An introduction to the theory of multiply periodic functions, by H.F. Baker.

ART. 9] integral fractions. 21
as h. dorn' = do-m'h = crdm n' - m'. dh,
this is the sane as
A1 + r-lm' = A ee.......................................)
where H = 7iro- (m' +- dlh - - OT (dh)2,
and this holds for arbitrary integer pairs h and rm'. Now any pair of integers
(ni, n2) can be uniquely written in the form (hl rd-lvm ', + rd,-lmn) by
choosing the integers rm/, m,' suitably, with the condition 0 ~ hl < rdl-l,
00
h0 $ h < rd.21; the terms of the doubly infinite series E A,, e27i'nldv can then
n= - o
be arranged in a finite number of sets according to the appropriate values of
hA and h2; namely, we have
co
A Le27rindlv A e27ri (h + rdl~'). diw
and
(h + rd-m'). cdw = d (h + rd-i'') w = r (r/ + - dh) w = rw (m' + - h);
thus, from (i), above,
00 7ri oo
Antld = A herin - r -(dh)2 e2ri'iw (m'+-hh +7rir (,m'+ -dh)
- o h72 '1 = - o
we introduce now the notation
(V, ~'; q' ) — ^ e27riv( + q') + rit (\ + q/)2 + 2iq (X + ')
where v, =(vl, 2v), denotes two independent variables, r is a symmetrical
matrix of type (2, 2), q is a row of any two constants, as is also q', and X
stands for two integers, each of which independently of the other takes all
integer values from - oo to + oo; it will be proved that if; when x1, x2 are any
two real quantities, the quadratic irx2 has its real part essentially negative
and not zero, this expression represents an integral function of v1, v,, and is
uniformly and absolutely converging; in terms of such functions, the integral
function p (w) is now shewn to be expressible in the form
() = ( B (rwr, r-id
where h,, h, are limited only by O h < rd~-l, O h2 < rd2-l, so that the
number of terms on the right is r2d'-ld2-l, the unknown constant Bh replacing
A h e- 1 (dh).
(e) As our defining equation was hypothetical it is necessary to shew
that the expression H (v, T; ) represents a function. Consider first the
q /LrY~-U VVC(4