We require the expansion for IW/>4: f(z) = Σ 1 ทะ 4- (w)nt ทะz 247-2 (iw) n Now we substitute back 2-21=w: f(z) = - 0047-2 1-2 (i (z-zi))^ ) 17-2; /> 4. 17=2 Now we are going to study one of the important transformations, the simplest after the linear and the inverse transformations, the Möbius transformation. First, let us briefly go over linear transformations. Ex 15.10: What is the laurent series of f(z) = (Z-1) (7-2) on the annulus A=√ZEα: 1 \ Z) > 1 A=D ทะเ -2i Thus, is valid only on the annular region 14. H The region here is a domain: the exterior of the circle centered at (0, 2i) of radius 4. We want a series expansion about to = di To do this we make a substitution Set W= Z-2i and look for the expansion in W, where | W | > 4 To make the series expension easier we manipulate again f(z) into a form similar to the series expansion for A-Z. We get: Now we use standard geometric series and 1 & (ix) n 4iw 4 (Win 1 *(Z) = 4iw (1-iw) 4 compute! 1 4iw (1-iw) - 4iw 1w/4 IWI>4

We require the expansion for IW/>4: f(z) = Σ 1 ทะ 4- (w)nt ทะz 247-2 (iw) n Now we substitute back 2-21=w: f(z) = - 0047-2 1-2 (i (z-zi))^ ) 17-2; /> 4. 17=2 Now we are going to study one of the important transformations, the simplest after the linear and the inverse transformations, the Möbius transformation. First, let us briefly go over linear transformations. Ex 15.10: What is the laurent series of f(z) = (Z-1) (7-2) on the annulus A=√ZEα: 1 \ Z) > 1 A=D ทะเ -2i Thus, is valid only on the annular region 14. H The region here is a domain: the exterior of the circle centered at (0, 2i) of radius 4. We want a series expansion about to = di To do this we make a substitution Set W= Z-2i and look for the expansion in W, where | W | > 4 To make the series expension easier we manipulate again f(z) into a form similar to the series expansion for A-Z. We get: Now we use standard geometric series and 1 & (ix) n 4iw 4 (Win 1 *(Z) = 4iw (1-iw) 4 compute! 1 4iw (1-iw) - 4iw 1w/4 IWI>4

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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