Complex analysis
65. How many roots has the equation z = f(z) in the circle |z| < 1, if for |z| ≤ 1 the function f(z) is analytic and satisfies the inequality |f(z)| < 1?
Complex analysis 65. How many roots has the equation z = f(z) in the circle |z|...
Problem 3: Consider the function f(2) = e2/ . (a) Determine the solutions to the equation f(2) =1 and sketch the locations of these points in the complex plane. (3 points) (b) Consider a circle in the complex plane described by |2 = 1 (unit circle). How many points satisfying f()1 are within the unit circle? Suppose you had considered a much smaller circle, say, described by 10-15. Now how many points are within this smaller circle? (3 points) Points...
Problem 1 Consider the composition f(w(z)) of two complex valued functions of a complex variable, f(w) and w(z), where z = x+iy and w=u+iv. Assume that both functions have continuous partial derivatives. Show that the chain rule can be written in complex form as of _ of ou , of Oz . . of az " dw dz * dw dz and Z of ou , of ou dw dz* dw ƏZ Show as a consequence that if f(w) is...
Complex Analysis (use the Liouville equation): Suppose that f(z)u, ) (, u) is an entire function such that 7u9n is bounded. Prove that fis constant Hint: Multiply f by an appropriate complex constant. Suppose that f(z)u, ) (, u) is an entire function such that 7u9n is bounded. Prove that fis constant Hint: Multiply f by an appropriate complex constant.
U3 is the notation for the group of 3rd roots of untity— U3={ a complex number z : z^3=1} Problem B. Define a function f: C GL2(R) by the following formula f(a+ib) = () a-b 1 (a) Check that f is a homomorphism. Is f injective? Is f surjective? (b) Verify that f takes the complex unit circle C into the group SO2(R) of rotation matrices (ossin) Prove that the resulting map sin cos f: C SO2(R) is an isomorphism....
Complex Analysis: . (a) Find a single function f(z) which has all of the following properties: f(z) is discontinuous at the origin z = 0, at z = 1, and at all points z with Arg(z) = 7/4, but f(z) is continuous at all other points of C; • f(z) has a simple zero at z = :i; and f(z) has a pole of order 3 at z = n. Justify that your function f(x) has each of the properties...
2- a) The real part of a complex function f(z) given as, u(x, y) = 3x?y - y. Iff(2) is an analytic function, find v(x,y) and f(z) (15p) b) Find the whether f(z) is analytic or not where f(z) = cos(x) +ie'sinx. (15p)
How are the n roots of z n arranged graphically in the complex plane? (Answer must be 1-2 sentences long)
How many and of which kind of roots does the equati f(x) = x3 + 2x2 + 4x + 8 have? A. 2 real; 1 complex B. 3 real C. 2 real; 2 complex D. 1 real; 2 complex
2. (a) Prove the product rule for complex functions. More specifically, if f(z) and g(z) f(z)g(z) is also analytic, and that analytic prove are that h(z) h'(z)f(z)9() f(z)g'(z) (You may use results from the multivariable part of the course without proof.) = nz"- for n e N = {1,2,3,...}. Your textbook establishes that S z"= dz (b) Let Sn be the statement is true. With the help of (a), show that if Sn is true, then Sn+1 is true. Why...
for complex variables 1. Find all complex roots of the following cubic equation. Write them in standard form z= a +ib where a and b are numerical values (round to 4 digits after decimal point). (a) 23 + 3z +1 = 0 (b) 223 – 622 + 2z+1 = 0