DOUBLE GAMMA FUNCTION. 
311 
/X — ± — if i l(w J I > j I(Wo) I , the upper or lower sign being taken accord- 
ing as I(wi) is positive or negative, 
771 . 
and /X = i — if | | > | 1 5 the upper or lower sign being taken accord- 
ing as I(w^) is positive or negative. 
§ 34. By means ot the preceding paragraphs we may now at once prove 
Weierstrass’ relation* 
~ ± -7n, 
the upper or lower sign being taken according as I is positive or negative, 
where and rj., are determined by tlie relations 
^(' + ^ 1 ) — 
^(2 + (^z) = + V-2- 
Take the expression for {(t) given in § 3il, and we tind by use of the formulae 
of § 22 that 
C{z) — ^(2 + ojJ = {in — nif^ — ia.2, + m^) + voi^, 
(Mo 
where 
Oil does 1 . . . 
m, = 0 , unless , Mie within the reg’ion b(-)unded by axes from 
^ — ojj does not J ^ 
the origin to — 1 and coo, in Avhich case nii = i 1, according as — 1 (cm.,) is positive 
or negative, and wio and ni^ are obtained by changing the signs of (i) both cmj and 
(Mo and (ii) respectiA ely in this formula. 
Thus 
^ V i 
(Ml Ur PA 
2 
(MjCMo 
log 
(Mj + (Mo 
(Mj (Mo 
-TTi 
CMjCMo 
[?/i — Dll 
— m-z + aig], 
the infinities in the double summation being ec^ual in absolute magnitude, and that 
A'alue of (Ml (Mo being taken AA'hich is on the same side of the real axis as ((Mi + cmo). 
And, similarly. 
Avhere 
(Mo 
V V'_ 
02 
(Mi(Mo 
i{z (mJ — {( 2 ) _ 7^0, 
[— Di + — UI 3 '] , 
, (Ml + (Mo , 
'Ittl 
(M, 
(Mo 
(Mi(Mo 
Avhere m has its usual meaning, and m 
/ is obtained from just as is from m. 
Hence — _p _p _ ^y,i^ _ nig')]. 
(Ml (Mo ^l(Mo 
* JoitUAX, ‘ Cours d’Aiialyse,’ p. 351; Fousytu, ‘Theory (A Functions,’ § 13'J. Again notice that 
each of the quantitie.s >/ and oj is double that usually taken. 
