 # The homework is from John B. Conway book Functions of One Complex V...

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The homework is from John B. Conway book Functions of One Complex Variable I Second edition. P67-68 - #5, 7-13, - 16, 19, 22 5. Let r(t) = exp << (-1+i)t~¹) - for 0 < t 1 and >(0) = 0. Show that y is a rectifiable path and find V(y). Give a rough sketch of the trace of y. 7. Show that y: [0, 1] C, defined by >(t) = t + it sin 1 / 1 for t * 0 and (0) = 0, is a path but is not rectifiable. Sketch this path. 8. Let 7 and a be the two polygons [1, i] and [1, 1 + i, i]. Express y and a as paths and calculate f, f and 1. f where f(z) = 12/2. 9. Define y: [0, 2=] C by x(t) = exp (int) where n is some integer (positive, negative, or zero). Show that Z 1 dz = 2win. y 10. Define x(t) = e" for 0 t 2w and find Sr z" dz for every integer n. 11. Let y be the closed polygon [1 - i, 1+i, - 1+i, - 1 - 1 - -i]. Find Z - 1 dz. y 12. Let dz where y: [0, w) C is defined by x(t) re". Show = that lim I(r) = 0. 00 13. Find fr z-+ dz where: (a) y is the upper half of the unit circle from + 1 to - 1: (b) y is the lower half of the unit circle from + 1 to - 1. 16. Show that if y and a are equivalent rectifiable paths then V(y) = V(o). 19. Let v(t) = 1 + e" for 051521 and find - 22. Let Y be a closed rectifiable curve in an open set G and a E G. Show that for n 2. i.(z - a) n dz = 0. P68 - #16, 20, 21 and P73 - #3, 4, 5, 7,9 16. Show that if y and a are equivalent rectifiable paths then V(y) = V(o). 20. Let Y(1) = 2e" for - 11 SIS IT and find fy (=2 - 1) 1 dz. 21. Show that if F1 and F2 are primitives for f: G C and G is open and connected then there is a constant e such that F1 (z) = c + F2(z) for each Z in G. 3. Suppose that y is a rectifiable curve in C and & is defined and continuous on (y). Use Exercise 2 to show that g(z) = is analytic on C-(y) - and = 4. (a) Prove Abel's Theorem: Let a, (z-a)" have radius of convergence 1 and suppose that E an converges to A. Prove that = (Hint: Find a summation formula which is the analogue of integration by parts.) (b) Use Abel's Theorem to prove that log 2 = 1 I-+++- 5. Give the power series expansion of log z about z = i and find its radius of convergence. 7. Use the results of this section to evaluate the following integrals: iz (a) 2 dz, x11==", = 0<1<2m; dz (b) , = 05152m; z-a sin (c) z³ Z dz, x(t) = e", 051521 (d) flow log dz, x(1) = 1+1e", and n 0. 9. Evaluate the following integrals: e-e- (a) dz where n is a positive integer and x(t) = e", 0 t 2w; 2" y dz (b) where n is a positive integer and Y(t) = +e",Osis2 ; n - y (c) / 2²+1 dz where x(t) = 2e", 0 t 2w. (Hint: expand (2²+1) - 1 by means of partial fractions); sin Z (d) dz where x(t) = e", OSIS21; z Z 1/m (e) dz where x(t) = 1 the", (z-1)" - 7

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