10 MASAMICHI TAKESAKI OUTLINE OF THE PROOF. Set f(a) = -11(00, ctec, for a fixed s GR. It follows that / has only simple poles at in, « 6 Z . Then integrate / along the following closed curve: i / 2 -R 3172- Letting R —• 4- oo ^nd applying the residue theorem, one concludes that p -is(t-i/2) ß -to(f + f/2) -i/(f + z/2) Ë 72) --ir(f + //2) y ' J Rewriting the above, one gets formula (23). Q.E.D. The above lemma, together with the spectral decomposition theorem, yields the follow- ing at once. LEMMA 1.11. For each sGR , r „—ist (24) es'2A1l2(A + e*)-1 = f" AH dt J-°°et + e -t Another easy consequence of Lemma 1.10 is the following LEMMA 1.12. Ifx, y G /.(Ö) satisfy the equation (25) (xitfj = Ï Ä - 1 / 2 ^ 1 / 2 ^ ) + e ^ A ' / ^ J A - ! / 2 ^ ) for every i t , i 2 e ß^Ä 1 ' 2 ) Ð ßÊÁ'1'2), then (26) e S/2y -S: Ä**÷Ä-**Á, s GR. With the spectral decomposition of Ä, Ä = ]"-_ ë^(ë), one may take L(L(Er ip)) as Ë in Lemma 1.10, where ^ r = E(r) - E(l/r), r 1, and may consider óá(÷) = Ä'áËÄ- , ÷ G L(Er$), á G C,
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