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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
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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Using the example, can you show the Laurent series for (2+3z)/(z^2+z^z^4), 0<abs(z)<1

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.
Transcribed Image Text: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<IZI<2} ? Sol: First, we write f a sum of 2 functions using a fraction expansion: 4 (2-1)(2-2) = 2²-2-2²-1 = Στη ทะเ ® - - partial A 2(₁-²) 2 (1 - 1) = - = ² ( ² )" = 811-를) the laurent series n=o Note: The first sum Σ (²)n exists only if | // K1E) 1219. The second sum 2 z-n exists only 1 =) < 1 => \ Z) > 1 A=D ทะเ -2i Thus, is valid only on the annular region 1</2/<2. Ex 15.11: Obtain the series expansion for f(z) = 7²214 valid in the region 1Z-zil >4. 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
Transcribed Image Text:Ex 15.10: What is the laurent series of f(z) = (Z-1) (7-2) on the annulus A=√ZEα: 1<IZI<2} ? Sol: First, we write f a sum of 2 functions using a fraction expansion: 4 (2-1)(2-2) = 2²-2-2²-1 = Στη ทะเ ® - - partial A 2(₁-²) 2 (1 - 1) = - = ² ( ² )" = 811-를) the laurent series n=o Note: The first sum Σ (²)n exists only if | // K1E) 1219. The second sum 2 z-n exists only 1 =) < 1 => \ Z) > 1 A=D ทะเ -2i Thus, is valid only on the annular region 1</2/<2. Ex 15.11: Obtain the series expansion for f(z) = 7²214 valid in the region 1Z-zil >4. 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
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