Complex Analysis — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

3. Attempt two parts from each question. 1. (a) (i) Solve ez = 3, where z is any complex number. (b) (c) (ii) Find the values of Log(1 + i)2 and Log(–1 + i)2 and justify the truth of Log(z1.z2) = Log(z1) + Log(z2) for z1, z2 ∈ C. Prove that if f′(z) = 0 everywhere in a domain D, then f(z) must be constant throughout D. Show that each of the following functions is nowhere analytic : 2. (a) Prove that if the function f(z) = u(x, y) + iv(x, y) be defined throughout some ∈-neighbourhood of a point z0 = x0 + iy0, and the following conditions hold true : (i) The first order derivatives of the functions u and v with respect to x and y exist everywhere in the neighbourhood; (b) (ii) Those partial derivatives are continuous at (x0, y0) and satisfy the Then f′(z) exists and f′(z) = ux Find the image of the domain x > 0, y > 0, xy < 0 under the mapping w = z2 (c) If z0 and w0 are points in the z and w planes, respectively, then prove that lim f z w = if and only if 0 = ∞ if and only if z 0 3. (a) Prove that if w(t) is piecewise continuous complex-valued function defined on an interval a ≤ t ≤ b, the : Find an upper bound for 3 ∫ , where C is the arc of the circle (b) Suppose that a function f(z) is continuous in a domain D. Prove that if f(z) has an antiderivative F(z) throughout D, then the value of the integral of f(z) along contours lying entire y in D and extending from any fixed point z1 to any other fixed point z2 is z z Use it to find the value if (c) State Cauchy-Goursat theorem. Use it to show that f z dz 0,= and C is the unit circle |z| = 1. and C is the unit circle |z – 2| = 2.‘ 1 4. (a) For a function f given to be analytic inside and on a positively oriented circle CR, centered at z0 and with radius R, if MR denotes the maximum value of n!M f z R f(z)| on CR, then prove that Hence, show that if f is an entire function such that | f(z) | ≤ A |z| for all z, where A is a fixed positive number, then f(z) = a1z, where a1 is a complex constant. ∫ , where C is the figure-eight contour formed by two (c) circles with centres at (0,0) and (2,0) respectively with radii 0.5 and 1.5 respectively, described in the positive direction, State and prove Liouville’s theorem. Apply this to show that if f(z) is entire and the function u(x, y) = Re[f(z)] is bounded above by u0 at all points (x, y) in the xy plane, then u(x, y) must be constant throughout the plane. 5. (a) Give all the Laurent series expansions in the powers of z for the function z 1 z and specify the domains in which those expansions are valid. (b) Suppose that zn = x n + iyn (n = 1,2,3,.....) and S = X + iY. Then, prove that = = = ∑ ∑ ∑ Also show that if n n n n n 1 n 1 n 1 z S iff x X and y Y. (c) State Taylor’s theorem and derive Taylor’s series representations for the following : = − + −LLL , when z ≠ 0. 6. (a) State Cauchy’s residue theorem and use it to evaluate the integral dz over the circle |z + 2| = 3 oriented anticlockwise.