EE_401_Hwk_1_090507

EE_401_Hwk_1_090507 - (i) y = 0; x : !" , + " ( )...

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Continues… 1 EE 401 Homework # 1 Due: Wed., 9/12/07 Jenkins 1. Find all values of z for which (a) sin z = 0 (b) cos z = 0 (c) cosh z = 0 (d) Log z = 0 2. Show whether or not the following functions are analytic: (a) f z ( ) = sin z (b) f z ( ) = e z (c) f z ( ) = xy + jy 3. For the function w = f z ( ) = z 1 2 (a) Is the function f multi-valued or single-valued? (b) If f is multi-valued, how many values does f have for a given z ? Give all values of f z 0 ( ) , for z 0 = r 0 exp j ! 0 ( ) . (c) Give two possible choices of branch cuts and the associated branches. (d) Give the branch point(s). 4. (a) For the function w = f z ( ) = cos z , and for each contour in the z plane described below, give the resulting contour in the w plane. (Give a sketch and a description.)

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Unformatted text preview: (i) y = 0; x : !" , + " ( ) , traversed from !" to + " . (ii) y = 1; x : !" , + " ( ) , traversed from !" to + " . (iii) y = ! 1; x : !" , + " ( ) , traversed from !" to + " . (b) For the domain of the finite z plane (entire z plane except z = ! ), what is the range of f ? 5. Show that d dz z 3 ( ) exists and find it. 2 6. Show that d dz z n ( ) = nz n ! 1 for n = positive integer. 7. ( Extra credit ) Prove that if f z ( ) = ! + j " is analytic ( and are real), then and are harmonic; in other words, show that and each satisfy Laplaces equation: ! 2 ! x 2 + ! 2 ! y 2 = ! 2 # ! x 2 + ! 2 ! y 2 = 0....
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This note was uploaded on 02/25/2009 for the course EE 401 taught by Professor Staff during the Fall '08 term at USC.

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EE_401_Hwk_1_090507 - (i) y = 0; x : !" , + " ( )...

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