Find analytic functions w 1 z and w 2 z that are real

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Find analytic functions w 1 ( z ) and w 2 ( z ) that are real for real z and for which the following functions are their respective real parts: (i) u 1 ( x,y ) = e x cos y , (ii) u 2 ( x,y ) = x ( x 2 + y 2 + 1) 2( x 2 + y 2 ) . From your answer to (ii) find a non-zero harmonic function that vanishes on the circle x 2 + y 2 = 1 and on the line y = 0. In case (i), find the images in the z -plane of the circles | w 1 | = ρ , for constant ρ . In case (ii), find the images in the w 2 -plane of the circles | z | = r , for constant r> 1. Q 10.23. (Q7(b), Paper I, 1996) In four of the following five cases there exists a bijective analytic map f : U V . In one case there is a topological reason why no such map is possible. Find a suitable f in the four cases and briefly explain the fifth. (i) U = { z C : ( z ) > 0 } , V = { z C : ( z ) > 0 } (ii) U = { z C : | z | < 1 } , V = { z C : ( z ) > 0 , ( z ) > 0 } (iii) U = { z C : 2 > | z | > 1 } , V = { z C : | z | < 1 } (iv) U = C \ { z R : z 0 } , V = C \ { z R : | z | ≥ 1 } (v) U = { z C : 0 < ( z ) < 1 } , V = C \ { z R : z 0 } . [ In case (iv) you may find it useful to consider the effect of a translation followed by the map z mapsto→ 1 /z . ] Q 10.24. (Q7(a), Paper I, 1995) Find the residue at each of the poles of the function f ( z ) = 1 z 2 (1 + z 4 ) 30
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in the complex plane. Q 10.25. (Q7, Paper II, 1999) You are asked to find the Laurent expansion about z = 0 for each of the following three functions. f 1 ( z ) = e 1 /z , f 2 ( z ) = z 1 / 2 , f 3 ( z ) = sinh z z 3 . In one case, you reply that you can not supply such an expansion. Why? In the other two cases, where there is a Laurent expansion, state the nature of the singularity at z = 0 and find its residue. Show that there is a function f analytic on C except, possibly, at finitely many points such that f ( z ) = summationdisplay n =1 z n for | z | > 1. Find any singularities of f ( z ) in the region | z | ≤ 1 and find the residues at those singularities. Q 10.26. (Q7(a), Paper II, 1995) What are the poles and associated residues of f ( z ) = (cosh z ) 1 in the complex z -plane? By considering a rectangular contour, or otherwise, evaluate the Fourier transform integraldisplay −∞ e ikx cosh x dx. Q 10.27. Show that, for a>b> 0, we have I ( a,b ) = integraldisplay 0 cos x ( x 2 + a 2 )( x 2 + b 2 ) dx = π 2( a 2 b 2 ) parenleftbigg 1 be b 1 ae a parenrightbigg . Find I ( a,a ) for a> 0 and check that I ( a,b ) I ( a,a ) as b a . Find I ( a,b ) for all real non-zero values of a and b . Q 10.28. (Q7(b), Paper II, 1993) By integrating around an appropriate closed curve in the complex plane cut along one half of the real axis, show that I ( a ) = integraldisplay 0 x a 1 1 + x + x 2 dx = 2 π 3 cos parenleftbigg 2 πa + π 6 parenrightbigg csc( πa ) if 0 <a< 2 and a negationslash = 1. Evaluate I (1) and show that I ( a ) I (1) as a 1. 31
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Q 10.29. (Q8, Paper IV, 1999) By interpreting the angle θ as the argument of a complex variable z , convert the real integral I ( α ) = integraldisplay 2 π 0 (1 + α cos θ ) 1 ( | α | < 1) into a contour integral in the z plane and hence evaluate it using the calculus of residues.
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  • Fall '08
  • Groah
  • Math, Analytic function, Q7, Cauchy, Lemma

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