An Introduction to the Theory of Multiply Periodic Functions
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(ii) When / is an elementary integral of the third kind, that is, is infinite at one place (a) like log 4, where t^ is the parameter for this place, and infinite at another place (/3) like — log fp, but not elsewhere, we have, whatever Z may be ij|^^;ij(..,^)cfe=iog|g, where Z{a), Z(^) denote the values of Z at these places, (iii) If the fundamental algebraic equation be y^-(4!x'-g.^x-gs) = 0, and R (x, y) = x/y, while Z is taken to be (y — mx — c)/(y — m^x — Co), we have Q = 3, and dljdt is in...finite only for ^ = oo ; putting, for the single place there occurring, x = t-\ y = -2t-'(l-ig,t'-...), we have f^=la-y,t' -...), 1 y — mx — c , 1 4- ^mt + \ct^ - ... ^ , , , log -^ = log = ^r^-- T— TT- = i (m - ?Wo) t + ..
^y-nioX-Co ^ l+lniot + Uot'- ... ^^ ^ NOTE ii] Converse of Abel's Theorem. 181 and ■(f) ■»^^" t-^ \ {m - mo); if then (^i, 3/1), (x^, 3/2) and (^3, 3/3) be the three intersections of the straight line y — mx-\- c with the cubic curve y- = ^x^ — g^x — g^, and (ci), {c^, (Ca) be arbitrary places, we have r(*^') xdx A*^) xdx A^3) xdx ^yi — y^ /-i ■ hej y he.) y he,) y ^^x,-x,~ ' where is a quantity unaltered by replacing the straight line y = mx + c by any other ; putting, as usual ^ y this is equivalent with r°° dx 1 r*' r 1 du, till) -\-K(v)-t(u-\-v)-irl ^ — ^ = constant, for arbitrary values of il and v ; by expansion in powers of u for small values of u we at once find the constant to be zero.
What to read after An Introduction to the Theory of Multiply Periodic Functions?
You can find similar books in the "Read Also" column, or choose other free books by Baker, H. F. (Henry Frederick), 1866-1956 to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
(ii) When / is an elementary integral of the third kind, that is, is infinite at one place (a) like log 4, where t^ is the parameter for this place, and infinite at another place (/3) like — log fp, but not elsewhere, we have, whatever Z may be ij|^^;ij(..,^)cfe=iog|g, where Z{a), Z(^) denote the values of Z at these places, (iii) If the fundamental algebraic equation be y^-(4!x'-g.^x-gs) = 0, and R (x, y) = x/y, while Z is taken to be (y — mx — c)/(y — m^x — Co), we have Q = 3, and dljdt is in...finite only for ^ = oo ; putting, for the single place there occurring, x = t-\ y = -2t-'(l-ig,t'-...), we have f^=la-y,t' -...), 1 y — mx — c , 1 4- ^mt + \ct^ - ... ^ , , , log -^ = log = ^r^-- T— TT- = i (m - ?Wo) t + ..
^y-nioX-Co ^ l+lniot + Uot'- ... ^^ ^ NOTE ii] Converse of Abel's Theorem. 181 and ■(f) ■»^^" t-^ \ {m - mo); if then (^i, 3/1), (x^, 3/2) and (^3, 3/3) be the three intersections of the straight line y — mx-\- c with the cubic curve y- = ^x^ — g^x — g^, and (ci), {c^, (Ca) be arbitrary places, we have r(*^') xdx A*^) xdx A^3) xdx ^yi — y^ /-i ■ hej y he.) y he,) y ^^x,-x,~ ' where is a quantity unaltered by replacing the straight line y = mx + c by any other ; putting, as usual ^ y this is equivalent with r°° dx 1 r*' r 1 du, till) -\-K(v)-t(u-\-v)-irl ^ — ^ = constant, for arbitrary values of il and v ; by expansion in powers of u for small values of u we at once find the constant to be zero.
What to read after An Introduction to the Theory of Multiply Periodic Functions?
You can find similar books in the "Read Also" column, or choose other free books by Baker, H. F. (Henry Frederick), 1866-1956 to read onlineMoreLess
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