Solve the following problems related to zeros of functions. (a) Find the number of zeros (counting multiplicity) of f(z) = 2z5 − 6z2 − z + 1 in D = {z : 1 < |z| < 2}. (Warning: simply applying Rouché's theorem won't suffice!) (b) Show that f(z) = z4 − 5z + 1 has no zeros in D = {z : |z| ≥ 2}, and evaluate ∫ |z| = 2 4z3 − 5z4 − 5z + 1 dz (c) Show that ez + 5z3 = −1 has three roots (counting multiplicity) in the unit disk D = {z : |z| < 1}.
![(20 pts) By completing this problem, you will have a better understanding why complex analysis is a suitable environment for studying series. (a) Consider the function f ( x ) = 1 1 + x 2 defined in R. Find its Taylor series in real-sense, i.e., as you learned in Calculus. (b) Notice that f(x) defined above is differentiable everywhere in R, but does its Taylor series converge pointwise everywhere in R? (c) What is the Taylor series expansion for f(z) = 1 1 + z 2 in D = {z ∈ C : |z| < 1}, and indicate what is its radius of convergence? (d) What is the Laurent series expansion for f(z) = 1 1 + z 2 in D = {z ∈ C : |z| > 1} ? (e) [no points] Can you explain what happens in (b)?](https://www.doubtrix.com/uploads/editor/1244849890IBkveboEpG.png)
![For the circuit shown in Figure 1, NMOS device M1 has the following properties: kn’(W/L) = 180 μA/V2, VT0 = 2 V, λ = 0.02 V-1, γ = 0.4 V1/2, 2ϕF = 0.7 V, and ID = 10 mA. Note that capacitor Cbig is a DC blocking cap, i.e., Cbig is an open at DC and a short at all other frequencies. Figure 1. Circuit for Problem 1. a) Determine the numerical value of bias voltage Vbias which is needed to achieve a drain current of ID = 10 mA. b) Draw the small-signal equivalent circuit. c) Analyze the small-signal circuit to find Rin, Rout, and Av = Vout/Vin. Find the analytic expression for each quantity first, before determining the numerical answers. d) Recall that gmb/gm = χ. Determine the numerical value of χ for this problem. [Note that this part has nothing to do with parts (b) and (c).] e) What is the threshold voltage of device M1? [Note that this part has nothing to do with parts (b) and (c).] Show your work, list all numerical values in proper engineering notation format with appropriate units and with at least two decimal places of precision.](https://www.doubtrix.com/js/ckeditor/filemanager/connectors/php/editor/1701531864-Rjgbdog0ehds.png)
![In Fig. 5, VDD = 2.5 V, VB1 = 1 V, RG = 100 kΩ, RD = 10 kΩ, R1 = 10 kΩ, (W/L)1 = (W/L)2 = (W/L)3 = (W/L)4 = (1/0.1). CC1∼CC3 are very large. (a) Determine the VB2 such that the output (Vout) swing is maximized while all MOSFETs operate in the saturation region. (b) For a dc input of Vin = 1 V, determine Vx, Vz and Vout. Fig. 5 (c) When Vin(t) = 1[V] + 10[mV]sin(ωt) is applied, determine Vout(t). (10)](https://www.doubtrix.com/uploads/editor/3770622967fpfOfEpcYQ.jpg)