A consequence of this is that a Laurent series may be used in cases where a Taylor expansion is not possible. 2 Calculating the Laurent series expansion To calculate the Laurent series we use the standard and modi ed geometric series which are 1 1 z = 8 >> >> > > >> >: X1 n=0 zn; jzj 1: (1) Here f(z) = 1 1 z is analytic everywhere apart from the singularity at, Compute answers using Wolfram’s breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history …
The Laurent series expansion is defined on a deleted neighborhood around a singularity, in this case, ${z: 0
If we choose to make the Laurent expansion about z0= 0, then r> 0 and R< 1. These limitations arise because f(z) diverges both at z= 0 and z= 1. A partial fraction expansion, followed by the binomial expansion of (1 ? z)?1, yields the Laurent series (11.49)1z(z?1)=?11?z?1z=?1z?1?z?z2?z3??=??n=?1?zn.can also try to nd a Laurent series expansion on other annuli. For instance the function is holomorphic on the annulus A 1;2(0) = 1 1 and so applying the geometric series expansion above to w= 1=z , we see that 1 z 1 = 1 z 1 1 1=z = 1 z ...Taylor Series, Cauchy–Riemann equations, Essential Singularity, Line Integral, Complex Analysis
