212HomeWork2sol - Physics 212A Homework #2 Name: 1....

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Physics 212A Homework #2 Name: ± 1. a)Starting from the Euler relation, derive the expression cot - 1 z = ² 2 Log B z z F ± Euler relation is E^(I z) = Cos[z] + I Sin[z] cotz = Cos @ z D ± Sin @ z D ±± TrigToExp - ± I ª z + ª ± z M ª z - ª ± z Solve @ y ± cotz, z D Solve::ifun : Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. ± :: z fi -± Log B - ± + y -± + y F> , : z fi -± Log B ± + y -± + y F>> TrigToExp @ ArcCot @ z DD 1 2 ± Log B 1 - ± z F - 1 2 ± Log B 1 + ± z F Simplify @ % D ±± PowerExpand 1 2 ± H Log @ -± + z D - Log @ ± + z DL ± b) Find the series expansion about z = 0 and an expansion valid for large / z / Series B Log B ² + y -² + y F , 8 y, 0, 8 <F Π 2 - y + y 3 3 - y 5 5 + y 7 7 + O @ y D 9
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s1 = Series B Log B ± + y -± + y F , 8 y, ¥ , 8 <F 1 y - 1 3 y 3 + 1 5 y 5 - 1 7 y 7 + O B 1 y F 17 ± 2 ± c) use this result to sum the series 1- 1 3 + 1 5 + .. and 1 - 1 3 ² 3 + 1 5 ² 3 ² 3 - 1 7 ² 3 ² 3 ² 3 +... . Confirm your results using Sum. ± Evaluating the series s1 at y -> 1 generates the sum 1- 1 3 + 1 5 + .. , s1=ArcCot[y], so the sum is ArcCot[1] ArcCot @ 1 D Π 4 ± The formula for the nth term is Table @H - 1 L ^ HH n LL ± HH 2 n + 1 LL , 8 n, 0, 5 <D : 1, - 1 3 , 1 5 , - 1 7 , 1 9 , - 1 11 > ± Using Sum, we get Sum @H - 1 L ^ HH n LL ± HH 2 n + 1 LL , 8 n, 0, ¥ <D Π 4 ± the same as using the ArcCot series. ± Next, we note that the series 1 - 1 3 ² 3 + 1 5 ² 3 ² 3 - 1 7 ² 3 ² 3 ² 3 is 3 H s1/.y-> 3 ) , so the sum should be Sqrt @ 3 D ArcCot @ Sqrt @ 3 DD Π 2 3 ± Verifying with sum, the nth term is Table @H - 1 L ^ HH n LL ± H 3^ H n L H 2 n + 1 LL , 8 n, 0, 6 <D : 1, - 1 9 , 1 45 , - 1 189 , 1 729 , - 1 2673 , 1 9477 > ± so the sum is Sum @H - 1 L ^ HH n LL ± H 3^ H n L H 2 n + 1 LL , 8 n, 0, ¥ <D 2 212HomeWork2sol.nb
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Π 2 3 ± 2. Use Manipulate to draw the image of a unit circle centered at zc=xc+I yc under the mapping Sin[z] . xc and yc are control parameters, so you will adjust a slider to specify the location of the circle in the z plane. Use ComplexExpand and Parametric plot ( a convenient way to parametrically describe the circle is z= xc +I yc + ª ² Θ ) . To prevent the plot from autscaling as you adjust the parameters, it is a good idea to use PlotRange to fix the range of the plot. Manipulate @ u = ComplexExpand @ Re @ Sin @ xc + I yc + E^ H I Θ LDDD ; v = ComplexExpand @ Im @ Sin @ xc + I yc + E^ H I Θ LDDD ; ParametricPlot @8 u, v < , 8 Θ , 0, 2 Π < , PlotRange 88 - 3, 3 < , 8 - 3, 3 <<D , 88 xc, 0 < , - 4, 4 < , 88 yc, 0 < , - 4, 4 <D ± 3. Show that f[z] = Sin[z] and f[z] = ª z are differentiable, but f[z] =Conjugate[z] is not. ± Extract the real and imaginary parts: 8 rp, ip < = ComplexExpand @8 Re @ Sin @ x + I y DD , Im @ Sin @ x + I y DD<D 8 Cosh @ y D Sin @ x D , Cos @ x D Sinh @ y D< ± The Cauchy-Riemann conditions are D @ ip, y D == D @ rp, x D True - D @ rp, y D == D @ ip, x D True ± so the CR conditions are satisfied for S[z] The derivatives D @ rp, x D Cos @ x D Cosh @ y D D @ rp, y D Sin @ x D Sinh @ y D D @ ip, x D - Sin @ x D Sinh @ y D 212HomeWork2sol.nb 3
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212HomeWork2sol - Physics 212A Homework #2 Name: 1....

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