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

5. (a) Find the Maclaurin series for the function f(z)=cos(z). ro) Find the Laurent series of the function f(z)= z(1+z2) , When O<lzl<1. (c) For the given function, f(Z)=Z=, show any singular point is a pole. Determine the order of each pole and find the corl.esponding residue. 6.. (a) State Cauchy Residue Theorem and use it to evaluate the integral of T==r around the circle z| = 2 in the counterclockwise. (b) Prove that (c) Find the value of the integral dz taken counterclockwise around the circle lz+2l=.3. : Elements of complex : COMMON PROG Mastl;in:urn Mar:ks : 90 I. (a) Let. f be a fu_nction defined at all the points z in some deleted neighborhood of a point zo and wo €C . Define !iEf(Z)Wo . Further, if f(Z)=Z, then show that i;#f (Z) does not exist® (b) show that lim (Z)8(z) =0 if lims (z) =0 and if there exists M > 0 such that |8 (z)) ≤ M for all z in some neighborhood Zo. (c) Let f be a function defined on a domain containing a neighborhood of a point zo. Then define the derivative of f at zo. Further, using the definition, show that the derivative of f (z) =z' is 2z. 2. (a) Using the Cauchy-Riemann equations, show that {(z) =|2" differentiable at z = 0 only. (b) let f: C→C be a function such that f(z) = 0 for constant function throughout D. all z in a domain D. show that f(z) must be a (c) Show that 3. (a) if W.(t) = u(t) +ivt) is a complex-valued function of a real variable t and w'(t) exists. Show that derivative of wt) with respect to t, evaluated at - (b) Use parametric representation for C to evaluate J.2+2 - dz, where C is the semicircle z= 2ei° (X≤O≤2T) (c) Without evaluating the integral, show that C73_1 7, when C is the arc of the circle z|=2 from z = 2 to z = 2i that lies in the first quadrant. 4. (a) By finding an antiderivative, evaluate the following integrals, where the path is any contour between the indicated limits of integration (b) State the Cauchy-Goursat theorem and hence apply it to evaluate J. (z)dz, where the contour C is the positivel oriented unit circle |z| = 1 and (c) Find the value of the integral of g(z) around the circle |z-i| = 2 in the positive sense when