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

ISBN-10: 1418167037

ISBN-13: 9781418167035

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Example text

13). We 10. return now to the Riemann surface, and consider upon it the function of (x) expressed by 8 (," '"-e,, n/' m -e. ^), (A 3 ), (A t ), so rendering (x). ' 1 m v^ U vf- e, t , the right and left' sides of (A 3 ) a similar statement r m holds for (A t ). The function (w e) is an integral function of t'/ and therefore on the Riemann surface, capable, that is, of v/< '", analytical taken by vf> e l at ; ' 1 , representation about any place of the surface by a series of integral powers of the parameter for that place, there being no negative powers hence, the ; number of places (x) where the function vanishes to the first order, if any, or the sum of the orders with which it vanishes, is given by taking the integral J_ fd 2-rriJ round the closed curves (AjA^A^A^ contour the two sides of (A t ) 1 ), (A 3 A t Ai~ give no contribution ; l A ~ t 1 )- Of the former the two sides of (A,) give theta functions.

Ft, ft. , q\$. - by Jirt', ft', ft') for all values , Finally, in view of subsequent work, detail the relations affecting o> and u>' it desirable to notice in is only. The relation ') u Uv Let now ft> , /O &>' - 1\ /w \ = (ft)', - to) /w \ = o)'o) - ft)ft)' = 0. ]21<2> ) 6 4 we are real, PI, Pi, 1\, ^2 \ tn t /e = have, since T, w) (a)', W/ /&J \ t,,t = &)o> ') (a>'w W = (T OW,,) t^t O \W/ ^ = 2io-a> = w (T - TO) o) ; < .

Necessarily capable of representation as a if jR,, R^ be the two values of the function for the conjugate places (x, y), (x, y), the functions R^ are at once seen to be rational functions of x only. + _R and y (/^ R,) Secondly, it is not with poles of the first order at two possible to construct a rational function arbitrary places (a:,, y,), (ara y2 ), unless these be conjugate places having 2 R , x l = x. and y = t l yt , in which case (x 1 a;,)" is such a function. t are suitably chosen constants, could be taken to be a function without and analytical on the surface, save for the periods 2wt [A, (,),, + A, (vfW , tori infinities, (cf.