Question

please solve this.thanks. Find all nulls and singularites of functins. Determine the types of singularity, calculate pole resiudums.

TASY.5 FUNCTIONS ON 010 SING OLA2ES FIND ALL AUL De TER NE 9INGULA2TY AND TYPES OF CALCULA T POLE EESTDUALS AoDITIONALLY iND LAURENT'S ARDUNO OeoER CACH- (+AしF, DE TERMING CONUEREENCE THC RAOUS OF 22 2-2 2-22 te) e) f 22 72+1 e R) fre

Answer 1

= CF (1 2) + 오e isa lated ingularity f(a) Las essertial Angularity at 2=0 -2 is e simp le pole st fre) (f.-2) = -2.

2 2 f(2)= 32 +으 (9-2) /2-19 2 (1-3) ( 2 P (t-2 (1-) - J +(1e 2)(2+1) 1 (4.2+ 1 48) 1 4 23 f(a)is mt analytic analytic for all other values of 2 2= 2 and 2=1 but at iss Lod ed inguloity 2= 2 and ?= ate In LowTent series expansion,+heye ane intinitely many negative pwers of 2. So. f(2) has an essenHal at Lim f(9)l im f(2)= 272 and prles if fra) 4 2 and / are Henre 2 (f.2) es 2- (1) 22 1

2 2 9 f(2)= 2-2 (a-2)(2t1) are the iselared ngulonitis 2-2 and . 2=-1 22 f(2)= (1- + 22 22 4 22 2 3 =I +(12)+(1-2+1) + + 3 + + 1 at 2=0 Las essentinl ngulani ty 2) Pole isoloted be an Let 2=a ingulasity if we say a s pale of Liom 27a f (2) pades of f(9) 2 and-I are (4,2) Yes 2+1 2 2 2 2-2 2-

1 (e)t 11 2- 2 2 2 20 2 1 1 2 + 2 + 27 ,2 + 4+6) -j2 roquloxite isolated 2= and 2- iY2 are ingulau'itres 2-0is vemevobfe 26 2 =ir and aie tr2) 1 res(1,ita)= 2 +1 Y2 + 72-142) (224 3) 2i2 (, 13) Tes 2i3 (S1-F (,i1) es
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