Exercises aw the contours γ-[0, i], σ [0, l] + [1,1]. Evaluate re z dz re z dz
Problem 4.9 (e) /(z) = and γ is parametrized by r(t), 0 z + t 1, and satisfies Imr(t)> 0, r(0) -4 + i, and γ(1) 6 + 2i (f) f(s) sin(z) and γ is some piecewise smooth path from 1 to π. 4.2 and the fact that the length of γ does not change under 4.9. Prove Proposi reparametrization. (Hint: Assume γ, σ, and τ are smooth. Start with the definition off, f, apply the chain rule to σ...
2. Evaluate Scf()dz for the following f() and C f(z) = zz2 and C is the se micircle z = 2e10, 0 a. θ π. b. fz)2an C i the circle lz -il 2. z2+4 2. Evaluate Scf()dz for the following f() and C f(z) = zz2 and C is the se micircle z = 2e10, 0 a. θ π. b. fz)2an C i the circle lz -il 2. z2+4
2 +1 (b) Evaluate the contour integral dz, 22 – 9 where I is the boundary of the square D = {z E C:-4 < Re(z) < 4, -4 < Im(z) < 4} traversed once counterclockwise.
Show that integral dz/(z-1-i)n+1 =0, if n does not equal 0 and 2 pi i if n = 0 for C the boundary of the square 0<=x<=2, 0<=y<=2, taken counterclockwise. [Hint: Use the fact that contours can be deformed into simpler shapes (like a circle) as long as the integrand is analytic in the region between them. After picking a simpler contour, integrate using parametrization.]
5.30. UITULU eur 5.39. Evaluate z dz when : >0 and C is the circle Izl = 3. 2 Ti I (z2 + 1)
Find! ! dz where C : |2|-l , clockwise Find zexp()dz where C is from to z- i along the axks Find! ! dz where C : |2|-l , clockwise Find zexp()dz where C is from to z- i along the axks
2. (a) Evaluate the contour integral z dz, where I is the circle 12 – 11 = 2 traversed once counterclockwise.
1. Evaluate the complex integral: ∫C [zRe(z) − z¯Im(z)]dz, where C is the line segment joining −1 to i. (z¯ = z bar) 2. Evaluate the complex integral: ∫ C [iz^2 − z − 3i]dz, where C is the quarter circle with centre the origin which joins −1 to i.
1 2 4 Evaluate Σ Σ (2i + i=0= 0
4. Evaluate the following integrals: f, where contour γ is a circle of radius 2 centered at the origin. z.İ f, -1-i,1-i,1+i,and-1+i. (z-0.1-1); where contour γ is the square with the four vertices ill) Jo (2+7 cos(e)) 4. Evaluate the following integrals: f, where contour γ is a circle of radius 2 centered at the origin. z.İ f, -1-i,1-i,1+i,and-1+i. (z-0.1-1); where contour γ is the square with the four vertices ill) Jo (2+7 cos(e))