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CM - Complex Methods Course P3 T W Krner o Small print The...

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Complex Methods Course P3 T. W. K¨orner September 18, 2007 Small print The syllabus for the course is defined by the Faculty Board Schedules (which are minimal for lecturing and maximal for examining). I should very much appreciate being told of any corrections or possible improvements and might even part with a small reward to the first finder of particular errors. This document is written in L A T E X 2 ε and should be accessible from my home page. My e-mail address is [email protected] . Contents 1 Complex differentiability is like real differentiability 2 2 Complex differentiability is not like real differentiability 3 3 Conformal mapping 7 4 Contour integration and Cauchy’s theorem 9 5 Applications of Cauchy’s theorem 12 6 Calculus of residues 15 7 Fourier transforms 17 8 Signals and such like 19 9 Miscellany 22 10 Exercises 24 1
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1 Complex differentiability is like real differ- entiability The complex numbers have algebraic properties which are very similar to those of the real numbers (formally they are both fields) except that there is no order on the complex numbers. This similarity means that we can define differentiability in the complex case in exactly the same way as we did in the real case. Definition 1.1. A function f : C C is differentiable at z with derivative f ( z ) if vextendsingle vextendsingle vextendsingle vextendsingle f ( z + h ) f ( z ) h f ( z ) vextendsingle vextendsingle vextendsingle vextendsingle 0 as | h | → 0 . Exactly the same proofs as in the real case produce exactly the same elementary properties of differentiation. Lemma 1.2. (i) The constant function given by f ( z ) = c for all z C is everywhere differentiable with derivative f ( z ) = 0 . (ii) The function given by f ( z ) = z for all z C is everywhere differen- tiable with f ( z ) = 1 . (iii) If f, g : C C are both differentiable at z , then so is f + g with ( f + g ) ( z ) = f ( z ) + g ( z ) . (iii) If f, g : C C are both differentiable at z , then so is their product f × g with ( f × g ) ( z ) = f ( z ) g ( z ) + f ( z ) g ( z ) . (iv) If f : C C is nowhere zero and f is differentiable at z , then so is 1 /f with (1 /f ) ( z ) = f ( z ) / ( f ( z )) 2 . (v) If f : C C is differentiable at z and g : C C is differentiable at f ( z ) then the composition g f is differentiable at z with ( g f ) ( z ) = f ( z ) g ( f ( z )) . (vi) If P ( z ) = N n =0 a n z n , then P is everywhere differentiable with deriva- tive given by P ( z ) = N n =1 na n z n 1 . The following extensive generalisation of part (iv) of Lemma 1.2 was proved in Analysis I (course C5). Theorem 1.3. Let a j C [0 j ] . Then, either n =0 a n z n converges for all z and we write R = , or there exists a real number R 0 such that n =0 a n z n converges for all | z | <R and diverges for all | z | >R . ( R is called the radius of convergence of n =0 a n z n .) 2
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If we write f ( z ) = n =0 a n z n for | z | <R , then f is differentiable at all z with | z | <R and f ( z ) = summationdisplay n =1 na n z n 1 .
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