Q5. a) Let f(z) be an analytic function on a connected open set D. If there...
Q5) Evaluate $c f(z) dz where C is the unit circle Iz| = 1 and f(2) is defined as follows a) f(z) = z2+z2+z_ b) f(x) = tan z c) f() = cosha
12. (a) Show that 1y dt By letting R o, deduce that the residue of f ) at t 0. at zoo by the equation f (z) dz is given by 2πί times (b) When zoe is an isolated singular point, define the residue of f () Show that (e)d2miRes () Coo (c) Use the above result to evaluate the integral Ca2 + 22 z where C is any positive contour enclosing the points z 0, tia, and check the...
1. if the real part of an analytic function, f(z), is given find the imaginary part, v(x, y) and f(z) as a function of x. (step by step) 2. Evaluate the following complex integral (step by step) 1. If the real part of an analytic function, f(z), is given as 2 - 12 (x2 + y2)2 find the imaginary part, v(x,y), and f(z) as a function of z. 2. Evaluate the following complex integral:
Exercise 3 Let f be an analytic function on D(0,1). Suppose that f(z) < 1 for all z € C and f() = 0. Show that G) . (Hint: use the function g(z) = f(2).)
3. Suppose f(z) is analytic on a region (i.e. open connected) D. Prove that if Im f(2), the imaginary part of f(z), attains its maximum value in D then f(2) must be constant on 3. Suppose f(z) is analytic on a region (i.e. open connected) D. Prove that if Im f(2), the imaginary part of f(z), attains its maximum value in D then f(2) must be constant on
f(D) CL/ D. If either 310 pts]. Let f be an analytic function defined on a domain or f(D) C C where f (D) denotes the range of f, L is any straight line and C is any circle in C, then show that f must be constant in D. f(D) CL/ D. If either 310 pts]. Let f be an analytic function defined on a domain or f(D) C C where f (D) denotes the range of f, L...
1. Let P(x) = 22020 – 3:2019 + 22 -3. (b) Compute the contour integral Scof(z)dz with f(z) := 2 fled with f(-) -- 2021 – 222020+2 P2) +, where C (0) is the circle 121 = 8 with positive orientation.
7. Let z x+y (a) Show that f(z) z3 is analytic. 4 marks Recall the Caucy-Riemann equations are: ди ди an d_ where f (z) -u(x, y) + iv(x, y). (b) Let x2 and y 1 such that z-2i is a solution to 2abi [3 marks] Determine a and b (c) Find all other solutions of 23-a + bi in polar form correct to 2 significant 3 marks] figures If you were not able to solve for a and b...
2. Let f(z) be the principal branch of i.е., f(z) exp@ Log(z)}. Co mpute (e)dz where C is the semicircle {et : 0 < θ < π
Consider the following probability density function: -x-1/2e-z/2 for x > 0. f(x) = the area under the curve (integral) is equal to one, then: i) Compute the mean of the function numerically based on the principle: rf (x) dr ES Where S is the set of values on which the function is defined i Compute the median y where: f(z) dz = Where m is the minimum value on which the function is defined. Consider the following probability density function:...