Complex Analysis Homework Help

2. Mandatory Question (simple computation) 10 points. Compute the following integrals with the help of the residue theorem. (Do not forget to say where and what type of singularities the corresponding complex functions have!) (a) I = = ["th (b) I 1 (tan² (t) + 2) (cotan² (t) + 2) 0 = x sin(x) = cos(x) - f x4-1 -∞ Kramers relations! (Do not forget to check the conditions!) dt (Hint: have a look at the Fourier transform on intervals), d dx with f the Cauchy principal value integral. Use the Kronig- 3. Mandatory Question (simple proof) 10 points. Find out how many zeros the following functions have in the given domains. Apply Rouché's Theorem! (a) h(z) = sin(z) + 4[z² − (3 + i)z + 3i] in the disc Ď(0, 2) (Hint: e² ≈ 7.4 and e-² ≈ 0.1); (b) h(z) = e²/2 +2e²+iz – 1 in the rectangle R = {x + iy € C|0 ≤ x ≤ 1, 0 ≤ y ≤ 2} (Hint: 2 √5 2.2 and e√5/2≈ 3.1); 4. Optional Question (advanced computation) 10 points. Compute the real series S = (-1)³ j¹ – 1 j=2 with the help of the residue theorem. Approach as follows: (a) Rewrite the summands in terms of residues of a function f(z) = g(z)/ sin(72). Where and of what type are the singularities? (b) Perform an ML-bound to rewrite the series in terms of a limit of a contour integral. You can use | sin(2)| ≥ | sin(|z|). Write explicitly what S in terms of the contour integral and the residues is. (c) Do not forget to write the result and simplify as much as possible (no decimal approximation, we want to see the exact result)!

3) matt invest 26500 in two accounts. The first earned a rate of 11% after a year and the secone lost 10% in the same time period. At he end of one year the total amount gained was -760. how much was invested in each account.

Triangle SUN has coordinates S(0,6), U(3,5), and N(3,0). On the accompanying grid, draw and label ASUN. Then, graph and state the coordinates of"As'U'N', the image of A SUN after a reflection in the y-axis. Given the coordinates of ABC, A (1,6), B (3,1), C (7,2). Reflect over theline horizontal line y = -2. Then give the coordinates of A'B'C'

Evaluate the contour integral \oint \frac{e^{p z}}{1+e^{z}} d z where p is a constant, for the contour shown below (i.e., starting at R on the positive real axis, going to R+2xi, then to –R+2Ti, then to –R and back to R, all in straight line segments). [4] (b) Now assume that p is real and 0 < p < 1. As R → ∞, what is the contribution of the vertical segments of the contour (i.e., R to R+2ñi and-R+ 2ni to – R) to the contour integral in part (a)? [1] (c) Use the results of parts (a) and (b) to evaluate \int_{-\infty}^{\infty} \frac{e^{p x}}{1+e^{x}} d x for 0 < p< 1. Hint: Consider how the contribution of the upper horizontal contour (i.e., from R+2xi to –R+ 2ni) in part (a) relates to the integral in part (c). [5]

Question 6 The area, A of a figure defined by some points, P1 and P2..., can be calculated from the coordinates of those points, using any of the following formulae: A=1 / 2 D\left(E i, N_{i+1}-E_{i+1}, N i\right) Where Ei and Ni are the easting and northing coordinates of the point Pi.The following table shows the coordinates to three points A, B and C. Difference = Divide by 2=

R1 (based on SW) How much does a woman's labor supply fall when she has an additional child?For this exercise you will need to download the data file fertility.Rda. The data are from the 1979 census and contain observations on more than 250,000 married women aged 21-35 with two or more children. The variables are: morekids =1 if mom had more than 2 children boylst =1 if lst child was a boy • boy2nd =1 if 2nd child was a boy samesex =1 if 1st two children same sex agem1age of mom at census black =1 if mom is black • hispan =1 if mom is Hispanic othrace =1 if mom is not black, Hispanic or white weeksml = mom's weeks worked in 1979 a) Regress weeks ml on more kids using OLS. On average, do women with more than two children work less than women with two children? How much less? b) Explain why the OLS regression you estimated is potentially subject to endogeneity bias. c) The data set contains the variable same sex, which is equal to 1 if the first two children are of the same sex and equal to zero otherwise. Are couples whose first two children of the same sex more likely to have a third child? Is the effect large? Is it statistically significant? d) Explain why same sex is a valid instrument for the instrumental variable regression of weeksml on more kids. e) Estimate the regression of weeksml on more kids using same sex as an instrument. How large is the effect of more kids on labor supply? f) Estimate the regression of weeks ml on more kids using same sex as an instrument,including age ml, black, his pan, and othrace in the labor supply regression (treating these variables as exogenous). Do the results change when you use these additional controls? Explain why or why not.

Complex numbers Multiple choice questions Q1 Given z² = 4(cos 3π/2 + i sin3π/2). which of the following is equal to z? Q2 Suppose that z1 and z2 are two complex numbers such that z1z2 = 1. Given that z,1 = re^θ, which of the following is equal to z2? Q3 Suppose a nonzero complex number z1 is multiplied by 1+i / 1-i to produce the complex number z2 Which of the following best describes z2 in relation to z1 ? Q4 Given that z^π =(-1)^3i, which of the following is equal to z? Q5 The Argand diagram below shows the complex number z. Which of the following best represents z²?

1. True or False. Justify your answer. \text { (a) }\{1,2\} \in\{\{1,2\},\{3,4\}\} \text { (b) }\{1,2,1,2,1\} \subseteq\{1,2\} \text { (c) For all sets } A \text { and } B, \text { if }|A| \leq|B| \text { , then } A \subseteq B \text { . } \text { (d) For all sets } A, B, \text { and } C, \text { if } A \neq B \text { , then } A \cap C \neq B \cap C \text { . } \text { (e) For all } x \in \mathbf{R},\lfloor 2 x\rfloor=2\lfloor x\rfloor \text { (f) Let } n \in \mathbb{Z}^{+}, \text {then }\left(3 n^{2}+3 n\right) \bmod 6=0 \text { . } \text { (g) Let } a, b \in \mathbb{Z} \text { . If } 10 \mid a b \text { , then } 10 \mid a \text { or } 10 \mid b \text { . } \text { (h) There is a set } A \text { such that }|\mathcal{P}(A)|=100 \text { . } The expressions Væ(P(x)→ Q(x)) and (VrP(x)) → (VxQ(x))logically equiva-arelent.

Sketch the function y = 3log 2 [- (x – 3)] – 1. Label the exact x-intercept, the location of the transformed x-intercept and asymptote.

2. (8 points) Show that the bilinear map w = maps points on the circle | z — 1 |= 1 to points on the line Re(w) = 12.