Explain briefly the relevance of this result to the

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Explain briefly the relevance of this result to the solution of Laplace’s equation via conformal mapping. Q 10.9. (Q7(b), Paper I, 1994) Find a conformal mapping f that sends the unit disc D = { z : | z | < 1 } onto the strip { z : π/ 2 < ( z ) < π/ 2 } for which f (0) = 0 and f (0) is real and positive. [Cambridge exams are often a conspiracy between examiner and examinee. If you choose the ‘obvious’ conformal maps then you will either get the answer at once or obtain one which is easily converted into the required one. If you get a f that sends the unit disc onto the strip { z : π/ 2 < ( z ) <π/ 2 } but which you can not bring to the right form do not worry unduly but do not go on to the rest of the question.] Find an analytic function h on D with the property that | h ( re ) | → e π/ 2 as r ր 1 for 0 <θ<π and | h ( re ) | → e π/ 2 as r ր 1 for π<θ< 0. Q 10.10. (Q7(a), Paper II, 1993) Suppose f has a pole of order k at z = 0. Show that the residue of f at 0 is 1 ( k 1)! d k 1 dz k 1 ( z k f ( z )) vextendsingle vextendsingle vextendsingle vextendsingle z =0 . Let r and s be analytic functions such that r (0) negationslash = 0 and s (0) = s (0) = 0, s ′′ (0) negationslash = 0. Show that the residue of r ( z ) /s ( z ) at z = 0 is 6 r (0) s ′′ (0) 2 r (0) s ′′′ (0) 3( s ′′ (0)) 2 . [My reason for including this is not to provide a formula for you to learn but to show that, once we move from simple poles, we can not expect simple ‘one size fits all’ methods for finding residues.] 26
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Q 10.11. (Q7, Paper II, 2000) Evaluate the integrals contintegraltext C f ( z ) dz , where C is the unit circle centred at the origin and f ( z ) is given by the following (a) sin z z , (b) sin z z 2 , (c) cosh z 1 z 3 , (d) 1 z 2 sin z , (e) 1 cos 2 z , (f) e 1 /z . Q 10.12. Use a result about integraltext −∞ e iλx / (1 + x 2 ) dx already obtained in the course to show that integraldisplay 0 cos mx 1 + x 2 dx = π 2 e m for m> 0. Hence evaluate integraldisplay 0 sin 2 x 1 + x 2 dx. Q 10.13. (Q8(b), Paper IV, 1994) Consider the integral I ( a ) = integraldisplay 2 π 0 (1 + a cos θ ) 2 where 0 < | a | < 1. By means of the substitution z = e , express I ( a ) as an integral around the contour | z | = 1 and hence show that I ( a ) = 2 π (1 a 2 ) 3 / 2 . [The examiner added that no credit would be given for answers obtained by real methods.] Q 10.14. (Q16, Paper II, 1997) By integrating a branch of (log z ) / (1 + z 4 ) about a suitable contour, show that integraldisplay 0 log x 1 + x 4 dx = π 2 8 2 , and evaluate integraldisplay 0 1 1 + x 4 dx. Q 10.15. (A golden oldie, last set as Q16, Paper I, 1998) Let I ( α ) = integraldisplay 0 x α ( x + 1) 3 dx, 27
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where α is real. Use real methods to find the range of α for which the integral converges. Use real methods to evaluate I (0) and I (1). Now consider the integral of z α / ( z + 1) 3 around a contour consisting of two circles of radius R and ǫ and straight lines on both sides of a cut along the positive real axis. What restrictions must be placed on α for the contributions from the circles to become negligible as r → ∞ and ǫ 0?
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