Transcription of SOLUTIONS TO HOMEWORK ASSIGNMENT # 7
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SOLUTIONS TO HOMEWORK ASSIGNMENT # 7
SOLUTIONS TO HOMEWORK ASSIGNMENT # 71. Determine the nature of all singularities of the following functionsf(z).(a)f(z) = cos 1/z.(b)f(z) =1z2sinz.(c)f(z) =zez2 :(a)z= 0 is the only singularity. It is an essential singularity since the Laurent seriesexpansion aboutz= 0,cos 1/z= 1 12!z2+14!z4+ ,has infinitely many negative powers ofz.(b) The singularities arez= 0 andz=n , n= 1, 2, ..The singularity atz= 0is a pole of order 3 sincez= 0 is a zero of order 3 follows easily fromthe Maclaurin series aboutz= 0 :z2sinz=z3 13!z5+15!z7+ = n=0( 1)n1(2n+ 1)!z2n+ singularitiesz=n , n= 1, 2, .. ,are simple poles since they are simple zerosofz2sinz.(c)z= 0 is a simple pole sincezez2 1=zz2+z4/2! +z6/3! + =1z+z3/2! +z5/3! + =1zg(z)whereg(z) is analytic atz= 0 andg(0)6= factg(0) = 1,although what simportant is just thatg(0)6= other singularities are the non-zero SOLUTIONS ofez2= 1,that isz= 2n ,wherenis a non-zero integer.
Another way to see this is that 1 zsinz = 1 z2 g(z) where g(z) = z sinz Now we could expand g(z) = z/sinz as a Taylor series about z = 0. But since g(z) is an
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