Advanced Calculus fi..

-+---+Advanced Calculus, Fifth EditionCase II. A finite number of negative powers of z - zo appeal: Thus we haveHere f (z) is said to have a pole of order N at zo.'we can write1f (2) = g(z), g(z) = a-N + a-N+~(z - ZO) + ..., (8.64)(Z- zoINso that g(z) is analytic for lz - zol < R2 and g(zo) # 0. Conversely, every functionf (z) representable in the form (8.64) has a pole of order N at zo. Poles are illustratedby rational functions of z, such as. twhich has poles of order 1 at f i and of order 3 at z = 1.The rational functionis called the principal part of f (z) at the pole zo. Thus f (z) - p(z) is analytic at zo.EXAMPLE 1ez cos zf (2) = - at z = 0.z3To obtain the Laurent series, we expand the numerator in a Taylor series:Hence--eZcosz 1 1 1A ....z z3 z2 3Here the first two terms form the principal part; the pole is of order 3.EXAMPLE 2We expand z/(z3 + 2) in a Taylor series about z = - 1 :z --- 1 4+-- IS+... .(Z + 1)2(z3 + 2) (Z + + 1The first two terms form the principal part; the pole is of order 2.