5. The following diagram shows the image of a closed C¹ curve y in C, together with the positions of two complex numbers

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5 The Following Diagram Shows The Image Of A Closed C Curve Y In C Together With The Positions Of Two Complex Numbers 1 (101.31 KiB) Viewed 11 times

5. The following diagram shows the image of a closed C¹ curve y in C, together with the positions of two complex numbers a and b. Im Y Re (a) Compute the winding numbers n(y; a) and n(y; b). (b) Let f: C\ {b} → C be a holomorphic function, whose Laurent series at b is +00 Σ ak(z-b)k. k=-∞0 Compute the following line integrals in terms of b and the Laurent series coefficients. (i) f(z)dz (ii) zf(z)dz (c) Let be a counterclockwise oriented simple closed C¹ curve in C, and let g be a meromorphic function on C such that g has no zeros or poles on image p. Suppose that g has 8 poles in the interior of p, counting multiplicities. If gop is the closed C¹ curve y whose image is shown above, (i) how many zeros in the interior of (ii) how many zeros in the interior of does g have, counting multiplicities? does g - a have, counting multiplicities?