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.
3. Suppose f(z) is analytic on a region (i.e. open connected) D. Prove that if Im f(2), the imaginary part of f(z), attains its maximum value in D then f(2) must be constant on
3. Suppose f(z) is analytic on a region (i.e. open connected) D. Prove that if Im f(2), the imaginary part of f(z), attains its maximum value in D then f(2) must be constant on
Problem (4) Let f(z) denote the function e a f(z) 1 - z Compute f (z) dz where y is any contour that encloses the origin but does not enclose the point z =1
Problem (4) Let f(z) denote the function e a f(z) 1 - z Compute f (z) dz where y is any contour that encloses the origin but does not enclose the point z =1
2. Let f(z) be the principal branch of i.е., f(z) exp@ Log(z)}. Co mpute (e)dz where C is the semicircle {et : 0 < θ < π
Let f(z) = ee^z . Find Re(f), Im(f) and |f|
gxercise 7.3.5. Provide an example or give a reason why the request is im- ossible. (a) A sequence (n)f pointwise, where each fn has at most a finite number of discontinuities but f is not integrable. b) A sequence (9n) g uniformly where each gn has at most a finite number of discontinuities and g is not integrable. c) A sequence (h)+h uniformly where each hn is not integrable but h is integrable.
gxercise 7.3.5. Provide an example or give...
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
Find a holomorphic function F(z) on Ω-{z I Izl < r} such that for any a E Ω, F(a) F(0)-Z dz. Suppose f(z) is entire and Ω is simply connected domain. Show lim 22-h2220
Find a holomorphic function F(z) on Ω-{z I Izl
Let f(z) be entire and such that Im f(z)S cfor all z. By considering the function ee show that f(z) is a constant. -if (z)
19. Which of the following statements are always true? (i) Re(2)Im(iz) 0 (ii) Re(iz)Im(z) = 0 (iii) z- Zi Im(z) = 0 (a) (ii) only (b) (i) only (c) (iii) only (d) (i) and (ii) only (e) (ii) and (ii) only