Question

Is it possible for an entire function w=f(z) to map the z-plane into the circle |w|<1? Fully justify your explanation.

Answer #1

(1) Find all functions f(z) that are analytic in the entire
complex plane and satisfy 2|sin(z)| ≥ |f(z)|.
(2) Find all functions f(z) that are analytic in the entire
complex plane and satisfy 2|f(z)| ≥ |sin(z)|.

The circle IZI = 2 is mapped onto the w plane by the
transformation w = (z+j)/(2z-j) Determine
(a) The image of the circle in the w plane
(b) The mapping of the region enclosed by IZI =2

Let omega be the unit disc |z|<1. Sketch the sets omega and
f(omega) (in the w-plane where w=f(z)=z-i.Likewise, where
f(z)=2z+3i.

1.Using only the definition of uniform continuity of a function,
show that f(z) = z^2 is uniformly continuous on the disk {z : |z|
< 2}.
2. Describe the image of the circle |z-3| = 1 under the mapping
w = f(z) = 5-2z. Be sure to show that your description is
correct.
Please show full explaination.

Describe the image of the circle |z − 3| = 1 under the Mobius
transformation w = f(z) = (z − i)/(z − 4).

for w=f(x,y,z)=2x^4y^2-6xz^3 use the tangent plane to
estimate f(-2.03,1.04,1.02)

9. Find a sum-of-products expression for F’ for the function
F(W, X, Y, Z) = X + YZ(W + X’)

Find a parametric representation
Circle in the plane z =1 with center (3, 2) and passing through
the origin.

Consider the function f : Z → Z defined by f(x) = x 2 . Is this
function one-to-one, onto, or neither? Give justification for your
claims that rely on definitions.
With explanation please

Complex analysis
For the function f(z)=1/[z^2(3-z)], find all possible Laurent
expansions centered at z=0.
then find one or more Laurent expansions centered at z=1.

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