Question

I need help with part a and b

5. (A step in the proof of the Cauchy-Goursat theorem.) At the top of page p. 153, the author indicates that dz0 and d: where C is a contour around one of the little squares in the proof, by invoking the fact that the functions 1 andz posses antiderivatives everywhere. But these results can be established without using the idea of antiderivatives. Consider the figure below with a rectangle identified in the complex plane with corners Let C be the positively oriented contour along with sides of the rectangle with compo- nents Ci. C2, Cs and Cs, as shown, C C +C2+C+C (a.b2) (az, b2) C2 (a.bi) (a2,bi) (a) Using basic contour integration along each of the contours (e.g. not antideriva tives) show that b) Similarly show that C2 C4

Answer 1

"(a) integral_c_1 dz = integral_a_1 + ib_1^a_2 + ib_2 = (a_2 + ib_2) - (a_1 + ib_1) integral_c_2 = integral_a_2 + ib_1^a_2 + ib_2 dz = (a_2 - a_2) + i(b_2 -b_1) integral_c_3 dz = integral_a_2 + ib_2^a_1 + ib_2 dz = (a_1 - a_2) + i(b_2 - b_2) = a_1 - a_2 integral_c_4 dz = integral_a_1 + ib_2^a_1 + ib_1 = (a_1 - a_1) + i(b_1 - b_2) = i(b_1 - b_2) so integral_c_1 dz + integral_c_2 dz + integral_c_3 dz + integral_c_4 dz = (a_2 - a_1) + i(b_2 - b_1) + (a_1 - a_2) + i(b_1 - b_2) = 0 (b) integral_c_1 z dz = integral_a_1^a_2(x + iy)dx = x^2/2

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