An Introduction To The Theory Of Multiply Periodic Functions by H. F. Baker

By H. F. Baker

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2 (0) - M . ; The ART. 10] 35 cross-ratio identity. fr also (p. 11), <:,, + <,;, if, n:i - n :; = - * i i r f w ; &** + /-> in particular, M= u*- a = U , then we have ? x" a + Uf - #=-22 a ri = It" "*, ,M r*'* U^- a (/>*' > + + it" 2 '" 2 M , (0) = W* 1' , />"*). r, Recall also the equation (p. 10) <;, = 'C = cr + u *^ ^^ + L ^ which gives K'l + **;! = px + pr^+(^x " " + M X" M ') A*-" + (,*" + ,**) v-". 1 ' *. These notations being made wherein clear, consider the function o 3 are branch places, as before, but (a^), (a;2 ), (ytj).

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.

P* i }a 2y f=f(x), . ' *-/(*), ,,*>-[* y Ja . y and F(x, z) We = 2X + shall \, (a; + z) + xz [2X, + X ,.. z*. }, 4 y < [* F- left side in (-) ' the identity shews that -R*'*, as a function an elementary integral of the third kind with logarithmic infinities ART. 6] connecting the integrals. 22 , are subject to the relation this leads to ; ,a + 2 ^a e r g u,', u rx - a = II*'" - [(z, \" Jo Now first let (z) s in the limit of - -=- 8 UA/ the limit of - I - (z, a)] dz. be in the neighbourhood of a particular place (z ), and express terms of the parameter of this place, and equate coefficients of the e power of this parameter; from uf> we obtain an expression which is and z x) [(z, x) ^ (z, a)] when t = 0; this we denote by which we denote by from p, (za ); u' c we *'" fj^(z ); dz we obtain the limit '" from IT*'" we obtain F^'"; from = for of [(z, x) (z, a)] -jj, is a certain rational function of (x) ; replacing now again which dt < .

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