Question

(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)|.

Answer #1

Fix an n>=2. What are all the entire functions f: C to C for
which f(z^n)=(f(z))^n for all z in C? Where C is the complex
numbers.

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.

Complex Analysis Proof - Prove: if f = u + iv is analytic in a
domain D, then u and v satisfy the Cauchy-Riemann equations in
D.

Let f(z) and g(z) be entire functions, with |f(z) - g(z)| < M
for some positive real number M and all z in C. Prove that f'(z) =
g'(z) for all z in C.

1- For the following functions, find all critical numbers
exactly.
f(x) = x − 2 sin x for −2π < x < 2π
f(x) = e^−x −e^−3x for x > 0
f(x) = x^5 − 2x^3

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

Find all the complex number z such that z^2=i

Complex Analysis
1) find an example of a non constant entire function f such
that
(a) supx∈R |f(x)| < ∞
(b) supy∈R |f(iy)| < ∞.

1.
Differentiate the functions:
a) f(x)=tan(1/x).
b) f(x)=sin(x^2)sin^2(x).

Find v(x,y) so that f(z) = 3x^2 +8xy - 3y^2 + iv(x,y) is
analytic

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