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EXERCISE: 4) Show that w = // maps the interior of the unit circle in the first quadrant to the exterior of the unit circle in the fourth quadrant: 4. 1

EXERCISE: 4) Show that w = // maps the interior of the unit circle in the first quadrant to the exterior of the unit circle in the fourth quadrant: 4. 1

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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Following the method, I need help with exercise part 1

EXERCISE: 1) Show that w = 1/2 maps the interior of the unit circle in the first quadrant to the exterior of the unit circle in the fourth quadrant: ¼, ħ 2) What is the image of the unit circle under / / ? 글 3) Write down how z → = = w maps the punctured plane alf of to itself land @ U{00} to itself) 11/12 Back to Ex 4.5. Let us study two images under z a) Find the image of the line z = t +i (1-t) under 1/2. Sol. Denote. W = U+iv, Z = x+iy => x=+₂ y=(1-t) 1
Transcribed Image Text:EXERCISE: 1) Show that w = 1/2 maps the interior of the unit circle in the first quadrant to the exterior of the unit circle in the fourth quadrant: ¼, ħ 2) What is the image of the unit circle under / / ? 글 3) Write down how z → = = w maps the punctured plane alf of to itself land @ U{00} to itself) 11/12 Back to Ex 4.5. Let us study two images under z a) Find the image of the line z = t +i (1-t) under 1/2. Sol. Denote. W = U+iv, Z = x+iy => x=+₂ y=(1-t) 1
1 w=U+iv= x+iy = u ft = µ² + √² 1₁-t = = = = = √²₂² A 1 (for Z+0₂ W‡0) <> xtiy= utiv u²+V² (1-t) (u( ² + √² ) = - V +iV Thus, plugging x=t₁ y = 1 - t, we obtain the following system: [ ²14²²+√² ) = u u²+V² [+(u²+√² ) = 4 <=> { ₁²+v² - t/u²+ √²³ ) = - V V²+V=0 <=> x²+√² = U-V (we add the equations) => u²-4+₁ v ² + V = 0 [u_ \ )² + (x + £ ) ² = = (complete the square in u and in v) Namely, the тар z W = =/ maps the line 7 = ++i(1-t) to the circle (in w-plane) centered at the point ( 2 ₁ - 12/1) with radius #: 1 )
Transcribed Image Text:1 w=U+iv= x+iy = u ft = µ² + √² 1₁-t = = = = = √²₂² A 1 (for Z+0₂ W‡0) <> xtiy= utiv u²+V² (1-t) (u( ² + √² ) = - V +iV Thus, plugging x=t₁ y = 1 - t, we obtain the following system: [ ²14²²+√² ) = u u²+V² [+(u²+√² ) = 4 <=> { ₁²+v² - t/u²+ √²³ ) = - V V²+V=0 <=> x²+√² = U-V (we add the equations) => u²-4+₁ v ² + V = 0 [u_ \ )² + (x + £ ) ² = = (complete the square in u and in v) Namely, the тар z W = =/ maps the line 7 = ++i(1-t) to the circle (in w-plane) centered at the point ( 2 ₁ - 12/1) with radius #: 1 )
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