# Explain briefly the relevance of this result to the

• Notes
• 100000464160110_ch
• 35

This preview shows pages 26–29. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

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
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?

This preview has intentionally blurred sections. Sign up to view the full version.

This is the end of the preview. Sign up to access the rest of the document.
• Fall '08
• Groah
• Math, Analytic function, Q7, Cauchy, Lemma

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern