1. Let C denote the line segment from the point -2+i to the point 1+3 and let f(z) = 2³+(3-2). where :=1+ip. Use the def

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1 Let C Denote The Line Segment From The Point 2 I To The Point 1 3 And Let F Z 2 3 2 Where 1 Ip Use The Def 1 (26.82 KiB) Viewed 41 times

1. Let C denote the line segment from the point -2+i to the point 1+3 and let f(z) = 2³+(3-2). where :=1+ip. Use the definition of a contour integral to calculate (2)dz 2. Let / be the branch of 2% such that a>0 and 0 < arg: <2r. Let C denote the semi-circular path :-2² (0505 =). (a) Show that the right hand limits at 8-0 of the real and imaginary parts of f1(0)2(6) exist and calculate their values (b) Calculate (1) dz (e) Why did we show that the right hand limits at #=0 of the real and imaginary parts of ƒ[={0}}=¹(✔) exist before calculating f(z) dz? 3. Let Cg denote the circle := R (05#≤2). Show that 4. Use Canchy's integral formula (or its extension) to calculate each of the following integrals (in each case the contour is positively oriented): (A) (b) COS Size (2² + 2² (2²-2²) ² 18 =-4 2/C₂ (2² + 1)(²³²-5) ² S 5. Suppose converges. Prove that 6. Calculate the Taylor series of f(1) 7. Write down the Laurent series of mil -7 converges about the point 2-3. Specify the region of validity. sin (2) about the point = 0; 8. Use division and/or multiplication of known power series to find the first four non-zero terms in the Laurent ecoh: expansion of in the region 0<x<*