To solve this problem the details calculations are given below
23. Consider the function w(z) = 2-2 (a) Where in the complex z-plane are the poles of w(z)? (b) ...
In(z) 3, Consider the function f(x)= (a) Find the Taylor series for r(z) at -e. b) What is the interval of convergence for this Taylor series? (c) Write out the constant term of your Taylor series from part (a). (Your answer should be a series!). (d) What can you say about the series you found in part (c), by interpreting it as the limit of your series as x → 0. (Does it converge? If so, what is the limit?)...
Complex Analysis:
= Define the function 22 f(z) 22 +1 For each annulus region given below, find the Laurent series of f(z) convergent in the region. (a) 0 < 12 – il < 2 (b) 1 < 121.
Q5. (a) Consider the region in the complex plane defined by: z = x+iy : 1, lul π/3. Draw this region in the z-plane and mark a few points on it of your choice (eg, A, B, C) Now, apply the conformal transformation w-e*. Plot the resulting region and mark the corresponding points (eg., A, B, C.) (b) What is the area (in arbitrary square units) of the figure in the z-plane? What is the area in the w-plane?
Derive the Laurent series expansion for the function (a) f(z) := z^2 sin (1/(z − 1 )) on the exterior |z − 1| > 0 of the unit disk centered at 1, and for the function (b) g(z) := 1 /(z^2 + z − 2) in the annular region 1 < |z − 1| < 3
4. The function in the extended complex plane is given by s(e) a) Find and characterize all the singular points of the function b) Find all Laurent series of f(z) with center zo = 0 c) Evaluate the integral f(z)dz ford: +-, counterclockwise d) Evaluate the integral s(-)d for C: -52, counterclockwise e) Evaluate the integral( f(z)dz for C:너_1, clockwise. 4
Q6. (20pts) Consider the function f(2)= cosh(z) (i) Let f(z) = Eno an izn be the Taylor series expansion of f(z) around z = 0. Determine aj, aj, and a. (ii) Let f(z) = 2n-obn: (z - 14" be the Taylor series expansion of f(z) around z = 1 Determine bo, bı, and b2. Simplify the resulting expressions as much as possible.
(Complex Analysis)
The linear mapping wFUz+p, where α, β e C maps the point ZFI+1 to the point wi-i, and the poin to the point w2-1i a) Determine α and β. b) Find the region in the w-plane corresponding to the upper half-plane Im(z) 20 in 9. the z-plane. Sketch the region in the w-plane. c) Find the region in the w-plane corresponding to the disk Iz 2 in the z-plane d) Find the fixed points of the mapping
The...
Consider the function f(x)-e a. Differentiate the Taylor series about 0 of f(x). b. Identify the function represented by the differentiated series c. Give the interval of convergence of the power series for the derivative. a. Choose the correct answer belovw 213 Ос. D. 2 41 61 b. The function represented by the differentiated series is Iill c. The interval of convergence of the power series for the derivative is Simplify your answer. Type an inequality or a compound inequality...
(Complex analysis)
Exercise 6 a) Show that the image of the half-plane y > c (c = const) in the z-plane 1 under the inversion mapping w--s the interior of a circle provided that C0 the inversion mapping w hen0? the inversion mapping w = z when c < 0? b) What is the image of the half-plane y > c (c -const) in the z-plane under c) What is the image of the half-plane y > c (cconst) in...
(2 points) Here are several points on the complex plane: The red point represents the complex number zı = and the blue point represents the complex number Z2 = The "modulus" of a complex number z = x+iy, written [z], is the distance of that number from the origin: z) = x2 + y2. Find the modulus of zi. |zıl = 61^(1/2) We can also write a complex number z in polar coordinates (r, 6). The angle is sometimes called...