Show that:
is an entire function. (Analytic everywhere)
Show that: is an entire function. (Analytic everywhere) sin(ze) sin(ze)
Show that the function defined by (2)= is analytic on the open unit disk {lz| < 1} and that |(rA) Remark: This function does not extend analytically to any larger open set than the unit disk +o0 as r-1 whenever A is a root of unity Show that the function defined by (2)= is analytic on the open unit disk {lz|
Create your own probability density function (that is, find a function that is everywhere positive and that integrates to 1 over its entire statespace), show that it is a probability density function, and find the corresponding cdf. Explain what your pdf might describe.
zea (zea-cos θ) ( )Z[e-ancosm]=z2e2a-2zeacos Show that () Ze cos no Show that +1 zea sin θ z2eza-2zea cos θ + 1
Problem 5. Let F(r,y) (e-v-v sinzy) ?-(ze-s + z sin zyj (1) Show that F is a gradient field. (2) Find a potential function f for it (3) Use the potential function f to evaluate F-ds, where x is the path x(t) = (t,t2) for 0sts1. (NO credit for any other method.)
Tutorial Group/Date/Time: Using the Cauchy-Riemann equations, show that f(z)-e' is fully analytic in the entire z-plane. 1. (40 marks)
дw Q-5): Find for the function: w = xey + ye? + ze* With x = t2 + v2 y=t+v z = sint + sin v
Hw2 Q1 Show that the function f(z) = z2 + z is analytic. Also find its derivative. (Hint: check CR Equations for Analyticity, and then proceed finding the derivative as shown in video 8 by any of the two rules shown in video 7] Q2 Verify that the following functions are harmonic i. u = x2 - y2 + 2x - y. ii. v=e* cos y. Q3 Verify that the given function is harmonic, and find the harmonic conjugate function...
Consider the following. x = sin zo, y = cos ze, isesi (a) Eliminate the parameter to find a Cartesian equation of the curve.
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).)
2. If ze = V15(cos 35° + i sin 35°) and Z2 = v5(cos 10° + i sin 10°), find 777, and write your answers in both trigonometric form and rectangular form. If rounding is necessary, round to three decimal places.