18 



APPENDIX 1 

 A. USEFUL INTEGRALS 



Certain useful results which can be obtained in a straightforward manner are stated 



below: 



Let 



/ {x\s,X) = f 



Vxd: 



' s — x {x — y) 



and let R denote "the real part of". Then 

 ; (x', s, X) = 2 sin"' |/^ 



(for X ' < o) I {x',s,X) = 2 sin"' y f 

 Y x' — s \ 



[29] 



) ( i/s-x i^s-x' +s - vY ^) 



[30] 



s + K^/I^r^-^tan-'/M) 



/(x;s,s) = 7r+i?[(/fj^(i7r)] = 



TT for s > X' > 



TT fl- l/-#^|/0'* ^' 



\ ¥ X'—S I 



> s 

 or x' < 



[31] 



/o )^(a;'-a:) 



dx = / (s-x', s,s) = TT (/or s > a;'< 0) [32] 



* Vs — a; dx 



IV 



(x'-x){x-t) (x 



73-T-[/(s-a;',s,s) -/(s-f,s,s)| 



[33] 



(t < \ ■jx I T - 



forQ<^x<s)-^^r^^y— 



f- s 



rs Vx_^ =^-\l(r,s,s)-I(t,s,s)] 



Jo Vs-x(t-x)(x-r) (t-T)^ ■■ 



/ . T<0\ '" \ \/ t _ l/Z^ 



= (f<>'- (<o)(7Z7)Lvt-« yr-. 



[34] 



