Explanation Concept : Draw the given region defined as x > 0 , y > 0 and x y < c . The equation x y = c represents rectangular hyperbola. Since x > 0 , y > 0 , it is first quadrant in x y -plane. Therefore, the given region is bounded by the rectangular hyperbola x y = c in the first quadrant. Since x , y are positive, so 0 < x y < c . Hence, c is a positive constant. The boundary of the region is non-negative x -axis, non-negative y -axis and the curve x y = c . Consider the analytic function f z = 1 2 c z 2 , where z = x + i y . Since f is a polynomial, it is an analytic function. Find the transformation: f z = 1 2 c x + i y 2 = 1 2 c x 2 + i 2 x y + i y 2 = 1 2 c x 2 − y 2 + i 1 c x y Hence, u = 1 2 c x 2 − y 2 and v = 1 c x y . The non-negative x -axis can be represented as x > 0 and y = 0 . Find the transformation of non-negative x -axis: f z = 1 2 c x + i 0 2 = 1 2 c x 2 − 0 2 + i 0 Hence, the image of non-negative x -axis is u = 1 2 c x 2 − 0 ≥ 0 and v = 0 . The image of positive non-negative u -axis in u v -plane. The non-negative y -axis can be represented as x = 0 and y > 0 . Find the transformation of non-negative y -axis: f z = 1 2 c 0 + i y 2 = 1 2 c 0 2 − y 2 + i 0 Hence, the image of non-negative y -axis is u = 1 2 c 0 2 − y 2 ≤ 0 and v = 0

10th Edition

ISBN: 9781119455929

Author: Kreyszig

Publisher: WILEY

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Chapter 17, Problem 39RQ

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To Find: an analytic map w=fz that maps the region x>0,y>0 and xy<c onto the strip 0<v<1 .

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Chapter 17, Problem 38RQ

Chapter 17, Problem 40RQ

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Chapter 17 Solutions

ADVANCED ENGINEERING MATHEMATICS (LL)

Ch. 17.1 - Prob. 1P Ch. 17.1 - Prob. 2P Ch. 17.1 - Prob. 3P Ch. 17.1 - Prob. 4P Ch. 17.1 - Prob. 5P Ch. 17.1 - Prob. 6P Ch. 17.1 - Prob. 7P Ch. 17.1 - Prob. 8P Ch. 17.1 - Prob. 9P Ch. 17.1 - Prob. 11P

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an analytic map w = f z that maps the region x > 0 , y > 0 and x y < c onto the strip 0 < v < 1 .

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To Find: an analytic map w=fz that maps the region x>0,y>0 and xy<c onto the strip 0<v<1 .

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To Find: an analytic map w=fz that maps the region x>0,y>0 and xy<c onto the strip 0<v<1 .

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Chapter 17 Solutions

an analytic map w = f z that maps the region x > 0 , y > 0 and x y < c onto the strip 0 < v < 1 .