Iv z mapsto z 1 inversion in unit circle followed by

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(iv) z mapsto→ z 1 . Inversion in unit circle followed by reflection in real axis. Takes circles and straight lines ‘not through the origin’ to circles and circles and lines ‘through the origin’ to straight lines. Takes C \ { 0 } to C \ { 0 } . [Note, in this course we are not interested in the ‘point at infinity’.] (v) z mapsto→ z α with α real and α > 0. [N.B. You must specify a branch!] Takes (appropriate) sectors (of the form { re : r > 0 , θ 1 > θ > θ 2 } ) to sectors with base angle multiplied by α . (vi) z mapsto→ exp z . Takes (appropriate) planks { z : θ 1 > z>θ 2 } to sectors. The map z mapsto→ log z [N.B. You must specify a branch!] does the reverse. We observe that maps of the type (i) to (iv) are M¨obius and together generate the M¨obius group. M¨obius maps were extensively discussed in the first year course ‘Algebra and Geometry’. Observe also that we do not really need maps of type (v) explicitly, since we can obtain them using maps of the type (iii) and (vi). The author strongly recommends constructing conformal maps in a large number of simple steps, as the composition of the simple maps given above, rather than trying to do everything at once. Example 3.4. Find a conformal map taking Ω = { z : z> 0 , z> 0 , | z | < 1 } to the unit disc D = { z : | z | < 1 } . Explain why the map z mapsto→ z 4 does not work. It should be noted that conformal mapping problems like Example 3.4 do not have unique solutions since there are non-trivial conformal maps of the disc into itself (for example rotation). In the early days of aviation, conformal mappings (of a very slightly more complicated kind) were used to find the flow of air past the wings of aeroplanes. The method depended on the following result. 8
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Lemma 3.5. Let Ω and Γ be open subsets of C and f : Ω Γ a conformal map. Set ˜ Ω = { ( x,y ) : x + iy Ω } , ˜ Γ = { ( x,y ) : x + iy Γ } and let parenleftbigg u v parenrightbigg : ˜ Ω ˜ Γ be the mapping given by f ( x + iy ) = u ( x,y ) + iv ( x,y ) . Then, if φ : ˜ Γ R is harmonic, so is ψ : ˜ Ω R where ψ ( x,y ) = φ ( u ( x,y ) ,v ( x,y )) . It must be admitted that the use of Lemma 3.5 and the general practice of conformal mapping at 1B level and substantially above it depends on the fact that, for the kind of Ω and Γ considered, the conformal map f : Ω Γ does, indeed, behave well near the boundaries. The reader is warned that, should she ever attend an advanced pure course on conformal maps or try to use theorems which merely guarantee the existence of such a map f without actually giving an explicit construction, this assumption can no longer be relied on 1 . 4 Contour integration and Cauchy’s theorem It is natural to define the integral of a function F : R C by integraldisplay b a F ( t ) dt = integraldisplay b a F ( t ) dt + i integraldisplay b a F ( t ) dt. In the course C9 (Analysis) it is shown that this definition produces an integral with all the properties we want. In addition, the following useful lemma is proved.
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