Question

5. (a) Show that the following improper real integral is absolutely convergent cos 2x dr I (1+?}% " (b) If CR is the semicircle of radius R in the upper half plane with centre at z = 0, show carefully that e2iz lim R00JCR (1+ z2)2 d% = 0 (c) Use residue calculus to evaluate the real integral I of part (a)

Answer 1

5. (a) CosC2dx O cos(2x) OD ban() d u -- C abnotutey conrergent Semicird or raouiu R haut in tAe b) CR at Z=O p Dun plan of with cuntre CR 2 R 11 R dz CR +222 ningSe ML +(2) d 2 CR arc leng th zl R ) 22 L222 (R2-02 L *R.

2i2 dl 2 (1+222 CR (R2-2 im eiz R Ra co R-co R2-172 Scziz CR C12 cl 2 R Co um R co e2iz t 22 CD 2i2 c) dz dz こ 1+2) -co cl2 CR 2i2 e d2. d2 22o) z-2i. i in C. pole ordar 2ニ1 Cet R e-,0) dz= 2xi d (z-i)2e2i2) T+22)2 C 2i2 Tt22)2 2/2 2=1 2iz 22)2 2i2 2i C dz d2 zti)/ 2 e2i2

2i 2 2ie-2 T2ij2 2 e-2 d 2 C2i) 27)3 3i 4-e2 2 T 3ス e d 2 2e2 C 4e2 e2i2 dz (I+22)2 3A 2C2 cos (22) ism (22) 2e2 Cit22)2 CD cos C22 d2 3R 202 CIt 22)2 sin(22) d2ニ0 C1+222 Cos(22) d2 3R 2e2 (1+22)2 3ズ cos(22 d 2 (122) 'cos(2) dx 2)2 2. 3ズ 4 e2