Question

6. (16 points) Sections 6.1-6.3 For each of the singularities of the following functions: i. find the residues ii. find the principal parts iii. classify the singularities as one of the types: removable, pole or essential sin 22 a. cosh b, C. 2 + 2 2² - 32 d. ze

Answer 1

any z n where tez) an 20 2-agonista, principal part contains finite number of terms We knon that if f is analytic in the annulus R, < 12-20 <R2 (where Rezo), then for in the annulus, f has a unique representation fers = zan(2- - 12-al- with Rer <R2 The toefficient प्प 1 in the Z-a Laurent Series f about is called residue f at iR a- Res [+(?); 9] In the expansion of series negative poners (7-9) 'os Balled principal part. that f removable at za singularity iff KT 9- K0, + pole of opder an to balt 1 with a kmon in the expansion of at z=a i n f 9-=0, KS, n+1. ir K7. f has essential singularity an iff at Za 9-k 0 for in finitely many ire prin apoy past contain in finite number of terms. [iff means if and only it]

Ict f(2)= Sin z 2 24 fez is analytic except at 2-0 at all points Then C that We kona S Sin (7) = z-z3 + 5! C Sin (22) + LO 2 51 31 Sin (22) 1 22 + 26 24 222 5! 0 <171 <3 the Non wefficient of 2 in the 2-0 of 2 expansion is sin z 24 of The residue Sin 72 at Z = 'yo. 모시 The principal part Since the 2z2 Sin 22 24 principal part contains termo so Sixz2 24 750 and the pole finitely many pole at is of orders 2. let FE) = loshZ 2 2 Here fz) is analytic at all C except at 240. points of on

knon that losh z = 1+ 22 76 + + 2 of / in - 22 4! + 24 22 + 6! the coefficient > Losuz Z2 22 is non the expansion of (Light 0 22 at Losht 22 Zo is O residue losnz 222 22 22 ii) The posincipal part of the posincipant past one (finite) term Sme losha 22 so contains at 20 a tosha pole feq) = Let (0) The singularity z 2 z Z+2 Z 2_3Z of fez) are given by * Z(7-3) = 0 2-0,3 2+2 Z+ 2 Non Z(7-3) z (-3) (1 - 2 / 2 (2+2) (--)? -32 Z+2 -37 Item +)+3) --} BU 12.4-237=+24 24+22 1 21 21

༩ | + 1 + ༄) ནི ༡ ཧ ན ། 7 ། <3 } (༣), ༽, A[༢ རྒྱ བ ནྟི ན ( ) ་ ཏེ བ ནི དྷ ༽ ， , ༈ ༔ - ༢- + རྣ༧ པ་）, ད༢། <3 , , ༢ + - ད23 }, ད?། <3 동 coefficient is The s ༽ the expansion ༈ +2 ༢-༡༢ pesidue of 2+2 ཧེ ༢༡ at z=ong 2 름 The principon pont of +2 <) - Since 2 32 ༢༢ ༢༢ principone post of 2731 (finite) terms 2+2 contains L so 7+༡ pole at ཤེགྱ. 2- ༢༢ Again f?） ༩+2 《-3） ཥ-+ 5 (2-3)（z- ཏུ 3) 23+ (2-3）ཉ(1+ ' )

(2-2005:{++33} (+45) 51 to 2 + (9037?. 125,31 << [ (+ + 2)2 = رہا۔ at-2+2?. 1ZICI →[25-32 1 + :34 5 z-3 + (2-3)+--} 12-31<3 coefficient of 1 in the Z-3 expansion of 2+2 Z(2-3) 3 who : The at z=33 residue of 7+2 7 (2-3) 옿 principal part of The Z+2 z (2-3) 5 3(2-3) since the principal part of nas eftermo) 1 terms 7+2 Z CZ-3) 50 Z+2 7(7-3) a pole at 2-3,

Let ford = zet Then fa) is analytic at all points of except at 720. we knon that 2 1 + 2 + z ༢། + ? t. 31 = 1 + + + Ha (29) 2 Zt a + Ze 2.1 31 23 0<2112 i (31) 22 0<IZIKA + 1 of is The of coefficient is 1 Z the expansion ze residue of Zet at zzo B B The principal part of ze ² is + (2) z (1173 Since the tous azt tal of ze h ze singularity at 750. pooincipal part terms many has infinitely has essential So