set_7_problem

# set_7_problem - z/a ) maps the semi-innite strip of width a...

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Problem Set # 7 MMAE: 501 Engineering Analysis I Kevin W. Cassel Mechanical, Materials and Aerospace Engineering Department Illinois Institute of Technology 10 West 32nd Street Chicago, IL 60616 cassel@iit.edu Problem Reference Problem # 1: Jeffrey, Section 17.1, Problem 9 Problem # 2: Jeffrey, Section 17.1, Problem 11 Problem # 3: Jeffrey, Section 17.1, Problem 16 Problem # 4: Problem # 5: c ± 2007 Kevin W. Cassel 1

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Problem # 1 Given that w = 1 /z , ﬁnd the image in the w -plane of the family of parallel straight lines y = x + c in the z -plane. Problem # 2 Find the linear fractional transformation that maps the circle deﬁned by the three points z 1 = i , z 2 = - i , and z 3 = 1 in the z -plane onto the three points w 1 = - 1, w 2 = 1, and w 3 = in the w -plane. Determine the region in the w -plane that corresponds to the region to the left of the given points in the z -plane when the points are traversed in the order z 1 ,z 2 to z 3 . Problem # 3 Prove that the function w = sin(
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Unformatted text preview: z/a ) maps the semi-innite strip of width a in the z-plane shown in the diagram on the left of the gure onto the upper half of the w-plane in the manner shown in the diagram on the right. Determine the images in the w-plane of the lines x = c and y = k . Problem # 4 Find a function ( x,y ) that is harmonic, i.e. a solution of Laplaces equation, in the unit disk | z | &lt; 1 and takes on the boundary values ( x,y ) = 0 , for z = e i , &lt; &lt; , ( x,y ) = 1 , for z = e i , &lt; &lt; 2 . 2 Use the mapping w = f ( x ) = i 1-z 1 + z to map the interior of the disk to the upper half plane Im ( w ) &gt; 0. Problem # 5 Find the solution ( x,y ) of the Laplace equation in the upper half-plane Im ( z ) &gt; that has the boundary values ( x, 0) = x, for-1 &lt; x &lt; 1 , and ( x, 0) = 0 , for | x | &gt; 1 . 3...
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## This note was uploaded on 09/24/2010 for the course MMAE 501 taught by Professor Kevincassel during the Spring '10 term at Illinois Tech.

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set_7_problem - z/a ) maps the semi-innite strip of width a...

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