present, by putting * = $/#,' M '\; 2 Ufi JfJ then in particular we find (A. )' "(-1 OJ ""1= ART. 10] Identical vanishing of theta function.
4i), (A 2 ) its 2 lH values agree, but at (A,), (A t ) it has factors e~ * *, e~ 2rlH ", where t is zero, as also, similarly, value taken by for it is H 3 = (/A). (#) The . ' 1 From this, X Ux ' U. II X x U. = thus equal to the constant Vr " \ log by the lemmas just preceding, we obtain K *. "' We + is Thus, putting / = Vrx " a + tV*"'i we have function T *, ^-^ ~ g ' (-- ( 29) that we may regard the arguments (MJ M2 '), and the arguments (/', 2"), as the independent variables, the places (x^, (#a ) and being functions of these hence, from the equation have proved (p.